A092561 -id:A092561 - cp's OEIS frontend

A092559 Numbers k such that 2^k + 1 is a semiprime.

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

Offset: 1

Zak Seidov, Feb 27 2004

nonn

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
From Giuseppe Coppoletta, May 09 2017: (Start)
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

			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.

Giuseppe Coppoletta, Table of n, a(n) for n = 1..57
C. Caldwell's The Top Twenty Wagstaff primes.
S. S. Wagstaff, Jr., The Cunningham Project.
Eric W. Weisstein, MathWorld: Catalan's Conjecture.

Cf. A085724, A092558, A092561, A092562, A000978, A107036, A066263.

Cf. A073936. - R. J. Mathar, Sep 08 2008

Select[Range@ 200, PrimeOmega[2^# + 1] == 2 &] (* Michael De Vlieger, May 09 2017 *)

isok(n) = bigomega(2^n+1) == 2; \\ Michel Marcus, Oct 05 2013

More terms from Cunningham project, Mar 23 2004
More terms from Don Reble, Mar 28 2010
a(49)-a(52) from Giuseppe Coppoletta, May 08 2017

A085724 Numbers k such that 2^k - 1 is a semiprime (A001358).

Original entry on oeis.org

4, 9, 11, 23, 37, 41, 49, 59, 67, 83, 97, 101, 103, 109, 131, 137, 139, 149, 167, 197, 199, 227, 241, 269, 271, 281, 293, 347, 373, 379, 421, 457, 487, 523, 727, 809, 881, 971, 983, 997, 1061, 1063

Offset: 1

Jason Earls, Jul 20 2003

Subsequence of A000430. Apart from 4, 9, and 49 composites in this sequence are greater than 1.9e7. - Charles R Greathouse IV, Jun 05 2013
1427 and 1487 are also terms. 1277 is the only remaining unknown below them. - Charles R Greathouse IV, Jun 05 2013
Among the known terms only 11, 23, 83 and 131 are in A002515, that is, they are the only known values for n such that (2^n - 1)/(2*n + 1) is prime. - Jianing Song, Jan 22 2019
Either a(n) is a prime, or the square of a Mersenne prime exponent. - M. F. Hasler, Jun 23 2025

			11 is a member because 2^11 - 1 = 23*89.

J. Earls, Mathematical Bliss, Pleroma Publications, 2009, pages 56-60. ASIN: B002ACVZ6O [From Jason Earls, Nov 22 2009]
J. Earls, "Cole Semiprimes," Mathematical Bliss, Pleroma Publications, 2009, pages 56-60. ASIN: B002ACVZ6O [From Jason Earls, Nov 25 2009]

S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Mersenne Number
Eric Weisstein's World of Mathematics, Semiprime

Cf. A092558, A092559, A092561, A092562.

SemiPrimeQ[n_]:=(n>1) && (2==Plus@@(Transpose[FactorInteger[n]][[2]])); Select[Range[100],SemiPrimeQ[2^#-1]&] (Noe)
Select[Range[1100],PrimeOmega[2^#-1]==2&] (* Harvey P. Dale, Feb 18 2018 *)
Select[Range[250], Total[Last /@ FactorInteger[2^# - 1, 3]] == 2 &] (* Eric W. Weisstein, Jul 28 2022 *)

issemi(n)=bigomega(n)==2
is(n)=if(isprime(n), issemi(2^n-1), my(q); isprimepower(n,&q)==2 && ispseudoprime(2^q-1) && ispseudoprime((2^n-1)/(2^q-1))) \\ Charles R Greathouse IV, Jun 05 2013

More terms from Zak Seidov, Feb 27 2004
More terms from Cunningham project, Mar 23 2004
More terms from the Cunningham project sent by Robert G. Wilson v and T. D. Noe, Feb 22 2006
a(41)-a(42) from Charles R Greathouse IV, Jun 05 2013

A092558 Numbers k such that 2^k +- 1 are both semiprimes.

Original entry on oeis.org

11, 23, 101, 167, 199, 347

Offset: 1

Zak Seidov, Feb 27 2004

nonn

Intersection of A092559 and A085724.
a(7), if it exists, is at least 41519. - Charles R Greathouse IV, Jun 05 2013
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

			11 is a term because 2^11 - 1 = 23*89 and 2^11 + 1 = 3*683.

Subsequence of A000040.

Cf. A092559, A085724, A092561, A092562.

is(n)=isprime(n) && n>7 && ispseudoprime((2^n+1)/3) && bigomega(2^n-1)==2 \\ Charles R Greathouse IV, Jun 05 2013

a(6) from Robert G. Wilson v, Apr 18 2006

A092562 Semiprimes of the form 2^k + 1.

Original entry on oeis.org

9, 33, 65, 129, 2049, 4097, 8193, 131073, 524289, 1048577, 8388609, 268435457, 2147483649, 4294967297, 1099511627777, 8796093022209, 2305843009213693953, 18446744073709551617, 604462909807314587353089, 4951760157141521099596496897

Offset: 1

Zak Seidov, Feb 27 2004

nonn

			2049 is a member because 2049 = 2^11+1 = 3*683.

Amiram Eldar, Table of n, a(n) for n = 1..40 (terms 1..30 from Vincenzo Librandi)

Corresponding k: A092559.

Cf. A000051, A092558, A092561.

IsSemiprime:=func; [s: n in [2..100] | IsSemiprime(s) where s is 2^n+1]; // Vincenzo Librandi, Sep 21 2012

Select[Table[2^n + 1, {n, 100}], PrimeOmega[#] == 2&] (* Vincenzo Librandi, Sep 21 2012 *)

a(n) = A000051(A092559(n)). - Amiram Eldar, Jun 18 2022

A102029 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.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Jun 23 2007

Semiprime analog of A061712. Extended by Stefan Steinerberger. Includes the subset Mersenne semiprimes A092561.

			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.

Donovan Johnson, Table of n, a(n) for n = 1..250

Cf. A000043, A000120, A000337, A000668, A001358, A007088, A061712, A085724, A089226, A089998, A089999, A091991, A092558, A092559, A092561, A092562, A081093, A102782, A110472, A110699, A110700.

Join[{4},Table[SelectFirst[Sort[FromDigits[#,2]&/@Permutations[ Join[ PadRight[{}, n,1],{0}]]],PrimeOmega[#]==2&],{n,2,40}]] (* Harvey P. Dale, Feb 06 2015 *)

A242832 Semiprimes of the form 2^k - 1 or 2^k - 2.

Original entry on oeis.org

6, 14, 15, 62, 254, 511, 2047, 16382, 262142, 1048574, 8388607, 4294967294, 137438953471, 562949953421311, 576460752303423487, 4611686018427387902, 147573952589676412927, 9671406556917033397649407, 1237940039285380274899124222, 158456325028528675187087900671

Offset: 1

Juri-Stepan Gerasimov, May 23 2014

nonn

Union of A092561 and 2*A000668.

Cf. A092561, A000668.

Select[Flatten[Table[{2^k - 2, 2^k - 1}, {k, 100}]], PrimeOmega[#] == 2 &] (* Alonso del Arte, May 26 2014 *)

More terms from Alonso del Arte, May 26 2014

A138104 2^(n-th semiprime) - 1.

Original entry on oeis.org

15, 63, 511, 1023, 16383, 32767, 2097151, 4194303, 33554431, 67108863, 8589934591, 17179869183, 34359738367, 274877906943, 549755813887, 70368744177663, 562949953421311, 2251799813685247, 36028797018963967

Offset: 1

Jonathan Vos Post, May 03 2008

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?

			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.

Cf. A000668, A001348, A001358, A085724, A092561.

2^#-1&/@Select[Range[100],PrimeOmega[#]==2&] (* Harvey P. Dale, Jun 26 2011 *)

a(n) = (2^A001358(n))-1.

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A092559 Numbers k such that 2^k + 1 is a semiprime.

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A085724 Numbers k such that 2^k - 1 is a semiprime (A001358).

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A092558 Numbers k such that 2^k +- 1 are both semiprimes.

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A092562 Semiprimes of the form 2^k + 1.

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A102029 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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A242832 Semiprimes of the form 2^k - 1 or 2^k - 2.

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A138104 2^(n-th semiprime) - 1.

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