A001348 -id:A001348 - cp's OEIS frontend

A103901 Mersenne primes p such that M(p) = 2^p - 1 is also a (Mersenne) prime.

3, 7, 31, 127

Offset: 1

Jonathan Sondow, Feb 20 2005

Same as exponents of double Mersenne primes. Only four terms are known.

			2^2 - 1 = 3 and 2^3 - 1 = 7 are Mersenne primes, so 3 is a member.

R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer-Verlag, NY, 2004, Sec. A3.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 3rd ed., Oxford Univ. Press, 1954, p. 16.
P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag, NY, 1996, Chap. 2, Sec. VII.

C. K. Caldwell, Mersenne Primes: Conjectures and Unsolved Problems
Eric Weisstein's World of Mathematics, Double Mersenne Number
Wikipedia, Mersenne prime

Cf. A000043, A000668, A001348, A077586, A103902.

A127208 Union of all n-step Lucas sequences, that is, all sequences s(1-n) = s(2-n) = ... = s(-1) = -1, s(0) = n and for k > 0, s(k) = s(k-1) + ... + s(k-n).

Original entry on oeis.org

1, 3, 4, 7, 11, 15, 18, 21, 26, 29, 31, 39, 47, 51, 57, 63, 71, 76, 99, 113, 120, 123, 127, 131, 191, 199, 223, 239, 241, 247, 255, 322, 367, 439, 443, 475, 493, 502, 511, 521, 708, 815, 843, 863, 943, 983, 1003, 1013, 1023, 1364, 1365, 1499, 1695, 1871, 1959

Offset: 1

T. D. Noe, Jan 09 2007

nonn

Noe and Post conjectured that the only positive terms that are common to any two distinct n-step Lucas sequences are the Mersenne numbers (A001348) that begin each sequence and 7 and 11 (in 2- and 3-step) and 5071 (in 3- and 4-step). The intersection of this sequence with the union of all the n-step Fibonacci sequences (A124168) appears to consist of 4, 21, 29, the Mersenne numbers 2^n-1 for all n and the infinite set of Eulerian numbers in A127232.

T. D. Noe, Table of n, a(n) for n=1..1000
Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4

Cf. A227885.

LucasSequence[n_,kMax_] := Module[{a,s,lst={}}, a=Join[Table[ -1,{n-1}],{n}]; While[s=Plus@@a; a=RotateLeft[a]; a[[n]]=s; s<=kMax, AppendTo[lst,s]]; lst]; nn=10; t={}; Do[t=Union[t,LucasSequence[n,2^(nn+1)]], {n,2,nn}]; t

Union(A000032, A001644, A073817, A074048, A074584, A104621, A105754, A105755,...)

A137410 a(n) = (5^n - 3)/2.

Original entry on oeis.org

-1, 1, 11, 61, 311, 1561, 7811, 39061, 195311, 976561, 4882811, 24414061, 122070311, 610351561, 3051757811, 15258789061, 76293945311, 381469726561, 1907348632811, 9536743164061, 47683715820311, 238418579101561, 1192092895507811, 5960464477539061, 29802322387695311, 149011611938476561

Offset: 0

Ctibor O. Zizka, Apr 15 2008

Sequence is a(n) = a(n;5,3,1) where a(n;A,B,r) = (A^n - B^r)/(A - B) for arbitrary integers A, B, r with A != B.
Primes of this form are sometimes of interest, examples:
A=2, B=1, r=1 gives A000225 and subsequence of primes: A001348,
A=3, B=1, r=1 gives A003462 and subsequence of primes: A028491,
A=3, B=2, r=1 gives A058481 and subsequence of primes: A014224,
A=4, B=1, r=1 gives A002450,
A=4, B=2, r=1 gives A083420,
A=4, B=2, r=2 gives A002446,
A=5, B=1, r=1 gives A003463 and subsequence of primes: A004061,
A=5, B=2, r=1 gives A037577.
Sum of n-th row of triangle of powers of 5: 1; 5 1 5; 25 5 1 5 25; 125 25 5 1 5 25 125; ... (cf. Examples). - Philippe Deléham, Feb 24 2014
Integer solutions to x^5 - (x+1)^5 -(x+2)^5 +(x+3)^5 = 5^m + 5^n (see Campbell and Zujev). - Michel Marcus, Mar 02 2016

			From _Philippe Deléham_, Feb 24 2014: (Start)
a(1) = 1;
a(2) = 5 + 1 + 5 = 11;
a(3) = 25 + 5 + 1 + 5 + 25 = 61;
a(4) = 125 + 25 + 5 + 1 + 5 + 25 + 125 = 311;
etc. (End)

M. F. Hasler, Table of n, a(n) for n = 0..100
Geoffrey B Campbell and Aleksander Zujev, On integer solutions to x^5 - (x+1)^5 -(x+2)^5 +(x+3)^5 = 5^m + 5^n, arXiv:1603.00080 [math.NT], 2016.
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A000040, A000225, A001348, A002446, A002450, A003462, A003463, A004061, A014224, A024049, A028491, A037577, A058481, A083420, A125831.

[(5^n-3)/2: n in [0..25]]; // Vincenzo Librandi, Mar 02 2016

LinearRecurrence[{6, -5}, {1, 11}, 25] (* Vincenzo Librandi, Mar 02 2016 *)
(5^Range[30]-3)/2 (* Harvey P. Dale, Feb 24 2017 *)

a(n) = (5^n - 3) / 2; \\ Michel Marcus, Mar 02 2016

a(n) = (5^n - 3)/2.
From Colin Barker, May 01 2012: (Start)
a(n) = 6*a(n-1) - 5*a(n-2).
G.f.: (-1+7*x)/((1-x)*(1-5*x)). (End)
a(n) = 5*a(n-1) + 6, a(1) = 1. - Philippe Deléham, Feb 24 2014
From Elmo R. Oliveira, Dec 11 2023: (Start)
a(n) = A024049(n)/2 - 1 = A125831(n) - 1.
E.g.f.: (1/2)*(exp(5*x) - 3*exp(x)). (End)

More terms from Michel Marcus, Mar 02 2016

Edited and missing term a(0) inserted by M. F. Hasler, Jul 10 2018

A140797 Numbers of the form (2^p^N-1)/(2^p^(N-1)-1), where N>0, p is prime.

Original entry on oeis.org

3, 5, 7, 17, 31, 73, 127, 257, 2047, 8191, 65537, 131071, 262657, 524287, 1082401, 8388607, 536870911, 2147483647, 4294967297, 137438953471, 2199023255551, 4432676798593, 8796093022207, 140737488355327, 9007199254740991, 18014398643699713, 576460752303423487

Offset: 1

Vladimir Shevelev, Jul 15 2008

nonn

Contains Fermat numbers A000215 (p=2) and Mersenne numbers A001348 (N=1). The terms of the sequence are either primes A000040 or overpseudoprimes A141232.

The values of A019320(n) for prime power n, sorted. This sequence is a subsequence of A064896, which means that all terms are sturdy numbers (A125121). It appears that the largest prime factor of each of these numbers is a sturdy prime (A143027). - T. D. Noe, Jul 21 2008

T. D. Noe, Table of n, a(n) for n=1..199
Vladimir Shevelev, Process of "primoverization" of numbers of the form a^n-1, arXiv:0807.2332 [math.NT], 2008.

Cf. A000040, A000215, A001348, A141232.

nmax[p_] := Which[p == 2, 6, p == 3, 4, True, 2];
Reap[Do[If[IntegerQ[k = (2^p^n-1)/(2^p^(n-1)-1)] && k<10^18, Print[{p, n, k}]; Sow[k]], {p, Prime[Range[17]]}, {n, 1, nmax[p]}]][[2, 1]] // Union (* Jean-François Alcover, Dec 10 2018 *)

Definition corrected by and more terms from T. D. Noe, Jul 21 2008

A152099 a(n) = (2^prime(n) - 1)(2^prime(n) + 1) = 2^(2prime(n)) - 1.

Original entry on oeis.org

15, 63, 1023, 16383, 4194303, 67108863, 17179869183, 274877906943, 70368744177663, 288230376151711743, 4611686018427387903, 18889465931478580854783, 4835703278458516698824703, 77371252455336267181195263, 19807040628566084398385987583

Offset: 1

Roger L. Bagula, Nov 24 2008

nonn

Idea resulted from seqfan posts by Artur Jasinski.

Cf. A000040, A000302, A001348, A034785, A098640.

Table[(2^Prime[n] - 1)*(2^Prime[n] + 1), {n, 1, 20}]

from sympy import prime
def A152099(n): return (1<<(prime(n)<<1))-1 # Chai Wah Wu, Jun 26 2023

a(n) = A001348(n) * A098640(n).
a(n) = A034785(n)^2 - 1.
a(n) = A000302(A000040(n)) - 1.

A230809 Primes p of the form 60n + 59 such that 2p + 1 is also prime.

Original entry on oeis.org

179, 239, 359, 419, 659, 719, 1019, 1439, 1499, 1559, 2039, 2339, 2399, 2459, 2699, 2819, 2939, 3299, 3359, 3539, 3779, 4019, 4919, 5039, 5279, 5399, 5639, 6899, 7079, 9059, 9419, 9479, 9539, 10799, 11519, 11579, 11699, 11939, 12119, 12899, 12959, 13619

Offset: 1

Arkadiusz Wesolowski, Oct 30 2013

nonn

Primes p such that 2*p + 1 divides Lucas(p) and Mersenne(p).

			179 is in the sequence since it is prime and 359 is a factor of both Lucas(179) and Mersenne(179) = 2^179 - 1.

Eric Weisstein's World of Mathematics, Lucas Number
Eric Weisstein's World of Mathematics, Mersenne Number

Subsequence of A142799, of A215850, and of A239548. Cf. A000032, A001348, A002515.

[p : p in [59..13619 by 60] | IsPrime(p) and IsPrime(2*p+1)];

forstep(p=59, 13619, 60, if(isprime(p)&&isprime(2*p+1), print1(p, ", ")));

A005384 INTERSECT A142799.

A002515 INTERSECT A215850.

A263686 Smallest prime factor of double Mersenne numbers.

Original entry on oeis.org

7, 127, 2147483647, 170141183460469231731687303715884105727, 338193759479, 231733529, 62914441, 295257526626031

Offset: 1

Arkadiusz Wesolowski, Oct 23 2015

A double Mersenne number is a Mersenne number of the form 2^(2^p - 1) - 1, where p is a Mersenne exponent (A000043).
From M. F. Hasler, Feb 28 2025: (Start)
The prime factors of Mersenne numbers 2^q - 1 must be of the form 2*q*k + 1.
The four smallest double Mersenne numbers (p = 2, 3, 5, 7 => q = 3, 7, 31, 127) are prime, so their smallest prime factor is equal to themselves, a(n) = M(q). This is equivalent to k = (2^(q-1)-1)/q, which is almost as large as M(q) itself: k = 1, 9 and 34636833 for the first three terms, and for q = 127, k has just three digits less than M(q) = a(4) itself. The prime p = 11 is not a Mersenne exponent.
The fifth term, a(5) = 2*(2^13-1)*k + 1 with k = 20644229 (which is prime) is the first proper divisor of the respective M(q), as are the next three, corresponding to p = 17, 19 and 31.
For p = 61, M(q) has 694127911065419642 digits, and so far no factor is known, but it is known that it has no factor less than 10^36. (End)

Double Mersennes Prime Search, History
Wikipedia, Double Mersenne number

Cf. A000043, A000668, A001348, A020639, A049479, A077586, A122094. Subsequence of A016047. Subsequence of A309130.

forprime(p=2,,q=2^p-1; !ispseudoprime(q) && next(); if(ispseudoprime(2^q-1), print1(2^q-1,", ");next()); forstep(r=2*q+1,+oo,2*q, !ispseudoprime(r) && next(); if(Mod(2,r)^q-1 == 0, print1(r,", ");next(2)))) \\ Jeppe Stig Nielsen, Aug 28 2019

a(n) = spf(MM(A000043(n))) = A049479(A000668(n)), where spf = A020639 is the smallest prime factor, A049479 = spf o M, M(p) = 2^p-1 = A000225(p), MM = M o M = A077585, A000668(n) = M(A000043(n)), A000043 are the Mersenne prime exponents. - M. F. Hasler, Mar 01 2025

A278741 Odd numbers k such that tau(k-1) is a prime.

Original entry on oeis.org

3, 5, 17, 65, 1025, 4097, 65537, 262145, 4194305, 268435457, 1073741825, 68719476737, 1099511627777, 4398046511105, 70368744177665, 4503599627370497, 288230376151711745, 1152921504606846977, 73786976294838206465, 1180591620717411303425, 4722366482869645213697

Offset: 1

Jaroslav Krizek, Nov 27 2016

nonn

tau(k) = A000005(k) = the number of divisors of k.
Conjecture: prime terms are in A249759: 3, 5, 17, 65537, ...
Supersequence of A256438 and A249759. Subsequence of {A009087(n) + 1}.

			Odd number 65 is in the sequence because tau(64) = 7 (prime).

Cf. A000005, A000040, A001348, A009087, A061286, A249759, A256438.

[n: n in[2..10000000] |  IsOdd(n) and IsPrime(NumberOfDivisors(n-1))];

isok(n) = (n % 2) && isprime(numdiv(n-1)); \\ Michel Marcus, Nov 27 2016

a(n) = A061286(n) + 1.
sigma(a(n)-1) = A001348(n), i.e., Mersenne numbers.
tau(a(n)-1) = A000040(n), i.e., all primes; a(n) = the smallest odd number k such that tau(a(n)-1) = prime(n) = A000040(n).

A296806 Take a prime, convert it to base 2, remove its most significant digit and its least significant digit and convert it back to base 10. Sequence lists primes that generate another prime by this process.

Original entry on oeis.org

13, 23, 31, 37, 43, 47, 59, 71, 79, 103, 127, 139, 151, 163, 167, 191, 211, 223, 251, 263, 271, 283, 331, 379, 463, 523, 547, 571, 587, 599, 607, 619, 631, 647, 659, 691, 719, 727, 739, 787, 811, 827, 839, 859, 907, 911, 967, 971, 991, 1031, 1039, 1051, 1063, 1087

Offset: 1

Paolo P. Lava, Paolo Iachia, Dec 21 2017

From an idea of Ken Abbott (see link).
From Paolo Iachia, Dec 21 2017: (Start)
Let us call these numbers "core of a prime".
Let C(q) be the core of a prime q.
Then C(q) = (q - 2^floor(log_2(q)) - 1)/2.
Examples: C(59) = (59 - 2^5 - 1)/2 = 13; C(71) = (71 - 2^6 - 1)/2 = 3; C(73) = (73 - 2^6 - 1)/2 = 4; C(251) = (251 - 2^7 - 1)/2 = 61.
0 <= C(q) <= 2^(floor(log_2(q)) - 1) - 1. The minimum (0) occurs when q = 2^n+1, with n > 2. Example: 17 = 2^4+1, C(17) = (17 - 2^4 - 1)/2 = 0.  The maximum is reached when q = 2^n-1 is a Mersenne prime. Example: 127 = 2^7 - 1, C(127) = (127 - 2^6 - 1)/2 = 31 = 2^5 - 1.
The last example is particularly interesting, as both the prime q and its core are Mersenne primes. The same holds for C(31) = 7 and for C(524247) = 131071, with 524247 = 2^19-1 and 131071 = 2^17-1, both Mersenne primes. Are there more such cases?
Note that the core of Mersenne number (prime or not) is a Mersenne number by definition. Counterexamples include C(8191) = 2047, with 8191 = 2^13 - 1, a Mersenne prime, but 2047 = 2^11 - 1 = 23*89, a Mersenne number not prime, and C(131071) = 32767 = 2^15 - 1 = 7*31*151, with 2 of its factors being Mersenne primes.
Primes whose binary expansion is of the form q = 1 0 ... 0 c_1 c_2 ... c_k 1 - with none or any number of consecutive 0's and with binary core c_1 c_2 ... c_k, k >= 0 - share the same core value. Let p = C(q), then we can write, in decimal form, q = (2p+1) + 2^n, for an appropriate n. While the property is true for p prime, it can be generalized to any positive integer.
Conjecture: for any positive integer p, there are infinitely many primes q for which there exists an integer n such that q-(2p+1) = 2^n. (End)

			13 in base 2 is 1101 and 10 is 2;
23 in base 2 is 10111 and 011 is 3;
31 in base 2 is 11111 and 111 is 7.

Iain Fox, Table of n, a(n) for n = 1..10000
Ken Abbott, Prime Cores, Number Theory group on LinkedIn.

Cf. A000030, A000225, A001348, A004676, A010879, A057195, A057196, A059242, A123250, A296807.

with(numtheory): P:=proc(q) local a,b,c,j,n,ok,x;  x:=5; for n from x to q do ok:=1; a:=convert(ithprime(n),base,2); b:=nops(a)-1; while a[b]=0 do b:=b-1; od; c:=0;
for j from b by -1 to 2 do c:=2*c+a[j]; od;if isprime(c) then x:=n; print(ithprime(n)); fi; od; end: P(10^6);
# simpler alternative:
select(t -> isprime(t) and isprime((t - 2^ilog2(t) - 1)/2), [seq(i,i=3..10^4,2)]); # Robert Israel, Dec 28 2017

Select[Prime[Range[200]],PrimeQ[FromDigits[Most[Rest[IntegerDigits[ #,2]]],2]]&] (* Harvey P. Dale, Jul 19 2020 *)

lista(nn) = forprime(p=13, nn, if(isprime((p - 2^logint(p, 2) - 1)/2), print1(p, ", "))) \\ Iain Fox, Dec 28 2017

from itertools import islice
from sympy import isprime, nextprime
def agen(): # generator of terms
    p = 7
    while True:
        if isprime(int(bin(p)[3:-1], 2)):
            yield p
        p = nextprime(p)
print(list(islice(agen(), 54))) # Michael S. Branicky, May 16 2022

Primes q such that C(q) = (q - 2^floor(log_2(q)) - 1)/2 is prime too.

A309130 Smallest prime factor of A077586(n).

Original entry on oeis.org

7, 127, 2147483647, 170141183460469231731687303715884105727, 47, 338193759479, 231733529, 62914441, 2351, 1399, 295257526626031, 18287, 106937, 863, 4703, 138863, 22590223644617

Offset: 1

Richard N. Smith, Jul 13 2019

A263686 is a subsequence.
Agrees with A263686 in the first four terms, but then the two sequences differ for the first time at n = 5, because prime(5) = 11 is not in A000043.
a(18) = A263686(9) is greater than 1.56*10^17*(2^61-1), see link.
a(n) = A077586(n) iff A077586(n) is prime, A077586(n) is prime for 1 <= n <= 4, but composite for 5 <= n <= 17. The status of A077586(18) = 2^(2^61-1)-1 is unknown. It is conjectured that A077586(n) is composite for all n >= 5.
a(20) = 456959, a(21) = 18384329, a(22) = 198839, a(23) = 2349023, a(24) = A263686(10) is greater than 1.25*10^16*(2^89-1).
Conjecture: All terms are in A122094 (all terms in A263686 are in A122094).
For examples related to that conjecture, see A322568. - Jeppe Stig Nielsen, Aug 29 2019
a(30) = 46559, a(32) = 23671, a(36) = 7151489, a(39) = 4698047, a(41) = 719, a(43) = 1440847, a(45) = 179689, a(47) = 11759383, a(48) = 23602441, a(50) = 9024439, a(51) = 28875361, a(52) = 6301423, a(54) = 2493983, a(56) = 33518137, a(59) = 6727783, a(66) = 95111, a(72) = 1439, a(73) = 99833, a(78) = 38119, a(81) = 26849, a(83) = 8258911, a(86) = 16173559, a(89) = 625343, a(93) = 9743. - Chai Wah Wu, Oct 16 2019

Double Mersennes Prime Search, History.
Eric Weisstein's World of Mathematics, Double Mersenne Number.

Cf. A077586, A263686, A001348, A049479.

A309130(n)=A020639(2^(2^prime(n)-1)-1) \\ For efficiency, use addprimes([large terms of this sequence]). - M. F. Hasler, Mar 01 2025

a(n) = A020639(A077586(n)).

a(n) = A049479(A001348(n)). - M. F. Hasler, Mar 01 2025

cp's OEIS Frontend

A103901 Mersenne primes p such that M(p) = 2^p - 1 is also a (Mersenne) prime.

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A127208 Union of all n-step Lucas sequences, that is, all sequences s(1-n) = s(2-n) = ... = s(-1) = -1, s(0) = n and for k > 0, s(k) = s(k-1) + ... + s(k-n).

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A137410 a(n) = (5^n - 3)/2.

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A140797 Numbers of the form (2^p^N-1)/(2^p^(N-1)-1), where N>0, p is prime.

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A152099 a(n) = (2^prime(n) - 1)*(2^prime(n) + 1) = 2^(2*prime(n)) - 1.

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A230809 Primes p of the form 60*n + 59 such that 2*p + 1 is also prime.

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Magma

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A263686 Smallest prime factor of double Mersenne numbers.

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A278741 Odd numbers k such that tau(k-1) is a prime.

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A152099 a(n) = (2^prime(n) - 1)(2^prime(n) + 1) = 2^(2prime(n)) - 1.

A230809 Primes p of the form 60n + 59 such that 2p + 1 is also prime.