A000668 -id:A000668 - cp's OEIS frontend

A145041 Primes p (A000043) such that 2^p-1 is prime (A000668) and congruent to 31 mod 6!.

5, 17, 89, 521, 4253, 9689, 9941, 11213, 19937, 21701, 859433, 1398269, 2976221, 3021377, 6972593, 32582657, 43112609, 57885161

Offset: 1

Artur Jasinski, Sep 30 2008

Mersenne numbers (with the exception of the first two) are congruent to 31, 127, 271, 607 mod 6!. This sequence is a subsequence of A000043.

Cf. A000043, A000668, A124477, A139484, A145038, A112633, A145041, A145042, A145044, A145045, A145046.

p = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 43112609}; a = {}; Do[If[Mod[2^p[[n]] - 1, 6! ] == 31, AppendTo[a, p[[n]]]], {n, 1, Length[p]}]; a
Select[MersennePrimeExponent[Range[48]], PowerMod[2, #, 720] == 32 &] (* Amiram Eldar, Oct 19 2024 *)

a(18) from Amiram Eldar, Oct 19 2024

A145042 Primes p (A000043) such that 2^p-1 is prime (A000668) and congruent to 127 mod 6!.

Original entry on oeis.org

7, 19, 31, 127, 607, 1279, 2203, 4423, 110503, 216091, 1257787, 20996011, 24036583

Offset: 1

Artur Jasinski, Sep 30 2008

nonn

Mersenne numbers (with the exception of the first two) are congruent to 31, 127, 271, 607 mod 6!. This sequence is a subsequence of A000043.

Cf. A000043, A000668, A124477, A139484, A145038, A112633, A145041, A145042, A145044, A145045, A145046.

p = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 43112609}; a = {}; Do[If[Mod[2^p[[n]] - 1, 6! ] == 127, AppendTo[a, p[[n]]]], {n, 1, Length[p]}]; a

A242998 Number of integers k such that R = (2^k*Q - Q - 1)/(Q + 1 - 2^k) is a prime number, when Q = A000668(n) is the n-th Mersenne prime.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 2, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

M. F. Hasler, Aug 17 2014

Related to the search of large primitive weird numbers: Kravitz has shown that 2^(k-1)*Q*R is a primitive weird number (cf. A002975) when Q > 2^k and R = (2^k*Q - Q - 1)/(Q + 1 - 2^k) both are prime. Here we count such primes for the special case where Q = 2^p - 1 is a Mersenne prime, p=A000043(n). For such Q one has R = 2^k - 1 + (2^k - 2)/(2^(p-k) - 1).
See A242025 for the resulting primes R, which however are there not listed in order of the p's.
This sequence gives the row lengths for the table A243003 whose rows hold the k-values leading to prime R, for a given Mersenne prime.

			For given p=A000043(n), the following k's yield a prime R:
p : k's (and resulting primes R, Q=2^p-1 and/or weird W=2^(k-1)*Q*R)
2 : -
3 : 2 (R=5, Q=7, W=70)
5 : 4 (R=29, Q=31, W=7192)
7 : 4 (R=17, Q=127, W=17272), 5 (R=41, Q=127, W=83312)
13 : 11 (R=2729, Q=8191, W=22889716736)
17 : 13 (R=8737, Q=131071, W=4690605371392)
19 : 16 (R=74897, W=1286718208049152), 17 (R=174761, W=6004730783793152)
31 : 16 (R=65537, W=2^15*(2^31-1)*R), 29 (R=715827881, W=2^28*(2^31-1)*R)
61 : 57 (R=153722867280912929, W=2^56*(2^61-1)*R)
89 : 78 (R=302379100949042568368129, W=2^77*(2^89-1)*R)
107 through 86243 : none.
107 through 3021377: none. _Robert Price_, Sep 05 2019
The present sequence lists the number of k's in each line.

S. Kravitz, A search for large weird numbers. J. Recreational Math. 9(1976), 82-85 (1977). Zbl 0365.10003
E. Weisstein, Weird numbers, on MathWorld - a Wolfram web ressource.

Cf. A258882 (PWN of the form 2^k*p*q), A000043 (Mersenne prime exponents), A000668.
Cf. A242025 (the primes R).
Row lengths of A242999 (values of p) and A243003 (values of k), cf. A242993 for the smallest possible k.
See also A320875 for more general solutions to R = (MQ-1)/(Q-M) = prime.

A000043 = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607,
   1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937,
   21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433,
   1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011,
   24036583, 25964951, 30402457, 32582657, 37156667, 42643801,
   43112609};
lst = {};
For[i = 1, i <= 28, i++,
  p = A000043[[i]];
  kc = 0;
  For[k = 1, k < p, k++,
   r = 2^k - 1 + (2^k - 2)/(2^(p - k) - 1);
   If[! IntegerQ[r], Continue[]];
   If[PrimeQ[r], kc++]];
  AppendTo[lst, kc]];
lst (* Robert Price, Sep 05 2019 *)

A242998(n,p=A000043[n])={sum(k=p\2+1, p-1, Mod(2, 2^(p-k)-1)^k==2 && ispseudoprime(2^k-1+(2^k-2)/(2^(p-k)-1)))}

Typo in definition corrected by Jens Kruse Andersen, Aug 27 2014

a(29)-a(37) from Robert Price, Sep 05 2019

A253851 Mersenne primes (A000668) of the form 2^sigma(n) - 1 for some n.

Original entry on oeis.org

7, 127, 8191, 2147483647, 170141183460469231731687303715884105727

Offset: 1

Jaroslav Krizek, Jan 16 2015

nonn

Numbers n such that 2^sigma(n) - 1 is a Mersenne primes are given in A253849.
Sequence of corresponding values of sigma(n) are given in A253850 and each term of this sequence must be a prime from the sequence of Mersenne exponents (A000043).
If a(6) exists, it must be bigger than A000668(43) = 2^30402457-1.

			Mersenne prime 2147483647 is in the sequence because there are two numbers n (16 and 25) with 2^sigma(n) - 1 = 2^31 - 1 = 2147483647.

Cf. A000043, A000203, A000668, A023195, A253849, A253850.

Set(Sort([(2^SumOfDivisors(n))-1: n in[1..10000] | IsPrime((2^SumOfDivisors(n))-1)]));

A080172 Final digit of n-th Mersenne prime A000668(n).

Original entry on oeis.org

3, 7, 1, 7, 1, 1, 7, 7, 1, 1, 7, 7, 1, 7, 7, 7, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 7, 7, 1, 7, 7, 1, 7, 1, 1, 1, 1, 1, 7, 7, 7, 1, 1, 7, 1, 1, 1

Offset: 1

Mark Dowdeswell, Feb 04 2003

Distribution of final digit for Mersenne primes appears (naturally) to be different from distribution for regular primes. Unconfirmed 49th, 50th and 51st digits in sequence are 1, 1, 1 (awaiting confirmation of 49th, 50th and 51st Mersenne primes).

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 16.

Great Internet Mersenne Prime Search (GIMPS), Distributed Computing Projects
Mersenne Primes, History, Theorems and Lists

Cf. A000668, A007652, A080173.

(Maple code from N. J. A. Sloane) # let s1 := list of terms in A000043
f:=n->if n mod 4 = 0 then 4 else n mod 4; fi; map(x->2^f(x)-1,s1);

Mod[2^MersennePrimeExponent[Range[48]]-1,10] (* Harvey P. Dale, Aug 09 2023; updated by Mark Dowdeswell, Sep 16 2024 *)

Updated by N. J. A. Sloane, Apr 01 2008
a(40)-a(47) from Ivan Panchenko, Apr 11 2018
a(48) from Mark Dowdeswell, Sep 16 2024

A135613 Initial digit of Mersenne primes A000668.

Original entry on oeis.org

3, 7, 3, 1, 8, 1, 5, 2, 2, 6, 1, 1, 6, 5, 1, 1, 4, 2, 1, 2, 4, 3, 2, 4, 4, 4, 8, 5, 5, 5, 7, 1, 1, 4, 8, 6, 1, 4, 9, 1, 2, 1, 3, 1, 2, 1, 3, 5

Offset: 1

Omar E. Pol, Mar 01 2008

			a(4) = 1 because the 4th Mersenne prime A000668(4) is 127 and the initial digit of 127 is 1.

Landon Curt Noll, Mersenne Prime Digits.

Cf. A000030, A000668, A077648, A080172.

lst = {* the list of terms in A000043 *}; f[n_] := Block[{pn = 2^n - 1}, Quotient[pn, 10^Floor[ Log[10, pn]] ]]; f@# & /@ lst (* Robert G. Wilson v, Apr 01 2008 *)
IntegerDigits[#][[1]]&/@(2^#-1&/@MersennePrimeExponent[Range[47]]) (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 04 2019 *)

a(n) = A000030(A000668(n)). - Omar E. Pol, Jul 04 2019

More terms from Robert G. Wilson v, Apr 01 2008
a(40)-a(44) from David Radcliffe, Jan 21 2016
a(45)-a(47) from Ivan Panchenko, Apr 11 2018
a(48) from Amiram Eldar, Oct 16 2024

A138816 Concatenation of initial digit of n-th Mersenne prime A000668(n), initial digit of n-th even superperfect number A061652(n) and initial digit of n-th perfect number A000396(n).

Original entry on oeis.org

326, 742, 314, 168, 843, 168, 521, 212, 212, 631, 181, 181, 632, 521, 155

Offset: 1

Omar E. Pol, Apr 01 2008

Also, concatenation of initial digit of n-th Mersenne prime A000668(n), initial digit of n-th superperfect number A019279(n) and initial digit of n-th perfect number A000396(n), if there are no odd superperfect numbers.

Also, concatenation of A135613(n), A138124(n) and A135617(n).

L. C. Noll, Mersenne Prime Digits and Names.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000396, A000668, A019279, A061652, A135613, A135617, A138124, A138817.

a(13)-a(15) from Robert Price, Jun 16 2019

A138864 First 3 digits of n-th Mersenne prime A000668(n).

Original entry on oeis.org

3, 7, 31, 127, 819, 131, 524, 214, 230, 618, 162, 170, 686, 531, 104, 147, 446, 259, 190, 285, 478, 346, 281, 431, 448, 402, 854, 536, 521, 512, 746, 174, 129, 412, 814, 623, 127, 437, 924, 125, 299, 122, 315, 124, 202, 169, 316

Offset: 1

Omar E. Pol, Apr 02 2008

L. C. Noll, Mersenne Prime Digits and Names.

Cf. A000668, A080173, A104511, A138863, A138864, A138865, A138866.

a(40)-a(47) from Ivan Panchenko, Aug 03 2018

A171255 Primes which are the average of two distinct Mersenne primes (A000668).

Original entry on oeis.org

5, 17, 19, 67, 79, 4099, 4111, 4159, 65537, 65539, 65551, 65599, 262147, 266239, 1073741827, 1073741839, 1073807359, 309485009821345068724785151

Offset: 1

M. F. Hasler, Mar 06 2010

nonn

The subsequence of primes in A171253, which equals A171254 minus its subsequence A000668.

			a(n) = A171253(n) for n=1,2,3, since all of these terms are prime. The term A171253(4) = 65 is the first element of A171252 to be composite, and therefore not included in the present sequence A171255.

Jonathan Vos Post, Sums of three Mersenne primes, and prime sums of three Mersenne primes, SeqFan list, Mar 5, 2010.

Cf. A171252, A171254 (includes elements of A000668).

select(isprime, concat(vector(#A00668,i,vector(i-1,j,A00668[i]+A00668[j])))/2) /* having defined A00668 as vector with initial terms of A000668. In PARI version 2.4.2, the syntax select( concat(...), x->isprime(x)) must be used. */

A171255 = A171253 intersect A000040 = A171254 \ A000668.

A117853 Decimal expansion of 2^30402457-1, the 43th Mersenne prime A000668(43).

Original entry on oeis.org

3, 1, 5, 4, 1, 6, 4, 7, 5, 6, 1, 8, 8, 4, 6, 0, 8, 0, 9, 3, 6, 3, 0, 3, 0, 2, 8, 6, 6, 4, 5, 4, 5, 1, 7, 0, 1, 2, 6, 5, 1, 9, 6, 5, 6, 2, 6, 2, 3, 2, 3, 8, 7, 0, 3, 1, 6, 3, 2, 3, 7, 1, 0, 7, 9, 5, 1, 3, 5, 3, 8, 7, 4, 4, 9, 0, 0, 6, 9, 3, 4, 6, 2, 0, 9, 4, 3, 8, 6, 2, 9, 4, 7, 5, 1, 7, 0, 2, 9, 6, 6, 3, 6, 2, 3

Offset: 9152052

Jacob Vecht, May 02 2006

The decimal expansion of 2^30402457-1. This is the 43rd known Mersenne prime, found at Dec 15, 2005 by the GIMPS project / Curtis Cooper and Steven Boone. The number has 9152052 digits; as a string, it occupies 9.24 MB.

			A000668(43) = 3154164756188460809363030286645451701265196562623...7411652943871.

Landon Curt Knoll, 2^30402457-1 is prime.
George Woltman, GIMPS GIMPS Home Page.

Cf. A000043 (Mersenne exponents), A000668.

RealDigits[10^N[30402457Log[10, 2] - 9152051, 111]][[1]]

Edited by Georg Fischer, Jul 19 2021

cp's OEIS Frontend

A145041 Primes p (A000043) such that 2^p-1 is prime (A000668) and congruent to 31 mod 6!.

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A145042 Primes p (A000043) such that 2^p-1 is prime (A000668) and congruent to 127 mod 6!.

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A242998 Number of integers k such that R = (2^k*Q - Q - 1)/(Q + 1 - 2^k) is a prime number, when Q = A000668(n) is the n-th Mersenne prime.

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A253851 Mersenne primes (A000668) of the form 2^sigma(n) - 1 for some n.

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Magma

A080172 Final digit of n-th Mersenne prime A000668(n).

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Maple

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A135613 Initial digit of Mersenne primes A000668.

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A138816 Concatenation of initial digit of n-th Mersenne prime A000668(n), initial digit of n-th even superperfect number A061652(n) and initial digit of n-th perfect number A000396(n).

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A138864 First 3 digits of n-th Mersenne prime A000668(n).

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A171255 Primes which are the average of two distinct Mersenne primes (A000668).

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