Search: a085724 -id:a085724
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A092559
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Numbers k such that 2^k + 1 is a semiprime.
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+10
18
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3, 5, 6, 7, 11, 12, 13, 17, 19, 20, 23, 28, 31, 32, 40, 43, 61, 64, 79, 92, 101, 104, 127, 128, 148, 167, 191, 199, 256, 313, 347, 356, 596, 692, 701, 1004, 1228, 1268, 1709, 2617, 3539, 3824, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239
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OFFSET
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1,1
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COMMENTS
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Thanks to the recently found factor of F_14 (see A093179), we know that 16384 is not in the sequence. First unknown: 16768. - Don Reble, Mar 28 2010
The big prime factors for "5807" and all smaller entries have been proved prime; the rest (as far as I know) are probable primes. - Don Reble, Mar 28 2010
As 3 divides 2^a(n) + 1 for any odd a(n), all odd terms are prime and they are exactly the Wagstaff numbers (A000978) or also the prime Jacobsthal indices (A107036).
All terms from a(51) onwards refer to probabilistic primality tests for 2^a(n) + 1 (see Caldwell's link for the list of the largest certified Wagstaff primes).
For the close relationship between this sequence and the Fermat numbers, see comments in A073936. The only difference is that here a term can be the square of a prime p, and by the Mihăilescu Theorem (also known as Catalan's conjecture, see link) that implies p = a(n) = 3. So, excluding a(1) = 3, they must coincide.
As for A073936, after a(57), the values 267017, 269987, 374321, 986191, 4031399 and 4101572 are also terms, but there still remains the remote possibility of some gaps in between. In addition, 13347311 and 13372531 are also terms, but are possibly much further along in the numbering (see comments in A000978).
(End).
The powers of 2 in this sequence (that correspond to semiprime Fermat numbers) are k = 2^m with m = 5, 6, 7, 8, and no more below 20. - Amiram Eldar, Jun 18 2022
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LINKS
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EXAMPLE
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11 is a term because 2^11 + 1 = 3 * 683.
3 is a term because 2^3 + 1 = 3^2.
10 is not a term because 2^10 + 1 = 5^2 * 41.
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MATHEMATICA
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PROG
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(PARI) isok(n) = bigomega(2^n+1) == 2; \\ Michel Marcus, Oct 05 2013
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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More terms from Cunningham project, Mar 23 2004
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STATUS
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approved
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A092561
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"Mersenne" semiprimes, semiprimes of the form 2^k-1.
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+10
8
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15, 511, 2047, 8388607, 137438953471, 2199023255551, 562949953421311, 576460752303423487, 147573952589676412927, 9671406556917033397649407, 158456325028528675187087900671, 2535301200456458802993406410751
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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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EXAMPLE
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2047 is a member because 2047 = 2^11 - 1 = 23*89.
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MATHEMATICA
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a := Select[Range[2, 120], ! PrimeQ[2^# - 1] && Length[Divisors[2^# - 1]] <= 4 &]; 2^a - 1 (* Stefan Steinerberger, Apr 12 2006 *)
Select[2^Range[0, 110]-1, PrimeOmega[#] == 2&] (* Harvey P. Dale, Feb 22 2013 *)
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CROSSREFS
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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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A092558
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Numbers k such that 2^k +- 1 are both semiprimes.
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+10
6
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OFFSET
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1,1
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COMMENTS
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2^41519 + 1 is the product of 3 and a composite number, so if a(7) exists, it exceeds 41519. - Jon E. Schoenfield, Feb 22 2022
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LINKS
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EXAMPLE
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11 is a term because 2^11 - 1 = 23*89 and 2^11 + 1 = 3*683.
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PROG
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(PARI) is(n)=isprime(n) && n>7 && ispseudoprime((2^n+1)/3) && bigomega(2^n-1)==2 \\ Charles R Greathouse IV, Jun 05 2013
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CROSSREFS
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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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A250288
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Numbers n such that the duodecimal repunit (12^n - 1)/11 is a semiprime.
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+10
3
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7, 13, 17, 37, 47, 73, 101, 131, 151, 167, 197, 241, 263
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OFFSET
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1,1
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COMMENTS
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First unknown term is 311.
If (12^n - 1)/11 is a semiprime, n must be prime or the square of a prime (A001248), but no n = prime squared is known which yields a semiprime value of (12^n - 1)/11. (Specifically, n must be the square of a prime in A004064, and must be at least 491401 = 701^2.)
No other known terms below 1000; the only other possible terms below 1000 are 449, 521, 571, 577, 613, 709, 751, 757, 769, 787, 853, 859, 887, 929, and 991.
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LINKS
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EXAMPLE
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a(1) = 7 so 1111111 = 46E * 2X3E (written in base 12).
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MATHEMATICA
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Select[Range[120], PrimeOmega[(12^# - 1)/11] == 2 &] (* Alonso del Arte, Dec 18 2014 *)
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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A278240
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Least number with the prime signature of 2^n - 1.
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+10
3
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1, 2, 2, 6, 2, 12, 2, 30, 6, 30, 6, 420, 2, 30, 30, 210, 2, 840, 2, 4620, 60, 210, 6, 60060, 30, 30, 30, 30030, 30, 60060, 2, 2310, 210, 30, 210, 38798760, 6, 30, 210, 1021020, 6, 180180, 30, 510510, 30030, 210, 30, 446185740, 6, 510510, 2310, 510510, 30, 240240, 30030, 9699690, 210, 30030, 6, 1203362940780, 2, 30, 60060, 510510, 30, 19399380, 6, 510510, 210
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OFFSET
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1,2
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LINKS
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FORMULA
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MATHEMATICA
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Table[Times @@ MapIndexed[(Prime@ First@ #2)^#1 &, #] &@ If[Length@ # == 1 && #[[1, 1]] == 1, {0}, Reverse@ Sort@ #[[All, -1]]] &@ FactorInteger[2^n - 1], {n, 69}] (* Michael De Vlieger, Nov 21 2016 *)
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PROG
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(PARI)
allocatemem(2^30);
for(n=1, 256, write("b278240.txt", n, " ", A278240(n)));
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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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A102029
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Smallest semiprime with Hamming weight n (i.e., smallest semiprime with exactly n ones when written in binary), or -1 if no such number exists.
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+10
2
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4, 6, 14, 15, 55, 95, 247, 447, 511, 1535, 2047, 7167, 12287, 32255, 49151, 98303, 196607, 393215, 983039, 1572863, 3145727, 6291455, 8388607, 33423359, 50331647, 117440511, 201326591, 528482303, 805306367, 1879048191, 3221225471
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listen;
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OFFSET
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1,1
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COMMENTS
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LINKS
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EXAMPLE
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a(1) = 4 because the first semiprime A001358(1) is 4 (base 10) which is written 100 in binary, the latter representation having exactly 1 one.
a(2) = 6 since A001358(2) = 6 = 110 (base 2) has exactly 2 ones.
a(4) = 15 since A001358(6) = 15 = 1111 (base 2) has exactly 4 ones and, as it also has no zeros, is the smallest of the Mersenne semiprimes.
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MATHEMATICA
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Join[{4}, Table[SelectFirst[Sort[FromDigits[#, 2]&/@Permutations[ Join[ PadRight[{}, n, 1], {0}]]], PrimeOmega[#]==2&], {n, 2, 40}]] (* Harvey P. Dale, Feb 06 2015 *)
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CROSSREFS
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Cf. A000043, A000120, A000337, A000668, A001358, A007088, A061712, A085724, A089226, A089998, A089999, A091991, A092558, A092559, A092561, A092562, A081093, A102782, A110472, A110699, A110700.
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KEYWORD
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easy,base,nonn
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AUTHOR
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STATUS
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approved
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A239638
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Numbers n such that the semiprime 2^n-1 is divisible by 2n+1.
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+10
1
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OFFSET
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1,1
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COMMENTS
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All terms are primes == 5 modulo 6 (A005384 Sophie Germain primes).
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LINKS
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EXAMPLE
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n = 11, 2^n -1 = 2047 = 23*89,
n = 23, 8388607 = 47*178481,
n = 131, 2722258935367507707706996859454145691647 = 263*10350794431055162386718619237468234569.
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MATHEMATICA
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Select[Range[4000], PrimeQ[2*# + 1] && PowerMod[2, #, 2*# + 1] == 1 &&
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PROG
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(PARI) is(n)=n%6==5 && Mod(2, 2*n+1)^n==1 && isprime(2*n+1) && ispseudoprime((2^n-1)/(2*n+1)) \\ Charles R Greathouse IV, Aug 25 2016
(Python)
from sympy import isprime, nextprime
while p < 10**6:
if (p % 6) == 5:
n = (p-1)//2
if pow(2, n, p) == 1 and isprime((2**n-1)//p):
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A250291
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Numbers n such that (2^n+1)/3 is a semiprime.
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+10
1
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29, 37, 41, 47, 49, 53, 67, 71, 73, 103, 107, 109, 139, 151, 179, 223, 229, 251, 269, 277, 311, 349, 353, 433, 457, 487, 503, 599, 601, 613, 619, 643, 739, 757, 827, 839, 1031, 1061, 1117
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OFFSET
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1,1
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COMMENTS
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First unknown: 1123.
If (2^n+1)/3 is a semiprime, n must be prime or the square of a prime; the only known square of a prime in this sequence is 49.
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LINKS
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EXAMPLE
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a(1) = 29 so (2^29+1)/3 = 178956971 = 59 * 3033169 is a semiprime.
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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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A363374
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Numbers k such that 2^k - 3 is a semiprime.
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+10
1
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8, 11, 13, 15, 17, 18, 21, 23, 25, 30, 32, 33, 34, 35, 36, 37, 40, 44, 54, 58, 60, 61, 71, 73, 92, 95, 101, 102, 106, 144, 160, 164, 183, 200, 209, 210, 216, 241, 244, 270, 273, 274, 281, 293, 309, 313, 344, 365, 422, 430, 461, 475, 477, 480, 504, 509, 556, 579, 597, 609, 612, 631, 650
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OFFSET
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1,1
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COMMENTS
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The numbers 717, 720, 759 are also terms with 713 being the only remaining unknown below them.
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LINKS
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EXAMPLE
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11 is a member because 2^11 - 3 = 2045 = 5 * 409 is a semiprime.
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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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A138104
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2^(n-th semiprime) - 1.
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+10
0
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15, 63, 511, 1023, 16383, 32767, 2097151, 4194303, 33554431, 67108863, 8589934591, 17179869183, 34359738367, 274877906943, 549755813887, 70368744177663, 562949953421311, 2251799813685247, 36028797018963967
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OFFSET
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1,1
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COMMENTS
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This is a semiprime analog of A001348 Mersenne numbers. The semiprimes in this sequence are the analogs of A000668 Mersenne primes (of form 2^p - 1 where p is a prime). a(n) is semiprime when a(n) is an element of A092561, which happens for values of n beginning 1, 3, 17, which is A085724 INTERSECTION A001358 and has no more values under 1000. Would someone like to extend the latter set of indices j of semiprimes k = A001358(j) such that (2^k)-1 is semiprime?
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LINKS
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FORMULA
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EXAMPLE
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a(1) = (2^4) - 1 = 15 because 4 is the 1st semiprime. Note that 15 = 3*5 is itself semiprime.
a(2) = (2^6) - 1 = 63 because 6 is the 2nd semiprime. Note that 63 = (3^2)*7 is not itself semiprime.
a(3) = (2^9) - 1 = 511 because 9 is the 3rd semiprime; and 511 = 7 * 73 is itself semiprime.
a(17) = (2^17)-1 = 562949953421311 = 127 * 4432676798593, itself semiprime.
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MATHEMATICA
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2^#-1&/@Select[Range[100], PrimeOmega[#]==2&] (* Harvey P. Dale, Jun 26 2011 *)
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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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