A000668 -id:A000668 - cp's OEIS frontend

A171254 Primes which are the average of any two (not necessarily distinct) Mersenne primes (A000668).

3, 5, 7, 17, 19, 31, 67, 79, 127, 4099, 4111, 4159, 8191, 65537, 65539, 65551, 65599, 131071, 262147, 266239, 524287, 1073741827, 1073741839, 1073807359, 2147483647, 2305843009213693951, 309485009821345068724785151

Offset: 1

M. F. Hasler, Mar 06 2010

nonn

The subsequence of primes in A171252, containing A000668 as a subsequence.

			a(n) = A171252(n) for n=1,...,6, since all of these terms are prime. The term A171252(7) = 65 is the first element of A171252 to be composite, and therefore not included in the present sequence A171252.

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

Cf. A171252, A171255 (excludes elements of A000668).

Select[Mean/@Tuples[2^MersennePrimeExponent[Range[10]]-1,2],PrimeQ]//Union (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 31 2019 *)

select(isprime, concat(vector(#A00668,i,vector(i,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. */

A171254 = A171252 intersect A000040.

A193864 Decimal expansion of 2^43112609 - 1, the 47th Mersenne prime A000668(47).

Original entry on oeis.org

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

Offset: 12978189

Kausthub Gudipati, Aug 07 2011

This number is very large, containing 12978189 digits. Edson Smith discovered this prime number on August 26, 2008 within the GIMPS framework. Landon Curt Noll calculated the decimal expansion of this prime number. It is a Mersenne prime.

This 47th Mersenne prime (cf. A000043) is remarkable since it was found before two smaller Mersenne primes, one in the following month (September 2008) and another one in April 2009. It remained the largest known prime until January 2013, when the 48th known Mersenne prime was found. - M. F. Hasler, May 22 2014

			316470269330255923143453723949(...12978129 digits omitted...)887478265780022181166697152511

GIMPS, List of Known Mersenne Prime Numbers
OEIS Wiki, Mersenne primes (with a list of similar sequences)

Cf. A000043 (main entry), A000668, A028335 (lengths).

IntegerDigits[2^43112609 - 1][[1 ;; 105]] (* T. D. Noe, Aug 09 2011 *)

A193864_list(Nmax)={default(realprecision,Nmax+5);digits(10^frac(43112609*log(2)/log(10))\.1^Nmax)} \\ Use digits(x)=eval(Vec(Str(x))) in older PARI versions. - M. F. Hasler, Mar 04 2012, updated May 22 2014

write("a193864.txt", 2^43112609 - 1) \\ Georg Fischer, Mar 19 2019

2^43112609 - 1.

Name changed by Georg Fischer, Mar 19 2019

A242993 Least k such that R = (2^k*Q-Q-1)/(Q+1-2^k) is prime, where Q = A000668(n) is the n-th Mersenne prime, or 0 if no such k exists.

Original entry on oeis.org

0, 2, 4, 4, 11, 13, 16, 16, 57, 78, 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

Kravitz has shown that 2^(k-1)*Q*R is a primitive weird number (cf. A002975) when Q > 2^k =: M+1 and R = (M*Q-1)/(Q-M) = M + (M^2-1)/(Q-M) both are prime. R cannot be an integer unless Q < M(M+1) which yields k > p/2 for Mersenne primes Q = 2^p-1. [Edited by M. F. Hasler, Nov 11 2018]

Sequence A242025 lists all primes R obtained in that way. Sequence A242998 gives the number of (k,R) for each Q in A000668. Sequence A242998 lists the primes p which give rise to a solution, with multiplicity, and A243003 lists the corresponding values of k. See the "main entry" A242025 for more information. - M. F. Hasler, Nov 11 2018

			For n = 2, Q = A000668(2) = 7, k = 2 yields the prime R = (2^k*Q-Q-1)/(Q+1-2^k) = 20/4 = 5 and the (smallest possible) weird number 2^(k-1)*Q*R = 2*7*5 = 70.
For n = 9, Q = A000668(9) = 2^61-1, k = 57 yields the prime R = 2^57-1 + (2^57-2)/(2^4-1) and the 53-digit primitive weird number 2^56*Q*R = 25541592347764814106588251084767772206406532903993344.
For n = 10, Q = A000668(10) = 2^89-1, k = 78 yields the prime R = 2^78-1 + (2^78-2)/(2^11-1) and the 74-digit primitive weird number 2^77*Q*R = 28283363272427014026275183563912621451964887156507346985599492888375328768.

S. Kravitz, A search for large weird numbers. J. Recreational Math. 9(1976), 82-85 (1977). Zbl 0365.10003

Cf. A242025 (primes R obtained in that way), A242998 (number of such k values for given n), A242999 (p=A000043(n) listed A242998(n) times), A243003 (all values of k).

Cf. A258882 (weird numbers of the form 2^k*p*q).

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 <= 25, 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 = k; Break[]]];
  AppendTo[lst, kc]];
lst (* Robert Price, Sep 05 2019 *)

a(n)={p=A000043[n]; for(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)) && return(k))}

Definition corrected by Jens Kruse Andersen, Aug 18 2014

a(28)-a(37) from Robert Price, Sep 05 2019

A330817 Numbers of the form 2^(2p+1)M_p^2, where M_p is a Mersenne prime, A000668, with Mersenne exponent p, A000043.

Original entry on oeis.org

288, 6272, 1968128, 528515072, 9005000365703168, 590286803193810649088, 151115150991626099228672, 42535295825503226685013029169053827072, 56539106072908298497625662716064949049646203797489239767322203013731319808

Offset: 1

Walter Kehowski, Jan 01 2020

nonn

Also numbers with power-spectral basis {M_p^2*(M_p+2)^2,(M_p^2-1)^2}.

The first factor of a(n) is A330818. The first element of the spectral basis of a(n) is A330819, and the second element is A330820.

			Since p=2 and M_2=3, then a(1)=2^(2*2+1)*3^3=288, and 288 has spectral basis {15^2, 2^6}, consisting of powers.

Walter Kehowski, Table of n, a(n) for n = 1..12

Cf. A000043, A000668, A132794, A133049, A330818, A330819, A330820.

A330817:=[]:
for www to 1 do
for i from 1 to 31 do
  #ithprime(31)=127
  p:=ithprime(i);
  q:=2^p-1;
  if isprime(q) then x:=2^(2*p+1)*q^2; A330817:=[op(A330817),x]; fi;
od;
od;
A330817;

2^(2 * (p = MersennePrimeExponent[Range[9]]) + 1) * (2^p - 1)^2 (* Amiram Eldar, Jan 03 2020 *)

A335874 Primes of the form q*2^h - 1, where q is a Mersenne prime (A000668).

Original entry on oeis.org

2, 5, 11, 13, 23, 47, 61, 191, 223, 383, 991, 3583, 3967, 6143, 16381, 63487, 253951, 262111, 786431, 917503, 1048447, 1048573, 4194271, 14680063, 16777183, 67108351, 260046847, 3758096383, 4261412863, 51539607551, 68718952447, 266287972351, 824633720831, 1065151889407, 1099503239167

Offset: 1

Antti Karttunen, Jun 28 2020

nonn

Primes p such that A331410(1+p) = 1. After 2, primes p for which A331410(p) = 2.

Cf. A000040, A000265, A000668, A331410.
After 2, primes in A335882.
Cf. also A334092.

A000265(n) = (n>>valuation(n,2));
isA000668(n) = (isprime(n)&&!bitand(n,1+n));
isA335874(n) = (isprime(n)&&isA000668(A000265(1+n)));

a(28)-a(35) from David A. Corneth, Jun 28 2020

A089065 Decimal expansion of 2^13466917 - 1, the 39th Mersenne prime A000668(39).

Original entry on oeis.org

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

Offset: 4053946

Cino Hilliard, Dec 20 2003

Since 2^13466917 ends in ...6259072, this sequence ends in ...6,2,5,9,0,7,1

Muniru A Asiru, Table of n, a(n) for n = 4053946..4055946
Peter Alfeld, Largest Known Prime.
OEIS Wiki, Mersenne primes (with a list of similar sequences)

Cf. A000043, A000668.

evalf[120](2^(13466917)-1); # Muniru A Asiru, Mar 20 2019

IntegerDigits[2^13466917 - 1][[;;100]] (* Paolo Xausa, Apr 05 2024 *)

digitsm39(n) = { default(realprecision,n); p10 = frac(m*log(2)/log(10)); b = 10^p10/10; for(j=1,n, v=b*10; d=floor(v); b=v-d; print1(d",") ) }

Edited by Georg Fischer, Mar 19 2019

A089578 Decimal expansion of 2^20996011 - 1, the 40th Mersenne prime A000668(40).

Original entry on oeis.org

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

Offset: 6320430

Cino Hilliard, Dec 29 2003

We can compute the digits of 2^p directly by noting that 2^p = 10^(p*log(2)/log(10)) = 10^(p*log_10(2)). This result is 10^(i+f) where i is the integer part and f the fractional part. Then 10^f will produce a decimal number i.d1d2d3d4... where i is an integer from 1 to 9 (zero cannot occur in i) and d1, d2 ... are the digits in the fractional part where 0 is allowed. So i is the first digit in 2^p, d1 the second, d2 the third etc. The expansion is self evident in the PARI program. This routine allows the direct computation of the digits of any base to a power: k^p = 10^(p*log_10(k)).

The 40th Mersenne prime found by GIMPS / Michael Shafer in 2003 is 1259768954503301...4065762855682047 = 2^20996011 - 1. The second PARI program below computes all digits. - Georg Fischer, Mar 18 2019

Muniru A Asiru, Table of n, a(n) for n = 6320430..6321430
GIMPS, List of Known Mersenne Prime Numbers
OEIS Wiki, Mersenne Primes (with a list of similar sequences)

Cf. A000043 (main entry), A000668, A028335 (lengths).

RealDigits[10^N[20996011Log[10, 2] - 6320430, 105]][[1]] (* Georg Fischer, Mar 19 2019 after Jakob Vecht in A117853 *)

\\ digits of the 40th Mersenne prime: 2^20996011 - 1
p = 20996011; digitsm40(n, p) = { default(realprecision,n); p10 = frac(p*log(2)/log(10)); v = 10^p10; for(j=1,n, d=floor(v); v=frac(v)*10; print1(d",") ) }
digitsm40(105,p)

write("a089578.txt", 2^20996011 - 1) \\ Georg Fischer, Mar 18 2019

Edited by Georg Fischer, Mar 19 2019

A138862 First two (i.e., the two most significant) digits of n-th Mersenne prime A000668(n).

Original entry on oeis.org

3, 7, 31, 12, 81, 13, 52, 21, 23, 61, 16, 17, 68, 53, 10, 14, 44, 25, 19, 28, 47, 34, 28, 43, 44, 40, 85, 53, 52, 51, 74, 17, 12, 41, 81, 62, 12, 43, 92, 12, 29, 12, 31, 12, 20, 16, 31

Offset: 1

Omar E. Pol, Apr 02 2008

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

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

Join[{3,7},FromDigits[Take[IntegerDigits[#],2]]&/@ (2^MersennePrimeExponent[ Range[ 3,47]]-1)] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 01 2020 *)

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

Definition clarified by Harvey P. Dale, Aug 01 2020

A138865 Last 3 digits of n-th Mersenne prime A000668(n).

Original entry on oeis.org

3, 7, 31, 127, 191, 71, 287, 647, 951, 111, 127, 727, 151, 127, 87, 7, 351, 71, 991, 607, 111, 551, 191, 471, 751, 511, 671, 207, 7, 311, 447, 887, 591, 527, 711, 151, 271, 791, 71, 47, 407, 247, 871, 871, 927, 751, 511

Offset: 1

Omar E. Pol, Apr 02 2008

L. C. Noll, Mersenne Prime Digits and Names.
Omar E. Pol, Los primos de Mersenne

Cf. A000668, A080173, A104511, A138862, A138863, A138864, A138866.

Array[Mod[2^MersennePrimeExponent[#] - 1, 1000] &, 45] (* Michael De Vlieger, Aug 05 2018 *)

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

A145044 Primes p (A000043) such that 2^p-1 is prime (A000668) and congruent to 271 mod 6!.

Original entry on oeis.org

13, 61, 2281, 3217, 23209, 44497, 132049, 13466917, 30402457, 42643801

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.

Select[MersennePrimeExponent[Range[47]], PowerMod[2, #, 6!] == 272 &] (* Amiram Eldar, Mar 22 2020 *)

a(10) from Amiram Eldar, Mar 22 2020

cp's OEIS Frontend

A171254 Primes which are the average of any two (not necessarily distinct) Mersenne primes (A000668).

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A193864 Decimal expansion of 2^43112609 - 1, the 47th Mersenne prime A000668(47).

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A242993 Least k such that R = (2^k*Q-Q-1)/(Q+1-2^k) is prime, where Q = A000668(n) is the n-th Mersenne prime, or 0 if no such k exists.

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A330817 Numbers of the form 2^(2*p+1)*M_p^2, where M_p is a Mersenne prime, A000668, with Mersenne exponent p, A000043.

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A335874 Primes of the form q*2^h - 1, where q is a Mersenne prime (A000668).

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A089065 Decimal expansion of 2^13466917 - 1, the 39th Mersenne prime A000668(39).

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A089578 Decimal expansion of 2^20996011 - 1, the 40th Mersenne prime A000668(40).

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A330817 Numbers of the form 2^(2p+1)M_p^2, where M_p is a Mersenne prime, A000668, with Mersenne exponent p, A000043.