A085724 -id:A085724 - 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

A092561 "Mersenne" semiprimes, semiprimes of the form 2^k-1.

Original entry on oeis.org

15, 511, 2047, 8388607, 137438953471, 2199023255551, 562949953421311, 576460752303423487, 147573952589676412927, 9671406556917033397649407, 158456325028528675187087900671, 2535301200456458802993406410751

Offset: 1

Zak Seidov, Feb 27 2004

nonn

			2047 is a member because 2047 = 2^11 - 1 = 23*89.

Amiram Eldar, Table of n, a(n) for n = 1..42

Corresponding k: A085724.

Cf. A000225, A092558, A092559, A092562.

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 *)

a(n) = A000225(A085724(n)). - Amiram Eldar, Jun 18 2022

One more term from Stefan Steinerberger, Apr 12 2006

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

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 *)

A250288 Numbers n such that the duodecimal repunit (12^n - 1)/11 is a semiprime.

Original entry on oeis.org

7, 13, 17, 37, 47, 73, 101, 131, 151, 167, 197, 241, 263

Offset: 1

Eric Chen, Dec 18 2014

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.

			a(1) = 7 so 1111111 = 46E * 2X3E (written in base 12).

Samuel Wagstaff, The Cunningham Project

Cf. A046413, A085724, A004064.

Select[Range[120], PrimeOmega[(12^# - 1)/11] == 2 &] (* Alonso del Arte, Dec 18 2014 *)

A278240 Least number with the prime signature of 2^n - 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 19 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..256

Cf. A000225, A046523.
Cf. A000043 (positions of 2's), A085724 (of 6's).
Cf. also A278242.

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 *)

allocatemem(2^30);
A046523(n) = my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]) \\ From Charles R Greathouse IV, Aug 17 2011
A278240(n) = A046523((2^n)-1);
for(n=1, 256, write("b278240.txt", n, " ", A278240(n)));

(define (A278240 n) (A046523 (A000225 n)))

a(n) = A046523(A000225(n)).

A239638 Numbers n such that the semiprime 2^n-1 is divisible by 2n+1.

Original entry on oeis.org

11, 23, 83, 131, 3359, 130439, 406583

Offset: 1

Zak Seidov, Mar 23 2014

All terms are primes == 5 modulo 6 (A005384 Sophie Germain primes).

a(8) >= 500000. - Max Alekseyev, May 28 2022

			n = 11, 2^n -1 = 2047 = 23*89,
n = 23, 8388607 = 47*178481,
n = 131, 2722258935367507707706996859454145691647 =  263*10350794431055162386718619237468234569.

Cf. A000043, A001348, A046051, A005384, A005420, A085724, A049479.

Select[Range[4000], PrimeQ[2*# + 1] && PowerMod[2, #, 2*# + 1] == 1 &&
PrimeQ[(2^# - 1)/(2*# + 1)] &] (* Giovanni Resta, Mar 23 2014 *)

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

from sympy import isprime, nextprime
A239638_list, p = [], 5
while p < 10**6:
    if (p % 6) == 5:
        n = (p-1)//2
        if pow(2,n,p) == 1 and isprime((2**n-1)//p):
            A239638_list.append(n)
    p = nextprime(p) # Chai Wah Wu, Jun 05 2019

a(5)-a(6) from Giovanni Resta, Mar 23 2014

a(7) from Eric Chen, added by Max Alekseyev, May 21 2022

A250291 Numbers k such that (2^k+1)/3 is a semiprime.

Original entry on oeis.org

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, 1123, 1217

Offset: 1

Eric Chen, Dec 24 2014

If (2^k+1)/3 is a semiprime, k must be prime or the square of a prime; the only known square of a prime in this sequence is 49.

a(42) >= 1259.

			a(1) = 29 so (2^29+1)/3 = 178956971 = 59 * 3033169 is a semiprime.

Samuel Wagstaff, The Cunningham Project.

Cf. A000978, A085724, A007583.

a(40)-a(41) from Max Alekseyev, Feb 25 2025

A363374 Numbers k such that 2^k - 3 is a semiprime.

Original entry on oeis.org

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

Offset: 1

Kevin P. Thompson, May 29 2023

nonn

The numbers 717, 720, 759 are also terms with 713 being the only remaining unknown below them.

			11 is a member because 2^11 - 3 = 2045 = 5 * 409 is a semiprime.

Cf. A085724.

Select[Range[700],PrimeOmega[2^#-3]==2&] (* Harvey P. Dale, Dec 14 2024 *)

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.

cp's OEIS Frontend

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

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A092561 "Mersenne" semiprimes, semiprimes of the form 2^k-1.

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

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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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A250288 Numbers n such that the duodecimal repunit (12^n - 1)/11 is a semiprime.

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A278240 Least number with the prime signature of 2^n - 1.

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A239638 Numbers n such that the semiprime 2^n-1 is divisible by 2n+1.

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A250291 Numbers k such that (2^k+1)/3 is a semiprime.

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