Search: a049992 -id:a049992
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A023645
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a(n) = tau(n)-1 if n is odd or tau(n)-2 if n is even.
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+10
28
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0, 0, 1, 1, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 3, 3, 1, 4, 1, 4, 3, 2, 1, 6, 2, 2, 3, 4, 1, 6, 1, 4, 3, 2, 3, 7, 1, 2, 3, 6, 1, 6, 1, 4, 5, 2, 1, 8, 2, 4, 3, 4, 1, 6, 3, 6, 3, 2, 1, 10, 1, 2, 5, 5, 3, 6, 1, 4, 3, 6, 1, 10, 1, 2, 5, 4, 3, 6, 1, 8, 4, 2, 1, 10, 3, 2, 3, 6, 1, 10, 3, 4, 3, 2, 3, 10, 1, 4, 5, 7, 1, 6, 1, 6
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OFFSET
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1,6
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COMMENTS
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Vertex-transitive graphs of valency 2 with n nodes.
Number of values of k such that n+2 divided by k leaves a remainder 2. - Amarnath Murthy, Aug 01 2002
Number of divisors of n that are less than n/2. - Peter Munn, Mar 31 2017, or equivalently, number of divisors of n that are greater than 2. - Antti Karttunen, Feb 20 2023
For n > 2, a(n) is the number of planar arrangements of equal-sized regular n-gons such that their centers lie on a circle and neighboring n-gons have an edge in common. - Peter Munn, Apr 23 2017
Number of partitions of n into two distinct parts such that the smaller divides the larger. - Wesley Ivan Hurt, Dec 21 2017
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REFERENCES
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CRC Handbook of Combinatorial Designs, 1996, p. 649.
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LINKS
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FORMULA
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G.f.: Sum_{k>0} x^(3*k) / (1 - x^k). - Michael Somos, Apr 29 2003.
a(n) = Sum_{ d|n, d < n/2 } 1. Cf. A296955.
G.f.: Sum_{k >= 3} x^k/(1 - x^k). (End)
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 5/2), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 08 2024
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EXAMPLE
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x^3 + x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + 2*x^9 + 2*x^10 + x^11 + 4*x^12 + ...
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MAPLE
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with(numtheory); f := n->if n mod 2 = 1 then tau(n)-1 else tau(n)-2; fi;
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MATHEMATICA
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Table[s = DivisorSigma[0, n]; If[OddQ[n], s - 1, s - 2], {n, 100}] (* T. D. Noe, Nov 18 2013 *)
Array[DivisorSigma[0, #] - 1 - Boole@ EvenQ@ # &, 104] (* Michael De Vlieger, Apr 25 2017 *)
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PROG
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(PARI) {a(n) = if( n<1, 0, numdiv(n) - 2 + n%2)} /* Michael Somos, Apr 29 2003 */
(PARI) a(n) = sumdiv(n, d, d < n/2); \\ Michel Marcus, Apr 01 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A014405
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Number of arithmetic progressions of 3 or more positive integers, strictly increasing with sum n.
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+10
18
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0, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 3, 0, 1, 5, 1, 0, 6, 0, 2, 7, 2, 0, 8, 2, 2, 9, 3, 0, 13, 0, 2, 11, 3, 4, 15, 0, 3, 13, 6, 0, 18, 0, 4, 20, 4, 0, 19, 2, 8, 18, 5, 0, 23, 6, 6, 20, 5, 0, 30, 0, 5, 25, 6, 7, 29, 0, 6, 24, 15, 0, 32, 0, 6, 34, 7, 4, 34, 0, 14, 31, 7, 0, 39, 9, 7, 31, 9, 0, 49, 5, 9, 33, 8, 10, 42, 0, 12
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OFFSET
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1,9
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LINKS
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FORMULA
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G.f.: Sum_{k >= 3} x^t(k)/(x^t(k) - x^t(k-1) - x^k + 1) = Sum_{k >= 3} x^t(k)/((1 - x^k) * (1 - x^t(k-1))), where t(k) = k*(k+1)/2 = A000217(k) is the k-th triangular number [Graeme McRae]. - Petros Hadjicostas, Sep 29 2019
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EXAMPLE
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E.g., 15 = 1+2+3+4+5 = 1+5+9 = 2+5+8 = 3+5+7 = 4+5+6.
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PROG
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(PARI) a(n)= t=0; st=0; forstep(s=(n-3)\3, 1, -1, st++; for(c=1, st, m=3; w=m*(s+c); while(w<n, w=w+s+m*c; m++); if(w==n, t++))); t \\ Rick L. Shepherd, Aug 30 2006
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CROSSREFS
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Cf. A000217, A007862, A014406, A014407, A023645, A047966, A049982, A049983, A049986, A049987, A049992, A129654, A240026, A240027, A307824, A320466, A325325, A325328.
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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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A049994
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a(n) is the number of arithmetic progressions of 4 or more positive integers, nondecreasing with sum n.
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+10
4
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0, 0, 0, 1, 1, 1, 1, 2, 1, 3, 1, 3, 1, 3, 3, 4, 1, 4, 1, 6, 3, 4, 1, 6, 4, 4, 3, 7, 1, 9, 1, 6, 3, 5, 7, 10, 1, 5, 3, 12, 1, 10, 1, 8, 10, 6, 1, 11, 4, 12, 4, 9, 1, 11, 9, 12, 4, 7, 1, 20, 1, 7, 9, 11, 10, 13, 1, 10, 4, 21, 1, 18, 1, 8, 14, 11, 7, 14, 1, 22, 8, 9, 1, 21, 12, 9, 5, 15, 1, 29, 8
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OFFSET
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1,8
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LINKS
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FORMULA
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PROG
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CROSSREFS
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Cf. A014405, A014406, A175676, A049980, A049981, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A049991, A049992, A049993, A127938, A321014.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A049993
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a(n) is the number of arithmetic progressions of 3 or more positive integers, nondecreasing with sum <= n.
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+10
2
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0, 0, 1, 2, 3, 6, 7, 9, 13, 16, 17, 24, 25, 28, 36, 40, 41, 51, 52, 58, 68, 72, 73, 87, 91, 95, 107, 114, 115, 134, 135, 141, 155, 160, 167, 189, 190, 195, 211, 223, 224, 248, 249, 257, 282, 288, 289, 316, 320, 332, 353, 362, 363, 392, 401, 413, 436, 443, 444, 484, 485, 492, 522, 533, 543, 578
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OFFSET
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1,4
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LINKS
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FORMULA
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G.f.: (g.f. of A049992)/(1-x). (End)
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CROSSREFS
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Cf. A007862, A014405, A014406, A049980, A049981, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A049991, A049992, A240026, A240027, A307824, A320466, A325325.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A049995
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Number of arithmetic progressions of 4 or more positive integers, nondecreasing with sum <= n.
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+10
0
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0, 0, 0, 1, 2, 3, 4, 6, 7, 10, 11, 14, 15, 18, 21, 25, 26, 30, 31, 37, 40, 44, 45, 51, 55, 59, 62, 69, 70, 79, 80, 86, 89, 94, 101, 111, 112, 117, 120, 132, 133, 143, 144, 152, 162, 168, 169, 180, 184, 196, 200, 209, 210, 221, 230, 242, 246, 253, 254, 274, 275, 282, 291, 302, 312, 325, 326, 336
(list;
graph;
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OFFSET
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1,5
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LINKS
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FORMULA
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G.f.: (g.f. of A049994)/(1-x). (End)
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CROSSREFS
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Cf. A014405, A014406, A049980, A049981, A049982, A049983, A049987, A049988, A049989, A049990, A049991, A049992, A049993, A049994, A127938.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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