Search: a114806 -id:a114806
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A288371
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Primes of the form k!9+1, where k!9 is the nonuple factorial number (A114806).
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+20
1
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2, 3, 5, 7, 11, 23, 37, 53, 71, 113, 137, 163, 191, 757, 2161, 2801, 51521, 1418561, 4093601, 42456961, 69509441, 105616001, 2420046721, 9160905601, 1270453824641, 2326680294401, 190787784140801, 509498986796801, 2805949277824001, 612940220628736001
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OFFSET
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1,1
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LINKS
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MATHEMATICA
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MultiFactorial[n_, k_] := If[n<1, 1, n*MultiFactorial[n-k, k]];
Select[Table[MultiFactorial[i, 9] + 1, {i, 0, 100}], PrimeQ[#]&]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A289755
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Primes of the form k!9-1, where k!9 is the nonuple factorial number (A114806).
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+20
1
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2, 3, 5, 7, 89, 439, 1609, 4373, 22679, 5445439, 152681759, 17893715839, 101636305971199, 12652843234348799, 266565181393279999, 4929089879840974847999, 16401565050020468398079999, 2263415976902824638935039999, 1692607074564424130419507199999
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OFFSET
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1,1
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LINKS
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MATHEMATICA
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MultiFactorial[n_, k_] := If[n<1, 1, n*MultiFactorial[n-k, k]];
Select[Table[MultiFactorial[i, 9] - 1, {i, 2, 100}], PrimeQ[#]&]
Select[Table[Times@@Range[n, 1, -9]-1, {n, 200}], PrimeQ] (* Harvey P. Dale, Sep 12 2019 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A045756
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Expansion of e.g.f. (1-9*x)^(-1/9), 9-factorial numbers.
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+10
22
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1, 1, 10, 190, 5320, 196840, 9054640, 498005200, 31872332800, 2326680294400, 190787784140800, 17361688356812800, 1736168835681280000, 189242403089259520000, 22330603564532623360000, 2835986652695643166720000, 385694184766607470673920000, 55925656791158083247718400000
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OFFSET
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0,3
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COMMENTS
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Nine-fold factorials of numbers 9k+1, k = 0, 1, 2, ... - M. F. Hasler, Feb 14 2020
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LINKS
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FORMULA
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a(n+1) = (9*n+1)(!^9) = Product_{k=0..n-1} (9*k+1), n >= 0.
E.g.f. (1-9*x)^(-1/9).
D-finite with recurrence: a(n) +(-9*n+8)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
a(n) = (-8)^n * sum_{k = 0..n} (9/8)^k * s(n + 1, n + 1 - k), where s(n, k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
a(n) = 9^n * Gamma(n + 1/9) / Gamma(1/9). - Artur Jasinski Aug 23 2016
a(n) ~ sqrt(2 * Pi) * 9^n * n^(n - 7/18)/(Gamma(1/9) * exp(n)). - Ilya Gutkovskiy, Sep 10 2016
Sum_{n>=0} 1/a(n) = 1 + (e/9^8)^(1/9)*(Gamma(1/9) - Gamma(1/9, 1/9)). - Amiram Eldar, Dec 21 2022
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MAPLE
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seq( mul(9*j+1, j=0..n-1), n=0..20); # G. C. Greubel, Nov 11 2019
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MATHEMATICA
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Table[9^n*Pochhammer[1/9, n], {n, 0, 20}] (* G. C. Greubel, Nov 11 2019 *)
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PROG
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(PARI) vector(21, n, prod(j=0, n-2, 9*j+1) ) \\ G. C. Greubel, Nov 11 2019
(Magma) [1] cat [(&*[9*j+1: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Nov 11 2019
(Sage) [product( (9*j+1) for j in (0..n-1)) for n in (0..20)] # G. C. Greubel, Nov 11 2019
(GAP) List([0..20], n-> Product([0..n-1], j-> 9*j+1) ); # G. C. Greubel, Nov 11 2019
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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a(0)=1 inserted; merged with A144772; formulas and programs changed accordingly by Georg Fischer, Feb 15 2020
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STATUS
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approved
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A288096
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Decimal expansion of m(9) = Sum_{n>=0} 1/n!9, the 9th reciprocal multifactorial constant.
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+10
10
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4, 0, 8, 1, 3, 7, 5, 5, 2, 0, 1, 6, 8, 8, 9, 8, 5, 4, 4, 0, 7, 1, 1, 0, 5, 1, 4, 6, 6, 0, 9, 6, 1, 0, 6, 9, 4, 6, 2, 6, 4, 1, 0, 0, 7, 7, 3, 1, 8, 6, 0, 7, 5, 8, 8, 4, 3, 4, 8, 5, 1, 7, 5, 1, 6, 7, 4, 9, 3, 4, 8, 7, 6, 3, 9, 0, 3, 3, 3, 5, 9, 9, 2, 1, 0, 5, 4, 2, 4, 2, 3, 0, 5, 7, 2, 0, 3, 5, 9, 0, 7, 4
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OFFSET
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1,1
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LINKS
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FORMULA
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m(k) = (1/k)*exp(1/k)*(k + Sum_{j=1..k-1} (gamma(j/k) - gamma(j/k, 1/k)) where gamma(x) is the Euler gamma function and gamma(a,x) the incomplete gamma function.
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EXAMPLE
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4.08137552016889854407110514660961069462641007731860758843485175...
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MATHEMATICA
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m[k_] := (1/k) Exp[1/k] (k + Sum[k^(j/k) (Gamma[j/k] - Gamma[j/k, 1/k]), {j, 1, k - 1}]); RealDigits[m[9], 10, 102][[1]]
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PROG
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(PARI) default(realprecision, 105); (1/9)*exp(1/9)*(9 + sum(k=1, 8, 9^(k/9)*(gamma(k/9) - incgam(k/9, 1/9)))) \\ G. C. Greubel, Mar 28 2019
(Magma) SetDefaultRealField(RealField(105)); (1/9)*Exp(1/9)*(9 + (&+[9^(k/9)*Gamma(k/9, 1/9): k in [1..8]])); // G. C. Greubel, Mar 28 2019
(Sage) numerical_approx((1/9)*exp(1/9)*(9 + sum(9^(k/9)*(gamma(k/9) - gamma_inc(k/9, 1/9)) for k in (1..8))), digits=105) # G. C. Greubel, Mar 28 2019
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A204659
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Numbers n such that n!9-1 is prime.
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+10
9
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3, 4, 6, 8, 15, 20, 23, 27, 30, 44, 51, 62, 80, 90, 95, 114, 129, 138, 150, 152, 156, 182, 201, 216, 293, 332, 342, 393, 411, 414, 419, 525, 668, 743, 800, 972, 1034, 1266, 1785, 1869, 2777, 3561, 3780, 4106, 4328, 4428, 4556, 4574, 4629, 5001, 5397, 6315
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OFFSET
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1,1
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COMMENTS
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a(1)-a(73) are proved prime by the deterministic test of pfgw. - Robert Price, Jun 14 2012
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LINKS
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MATHEMATICA
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MultiFactorial[n_, k_] := If[n < 1, 1, n*MultiFactorial[n - k, k]];
Select[Range[1000], PrimeQ[MultiFactorial[#, 9] - 1] & ] (* Robert Price, Apr 19 2019 *)
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PROG
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(PARI) for(n=0, 9999, isprime(prod(i=0, (n-2)\9, n-9*i)-1)& print1(n", "))
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CROSSREFS
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Cf. A204657, A204658, A204660, A204661, A204662, A204663, A204664, A156165, A156167, A085150, A085148, A085146, A037083, A080778, A002981.
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KEYWORD
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nonn,hard
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AUTHOR
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EXTENSIONS
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Extended b-file adding a(74)-a(81) using data from Ken Davis link by Robert Price, Apr 19 2019
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STATUS
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approved
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A204660
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Numbers n such that n!9+1 is prime.
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+10
9
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0, 1, 2, 4, 6, 10, 11, 12, 13, 14, 16, 17, 18, 19, 21, 24, 25, 32, 40, 43, 48, 49, 50, 57, 60, 71, 73, 82, 83, 86, 97, 105, 114, 121, 142, 147, 159, 168, 195, 205, 210, 212, 233, 262, 288, 289, 300, 309, 316, 323, 356, 403, 447, 505, 514, 553, 735, 739, 777
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OFFSET
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1,3
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COMMENTS
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a(1)-a(106) verified prime by deterministic test of PFGW. - Robert Price, Jun 18 2012
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LINKS
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MATHEMATICA
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MultiFactorial[n_, k_] := If[n < 1, 1, n*MultiFactorial[n - k, k]];
Select[Range[0, 1000], PrimeQ[MultiFactorial[#, 9] + 1] & ] (* Robert Price, Apr 19 2019 *)
Select[Range[0, 800], PrimeQ[Times@@Range[#, 1, -9]+1]&] (* Harvey P. Dale, Aug 19 2021 *)
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PROG
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(PARI) for(n=0, 9999, isprime(prod(i=0, (n-2)\9, n-9*i)+1)& print1(n", "))
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CROSSREFS
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Cf. A204657, A204658, A204659, A204661, A204662, A204663, A204664, A156165, A156167, A085150, A085148, A085146, A037083, A080778, A002981.
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KEYWORD
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nonn,hard
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AUTHOR
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STATUS
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approved
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A288327
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Decuple factorial, 10-factorial, n!10, n!!!!!!!!!!.
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+10
8
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1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 24, 39, 56, 75, 96, 119, 144, 171, 200, 231, 528, 897, 1344, 1875, 2496, 3213, 4032, 4959, 6000, 7161, 16896, 29601, 45696, 65625, 89856, 118881, 153216, 193401, 240000, 293601, 709632, 1272843, 2010624, 2953125, 4133376
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n)=1 for n < 1, otherwise a(n) = n*a(n-10).
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EXAMPLE
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a(13) = 13 * 3 * 1 = 39.
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MAPLE
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a:= n-> `if`(n<1, 1, n*a(n-10)); seq(a(n), n=0..50); # G. C. Greubel, Aug 22 2019
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MATHEMATICA
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MultiFactorial[n_, k_]:=If[n<1, 1 , n*MultiFactorial[n-k, k]];
Table[MultiFactorial[i, 10], {i, 0, 100}]
Table[Times@@Range[n, 1, -10], {n, 0, 50}] (* Harvey P. Dale, Aug 11 2019 *)
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PROG
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(PARI) a(n)=if(n<1, 1, n*a(n-10));
(Magma) b:=func< n | n le 10 select n else n*Self(n-10) >;
(Sage)
def a(n):
if (n<1): return 1
else: return n*a(n-10)
(GAP)
a:= function(n)
if n<1 then return 1;
else return n*a(n-10);
fi;
end;
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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A129116
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Multifactorial array: A(k,n) = k-tuple factorial of n, for positive n, read by ascending antidiagonals.
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+10
3
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1, 1, 2, 1, 2, 6, 1, 2, 3, 24, 1, 2, 3, 8, 120, 1, 2, 3, 4, 15, 720, 1, 2, 3, 4, 10, 48, 5040, 1, 2, 3, 4, 5, 18, 105, 40320, 1, 2, 3, 4, 5, 12, 28, 384, 362880, 1, 2, 3, 4, 5, 6, 21, 80, 945, 3628800, 1, 2, 3, 4, 5, 6, 14, 32, 162, 3840, 39916800, 1, 2, 3, 4, 5, 6, 7, 24, 45, 280
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OFFSET
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1,3
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COMMENTS
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The term "Quintuple factorial numbers" is also used for the sequences A008546, A008548, A052562, A047055, A047056 which have a different definition. The definition given here is the one commonly used. This problem exists for the other rows as well. "n!2" = n!!, "n!3" = n!!!, "n!4" = n!!!!, etcetera. Main diagonal is A[n,n] = n!n = n.
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LINKS
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FORMULA
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A(k,n) = n!k.
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EXAMPLE
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Table begins:
k / A(k,n)
1.|.1.2.6.24.120.720.5040.40320.362880.3628800... = A000142.
2.|.1.2.3..8..15..48..105...384....945....3840... = A006882.
3.|.1.2.3..4..10..18...28....80....162.....280... = A007661.
4.|.1.2.3..4...5..12...21....32.....45.....120... = A007662.
5.|.1.2.3..4...5...6...14....24.....36......50... = A085157.
6.|.1.2.3..4...5...6....7....16.....27......40... = A085158.
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MAPLE
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A:= proc(k, n) option remember; if n >= 1 then n* A(k, n-k) elif n >= 1-k then 1 else 0 fi end: seq(seq(A(1+d-n, n), n=1..d), d=1..16); # Alois P. Heinz, Feb 02 2009
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MATHEMATICA
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A[k_, n_] := A[k, n] = If[n >= 1, n*A[k, n-k], If[n >= 1-k, 1, 0]]; Table[ A[1+d-n, n], {d, 1, 16}, {n, 1, d}] // Flatten (* Jean-François Alcover, May 27 2016, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A297707
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a(n) = Product_{k=1..n-1} n!k, where n!k is k-tuple factorial of n.
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+10
3
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1, 2, 18, 768, 90000, 44789760, 30494620800, 121762322841600, 393644011735296000, 5618427494400000000000, 107587910030480590233600000, 5951222311476064581656248320000, 176804782652901880753915871232000000, 69819090744423637487544223697731584000000
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OFFSET
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1,2
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COMMENTS
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What is the least n > 2 for which a(n) - prevprime(a(n)) is a composite number? If such a number n exists, it is greater than 250.
The least n for which nextprime(a(n)) - a(n) is a composite number is 158.
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LINKS
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FORMULA
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a(n) = Product_{t=1..n-1} (Product_{k=0..floor((n-1)/t)} (n-t*k)).
a(n) = (n^(n-1))*Product_{k=1..n-1} k^tau(n-k).
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EXAMPLE
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a(2) = (2!1) = (2*1) = 2;
a(3) = (3!1)*(3!2) = (3*2*1)*(3*1) = 18;
a(4) = (4!1)*(4!2)*(4!3) = (4*3*2*1)*(4*2)*(4*1) = 768;
a(5) = (5!1)*(5!2)*(5!3)*(5!4) = (5*4*3*2*1)*(5*3*1)*(5*2)*(5*1) = 90000.
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MAPLE
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b:= proc(n, k) option remember; `if`(n<1, 1, n*b(n-k, k)) end:
a:= n-> mul(b(n, k), k=1..n-1):
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MATHEMATICA
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Array[(#^(# - 1)) Product[k^DivisorSigma[0, # - k], {k, # - 1}] &, 13] (* Michael De Vlieger, Jan 04 2018 *)
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PROG
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(PARI) a(n) = (n^(n-1))*prod(k=1, n-1, k^numdiv(n-k)); \\ Michel Marcus, Dec 02 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A114423
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Multifactorial array read by ascending antidiagonals.
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+10
1
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1, 2, 1, 6, 2, 1, 24, 3, 2, 1, 120, 8, 3, 2, 1, 720, 15, 4, 3, 2, 1, 5040, 48, 10, 4, 3, 2, 1, 40320, 105, 18, 5, 4, 3, 2, 1, 362880, 384, 28, 12, 5, 4, 3, 2, 1, 3628800, 945, 80, 21, 6, 5, 4, 3, 2, 1, 39916800, 3840, 162, 32, 14, 6, 5, 4, 3, 2, 1
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OFFSET
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1,2
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COMMENTS
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The columns are n!, n!!, n!!!, ... n!k for n >= 1, k >= 1.
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LINKS
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FORMULA
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M(n,k) = n!k.
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EXAMPLE
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Table M begins:
n / M(n,k)
1.|...1...1...1...1...1
2.|...2...2...2...2...2
3.|...6...3...3...3...3
4.|..24...8...4...4...4
5.|.120..15..10...5...5
6.|.720..48..18..12...6
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MATHEMATICA
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NFactorialM[n_, m_] := Block[{k = n, p = Max[1, n]},
While[k > m, k -= m; p *= k]; p];
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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