A002515 -id:A002515 - cp's OEIS frontend

A054723 Prime exponents of composite Mersenne numbers.

11, 23, 29, 37, 41, 43, 47, 53, 59, 67, 71, 73, 79, 83, 97, 101, 103, 109, 113, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331

Offset: 1

Jeppe Stig Nielsen, Apr 20 2000

Primes p such that 2^p-1 is composite.
No proof is known that this sequence is infinite!
Assuming a conjecture of Dickson, we can prove that this sequence is infinite. See Ribenboim. - T. D. Noe, Jul 30 2012
A002515 \ {3} is a subsequence. Any proof that A002515 is infinite would imply that this sequence is infinite. - Jeppe Stig Nielsen, Aug 03 2020

			p=29 is included because 29 is prime, but 2^29-1 is *not* prime.

Paulo Ribenboim, The New Book of Prime Number Records, Springer, 1996, p. 378.

Vincenzo Librandi, Table of n, a(n) for n = 1..2974
Charles B. Barker, Proof that the Mersenne number M167 is composite, Bull. Amer. Math. Soc. 51 (1945), 389.
H. S. Uhler, Note on the Mersenne numbers M157 and M167, Bull. Amer. Math. Soc. 52 (1946), 178.

Complement of A000043 inside A000040.

Cf. A016027.

[p: p in PrimesUpTo(350) | not IsPrime(2^p-1)];  // Bruno Berselli, Oct 11 2012

Select[Prime[Range[70]], ! PrimeQ[2^# - 1] &] (* Harvey P. Dale, Feb 03 2011 *)
Module[{nn=15,mp},mp=MersennePrimeExponent[Range[nn]];Complement[ Prime[ Range[ PrimePi[Last[mp]]]],mp]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 10 2019 *)

forprime(p=2, 1e3, if(!isprime(2^p-1), print1(p, ", "))) \\ Felix Fröhlich, Aug 10 2014

Offset corrected by Arkadiusz Wesolowski, Jul 29 2012

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

A101794 Numbers k such that 4k-1, 8k-1, 16k-1 and 32k-1 are all primes.

Original entry on oeis.org

45, 90, 675, 885, 3030, 4290, 6870, 13410, 14460, 15855, 17850, 18675, 20625, 21885, 25350, 26820, 26925, 28230, 30525, 30705, 31710, 31785, 33375, 34860, 41685, 41940, 57435, 63420, 63570, 71805, 74025, 78585, 83865, 85230, 93075

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 16 2004

			4*45-1 = 179, 8*45-1 = 359, 16*45-1 = 719 and 32*45-1 = 1439 are primes, so 45 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002515, A101795, A101796, A101797, A101798.
Subsequence of A005099, A005122 and A101790.
Subsequence: A101994.

Select[Range[10^5], And @@ PrimeQ[2^Range[2, 5]*# - 1] &] (* Amiram Eldar, May 13 2024 *)

is(k) = isprime(4*k-1) && isprime(8*k-1) && isprime(16*k-1) && isprime(32*k-1); \\ Amiram Eldar, May 13 2024

A101994 Numbers k such that 4k-1, 8k-1, 16k-1, 32k-1 and 64*k-1 are all primes.

Original entry on oeis.org

45, 13410, 15855, 31710, 31785, 63570, 74025, 85230, 151830, 202635, 267300, 280665, 399675, 405405, 455250, 466560, 478170, 480240, 511335, 534600, 539475, 561330, 569520, 589305, 666945, 716460, 743160, 748215, 766785, 799350, 860835

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 23 2004

			4*45-1 = 179, 8*45-1 = 359, 16*45-1 = 719, 32*45-1 = 1439 and 64*45-1 = 2879 are primes, so 45 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002515, A101995, A101996, A101997, A101998, A101999.

Subsequence of A005099, A005122, A101790 and A101794.

Select[Range[10^6], And @@ PrimeQ[2^Range[2, 6]*# - 1] &] (* Amiram Eldar, May 13 2024 *)

is(k) = isprime(4*k-1) && isprime(8*k-1) && isprime(16*k-1) && isprime(32*k-1) && isprime(64*k-1); \\ Amiram Eldar, May 13 2024

A158034 Integers n for which f = (4^n - 2^n + 8n^2 - 2) / (2n * (2n + 1)) is an integer.

Original entry on oeis.org

3, 11, 23, 83, 131, 179, 191, 239, 243, 251, 359, 419, 431, 443, 491, 659, 683, 719, 743, 891, 911, 1019, 1031, 1103, 1223, 1439, 1451, 1499, 1511, 1539, 1559, 1583, 1811, 1931, 2003, 2039, 2063, 2211, 2339, 2351, 2399, 2459, 2511, 2543, 2699, 2819, 2903

Offset: 1

Reikku Kulon, Mar 11 2009

Superset of A002515; 2n + 1 is prime. A recursive search for members of this sequence results in the infinite series of very large primes A145918. Most members of this sequence are also prime, but five members less than 10000 are composite:
.. . 243 = 3^5
.. . 891 = 3^4 * 11
. . 1539 = 3^4 * 19
. . 2211 = 3 * 11 * 67
. . 2511 = 3^4 * 31
The polygonal number with f sides of length 2n + 1 is (2^n - 1)(2^(n - 1)).
Contribution from Reikku Kulon, May 19 2009: (Start)
The average difference between successive composite terms gradually increases, remaining near their order of magnitude. Roughly 3% of all primes less than 20 billion belong to this sequence or the 2n + 1 sequence. The interval between composite terms 12228632879 and 13169544651 contains 1113606 primes, accounting for 2.75% of the primes in the interval and 1.42% of the primes between 24457265759 and 26339089303.
Prime factors are most often congruent to 3 (mod 4), but some factors are congruent to 1 (mod 4), especially when a term has an even number of not necessarily distinct factors. The most common factor is 3, and often a large power of 3 is a divisor. 5, 7, 13, and 17 are never factors.
The ones digit of composite terms is most often 1, and becomes progressively more likely to be 1. It is never 5. It cannot be 7, because 2n + 1 would then be divisible by 5. The lack of solutions with n divisible by 5 appears crucial to the consistent primality of 2n + 1.
The tens digit is odd if the ones digit is 1 or 9; it is even if the ones digit is 3. This is a consequence of congruence to 3 (mod 4).
The most common least significant two digits of composite terms are 51.
The least significant digits of prime terms do not follow an obvious distribution.
This is the simplest and possibly most productive member of a family of similar sequences defined by f = (s + 8n^2 - 2) / (2n * (2n + 1)), where s is pronic. For these sequences, 2n + 1 is dominated by primes.
=====================================
Large sequences of consecutive primes
=====================================
.   Initial term    Primes   Predecessor     Successor          Gap
.   ---------------------------------------------------------------
.     1529648303    157285    1529648231    1639846391    110198160
.     3832649339    473045    3832647111    4193496803    360849692
.     5897103683    411434    5897102751    6223464171    326361420
.     6543227423    445293    6543226251    6899473631    356247380
.     8126586971    913506    8126586711    8871331491    744744780
.     9533381219    689395    9533380131   10103115231    569735100
.    11576086883    708712   11576086731   12171829419    595742688
.    12228633251   1113606   12228632879   13169544651    940911772
.    21315457451   2328623   21315457251   23375077119   2059619868
(End)

			ngon(f, k) = k * (f * (k - 1) / 2 - k + 2)
. . . 3 = (4^3 - 2^3 + 8 * 9 - 2) / (6 * 7)
. . . . = (2 * 28 + 70) / 42
. . 126 = (2 * 28 + 70)
.. . 28 = (2^3 - 1) * 2^2
. . . . = 126 - 70 - 28
. . . . = 7 * (18 - 10 - 4)
. . . . = 7 * (3 * 6 - 3 * 3 - 5)
. . . . = 7 * (3 * 3 - 7 + 2)
.. 8287 = (4^11 - 2^11 + 8 * 121 - 2) / (22 * 23)
. . . . = (2 * 2096128 + 966) / 506
4193222 = (2 * 2096128 + 966)
2096128 = (2^11 - 1) * 2^10
. . . . = 4193222 - 2096128 - 966
. . . . = 23 * (182314 - 91136 - 42)
. . . . = 23 * (8287 * 22 - 8287 * 11 - 21)
. . . . = 23 * (8287 * 11 - 23 + 2)
Coincidentally, 8287 = 129 * 64 + 31 = 257 * 32 + 63 is prime, and may be the largest value of f that is.
1031 = 257 * 4 + 3 and 2063 = 1031 * 2 + 1 are both members of this sequence, 4127 = 2063 * 2 + 1 is prime, and 8287 = (4127 + 16) * 2 + 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 112.

Cf. A002515 (Lucasian primes)
Cf. A145918 (exponential Sophie Germain primes)
Cf. A139601 (polygonal numbers)
Cf. A046318, A139876 (related to composite members 243, 891, 1539, and 2511)
Cf. A060210, A002034, A109833, A136801 (their factors)
Cf. A039506 (3, 8287)
Cf. A006516 (2^n - 1)(2^(n - 1))
Cf. A000051 (Fermat numbers), A019434 (Fermat primes)
Cf. A142291 (prime sequence 257, 1031, 2063, 4127)
Cf. A235540 (nonprimes), A002943.

a158034 n = a158034_list !! (n-1)
a158034_list = [x | x <- [1..],
                    (4^x - 2^x + 8*x^2 - 2) `mod` (2*x*(2*x + 1)) == 0]
-- Reinhard Zumkeller, Jan 12 2014

A101790 Numbers k such that 4k-1, 8k-1 and 16*k-1 are all primes.

Original entry on oeis.org

3, 45, 90, 180, 255, 258, 363, 378, 453, 483, 615, 675, 705, 873, 885, 978, 1350, 1533, 1770, 1788, 2673, 2793, 2868, 3030, 3225, 3240, 4203, 4290, 4548, 4830, 4998, 5103, 5253, 5295, 5568, 5775, 5955, 6060, 6138, 6870, 7383, 7713, 8133, 8370, 8580, 9000

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 16 2004

			4*3 - 1 = 11, 8*3 - 1 = 23 and 16*3 - 1 = 47 are primes, so 3 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002515, A101791, A101792, A101793.
Subsequence of A005099 and A005122.
Subsequences: A101794, A101994.

[n: n in [0..10000] | IsPrime(4*n-1) and IsPrime(8*n-1) and IsPrime(16*n-1)]; // Vincenzo Librandi, Nov 17 2010

Select[Range[10^4], And @@ PrimeQ[2^Range[2, 4]*# - 1] &] (* Amiram Eldar, May 12 2024 *)

is(k) = isprime(4*k-1) && isprime(8*k-1) && isprime(16*k-1); \\ Amiram Eldar, May 12 2024

A101793 Primes of the form 16k-1 such that 4k-1 and 8*k-1 are also primes.

Original entry on oeis.org

47, 719, 1439, 2879, 4079, 4127, 5807, 6047, 7247, 7727, 9839, 10799, 11279, 13967, 14159, 15647, 21599, 24527, 28319, 28607, 42767, 44687, 45887, 48479, 51599, 51839, 67247, 68639, 72767, 77279, 79967, 81647, 84047, 84719, 89087, 92399, 95279, 96959, 98207

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 16 2004

			4*3-1 = 11, 8*3-1 = 23 and 16*3-1 = 47 are primes, so 47 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002515, A101790, A101791, A101792.
Subsequence of A127576.
Subsequences: A101797, A101997.

16#-1&/@Select[Range[10000],AllTrue[{4#-1,8#-1,16#-1},PrimeQ]&] (* Harvey P. Dale, Jun 13 2015 *)

is(k) = if(k % 16 == 15, my(m = k\16 + 1); isprime(4*m-1) && isprime(8*m-1) && isprime(16*m-1), 0); \\ Amiram Eldar, May 13 2024

a(n) = 16*A101790(n) - 1 = 4*A101791(n) + 3 = 2*A101792(n) + 1. - Amiram Eldar, May 13 2024

A101995 Primes of the form 4k-1 such that 8k-1, 16k-1, 32k-1 and 64*k-1 are also primes.

Original entry on oeis.org

179, 53639, 63419, 126839, 127139, 254279, 296099, 340919, 607319, 810539, 1069199, 1122659, 1598699, 1621619, 1820999, 1866239, 1912679, 1920959, 2045339, 2138399, 2157899, 2245319, 2278079, 2357219, 2667779, 2865839

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 23 2004

			4*45-1 = 179, 8*45-1 = 359, 16*45-1 = 719, 32*45-1 = 1439 and 64*45-1 = 2879 are primes, so 179 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002515, A101794.
Cf. A101994, A101996, A101997, A101998, A101999.
Subsequence of A002145, A101791 and A101795.

4 * Select[Range[10^5], And @@ PrimeQ[2^Range[2, 6]*# - 1] &] - 1 (* Amiram Eldar, May 13 2024 *)

is(k) = if(k % 4 == 3, my(m = k\4 + 1); isprime(4*m-1) && isprime(8*m-1) && isprime(16*m-1) && isprime(32*m-1) && isprime(64*m-1), 0); \\ Amiram Eldar, May 13 2024

a(n) = 4*A101994(n) - 1. - Amiram Eldar, May 13 2024

A158035 a(n) = 2*A158034(n) + 1, prime numbers p for which f = (2^p - 2^((p - 1) / 2 + 1) + 4p^2 - 8p) / (2p^2 - 2p) is an integer.

Original entry on oeis.org

7, 23, 47, 167, 263, 359, 383, 479, 487, 503, 719, 839, 863, 887, 983, 1319, 1367, 1439, 1487, 1783, 1823, 2039, 2063, 2207, 2447, 2879, 2903, 2999, 3023, 3079, 3119, 3167, 3623, 3863, 4007, 4079, 4127, 4423, 4679, 4703, 4799, 4919, 5023, 5087, 5399, 5639

Offset: 1

Reikku Kulon, Mar 11 2009

(p - 1) / 2 is often prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 112.

Cf. A158034.
Cf. A002515 (Lucasian primes).
Cf. A145918 (exponential Sophie Germain primes).
Cf. A046318, A139876 (related to composite members of A158034: 243, 891, 1539, and 2511).

A158035 := proc(n) local i,am,p,tren;
am := [ ]:
for i from 2 to n do
  p := ithprime(i):
  tren := (2^(p) - 2^((p - 1) / 2 + 1) + 4*p^(2) - 8*p) / (2*p^(2) - 2*p):
  if (type( tren, 'integer') = 'true') then
    am := [op(am),p]:
  fi
od; RETURN(am) end:
A158035(740); # Jani Melik, May 06 2013

Select[Prime[Range[800]],IntegerQ[(2^#-2^((#-1)/2+1)+4#^2-8#)/(2#^2-2#)]&] (* Harvey P. Dale, Nov 08 2017 *)

A101795 Primes of the form 4k-1 such that 8k-1, 16k-1 and 32k-1 are also primes.

Original entry on oeis.org

179, 359, 2699, 3539, 12119, 17159, 27479, 53639, 57839, 63419, 71399, 74699, 82499, 87539, 101399, 107279, 107699, 112919, 122099, 122819, 126839, 127139, 133499, 139439, 166739, 167759, 229739, 253679, 254279, 287219, 296099

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 16 2004

			4*45-1 = 179, 8*45-1 = 359, 16*45-1 = 719 and 32*45-1 = 1439 are primes, so 179 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A002515, A101794, A101796, A101797, A101798.
Subsequence of A002145 and A101791.
Subsequence: A101995.

Select[Table[4n-1,{n,75000}],AllTrue[(#+1)*{1,2,4,8}-1,PrimeQ]&] (* Harvey P. Dale, Apr 23 2019 *)

is(k) = if(k % 4 == 3, my(m = k\4 + 1); isprime(4*m-1) && isprime(8*m-1) && isprime(16*m-1) && isprime(32*m-1), 0); \\ Amiram Eldar, May 13 2024

a(n) = 4*A101794(n) - 1. - Amiram Eldar, May 13 2024

cp's OEIS Frontend

A054723 Prime exponents of composite Mersenne numbers.

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

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A101794 Numbers k such that 4*k-1, 8*k-1, 16*k-1 and 32*k-1 are all primes.

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A101994 Numbers k such that 4*k-1, 8*k-1, 16*k-1, 32*k-1 and 64*k-1 are all primes.

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A158034 Integers n for which f = (4^n - 2^n + 8n^2 - 2) / (2n * (2n + 1)) is an integer.

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A101790 Numbers k such that 4*k-1, 8*k-1 and 16*k-1 are all primes.

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A101793 Primes of the form 16*k-1 such that 4*k-1 and 8*k-1 are also primes.

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A101794 Numbers k such that 4k-1, 8k-1, 16k-1 and 32k-1 are all primes.

A101994 Numbers k such that 4k-1, 8k-1, 16k-1, 32k-1 and 64*k-1 are all primes.

A101790 Numbers k such that 4k-1, 8k-1 and 16*k-1 are all primes.

A101793 Primes of the form 16k-1 such that 4k-1 and 8*k-1 are also primes.

A101995 Primes of the form 4k-1 such that 8k-1, 16k-1, 32k-1 and 64*k-1 are also primes.

A101795 Primes of the form 4k-1 such that 8k-1, 16k-1 and 32k-1 are also primes.