cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Views

Author

M. F. Hasler, Mar 06 2010

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

  • PARI
    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. */

Formula

A171255 = A171253 intersect A000040 = A171254 \ A000668.

A171251 Sums of two (not necessarily distinct) Mersenne primes (A000668).

Original entry on oeis.org

6, 10, 14, 34, 38, 62, 130, 134, 158, 254, 8194, 8198, 8222, 8318, 16382, 131074, 131078, 131102, 131198, 139262, 262142, 524290, 524294, 524318, 524414, 532478, 655358, 1048574, 2147483650, 2147483654, 2147483678, 2147483774, 2147491838
Offset: 1

Views

Author

M. F. Hasler, Mar 06 2010

Keywords

Comments

All terms are even, since all terms of A000668 are odd. This motivates to introduce A171252 = (1/2) * A171251, see there for further information.

Links

Crossrefs

Cf. A000668, A171252.

Programs

  • Mathematica
    With[{mps=Select[2^Range[50]-1,PrimeQ]},Union[Total/@Tuples[mps,2]]] (* Harvey P. Dale, Aug 11 2012 *)
With[{mps=2^MersennePrimeExponent[Range[10]]-1},Union[Total/@Tuples[mps,2]]] (* Harvey P. Dale, Mar 20 2025 *)
  • PARI
    concat(vector(#A000668,i,vector(i,j,A000668[i]+A000668[j]))) /* having defined A000668 as vector with initial terms of A000668 */

Formula

A171251(n) = 2*A171252(n) = A000668(i) + A000668(j) where n = i*(i-1)/2+j.

A171253 Semi-sums (average) of any two distinct Mersenne primes (A000668).

Original entry on oeis.org

5, 17, 19, 65, 67, 79, 4097, 4099, 4111, 4159, 65537, 65539, 65551, 65599, 69631, 262145, 262147, 262159, 262207, 266239, 327679, 1073741825, 1073741827, 1073741839, 1073741887, 1073745919, 1073807359, 1074003967
Offset: 1

Views

Author

M. F. Hasler, Mar 06 2010

Keywords

Comments

Subsequence of A171252, excluding the terms of A000668 itself.

Examples

			a(1) = (A000668(2)+A000668(1))/2, a(2) = (A000668(3)+A000668(1))/2, a(3) = (A000668(3)+A000668(2))/2, a(4) = (A000668(4)+A000668(1))/2, ,...

Links

Crossrefs

Cf. A171252 (includes terms of A000668), A171255 (primes in this sequence).

Programs

Formula

a(n) = A171252(A014132(n)).
A171253 = A171252 \ A000688.

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

Original entry on oeis.org

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

Views

Author

M. F. Hasler, Mar 06 2010

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

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

Formula

A171254 = A171252 intersect A000040.

A174055 Sums of three Mersenne primes.

Original entry on oeis.org

9, 13, 17, 21, 37, 41, 45, 65, 69, 93, 133, 137, 141, 161, 165, 189, 257, 261, 285, 381, 8197, 8201, 8205, 8225, 8229, 8253, 8321, 8325, 8349, 8445, 16385, 16389, 16413, 16509, 24573, 131077, 131081, 131085, 131105, 131109, 131133, 131201, 131205, 131229, 131325, 139265, 139269, 139293, 139389, 147453, 262145, 262149
Offset: 1

Views

Author

Jonathan Vos Post, Mar 06 2010

Keywords

Examples

			a(1) = 3 + 3 + 3 = 9. a(2) = 3 + 3 + 7 = 13. a(3) = 3 + 7 + 7 = 17. a(4) = 7 + 7 + 7 = 21. a(5) = 3 + 3 + 31 = 37. a(6) = 3 + 7 + 31 = 41.

Links

Crossrefs

Cf. A000668, A155877, A166484, A171251, A171252, A171253, A171254, A171255, A174056, A174057.

Programs

  • Maple
    N:= 10^6: # to get all terms <= N
for n from 1 while numtheory:-mersenne([n]) < N do od:
S:= {seq(numtheory:-mersenne([i]),i=1..n-1)}:
sort(convert(select(`<=`,{seq(seq(seq(s+t+u,s=S),t=S),u=S)},N),list)); # Robert Israel, Mar 02 2016

Formula

A000668(i) + A000668(j) + A000668(k), with integers i,j,k not necessarily distinct. The subsequence of prime sums of three Mersenne primes is A174056.

Extensions

More terms from Max Alekseyev, Oct 15 2012
Edited by Robert Israel, Mar 02 2016

A174056 Prime sums of three Mersenne primes. Primes in A174055.

Original entry on oeis.org

13, 17, 37, 41, 137, 257, 2147483777, 162259895799233006081715459850241
Offset: 1

Views

Author

Jonathan Vos Post, Mar 06 2010

Keywords

Comments

Sums of five Mersenne primes can also be prime (though, obviously sums of an even number of Mersenne primes are even).
3 + 3 + 3 + 3 + 7 = 19
3 + 3 + 3 + 7 + 7 = 23
3 + 7 + 7 + 7 + 7 = 31
3 + 3 + 3 + 3 + 31 = 43
3 + 3 + 3 + 7 + 31 = 47
7 + 7 + 7 + 7 + 31 = 59
3 + 3 + 3 + 31 + 31 = 71
3 + 7 + 7+ 31 + 31 = 79
That sequence of sums of five Mersenne primes 19, 23, 31, 43, 47, 59, 71, 79, ... is A269666.
No other terms < 10^1000. Conjecture: these are all the terms. - Robert Israel, Mar 02 2016

Examples

			a(1) = 3 + 3 + 7 = 13. a(2) = 3 + 7 + 7 = 17. a(3) = 3 + 3 + 31 = 37. a(4) = 3 + 7 + 31 = 41. a(5) = 3 + 7 + 127 = 137. a(6) = 3 + 127 + 127 = 257.

Crossrefs

Cf. A000668, A171251, A171252, A171253, A171254, A171255, A174055, A174057, A269666.
Cf. A155877 (sums of three Fermat numbers).
Cf. A166484 (prime sums of three Fermat numbers).

Programs

  • Maple
    N:= 10^1000: # to get all terms <= N
for n from 1 while numtheory:-mersenne([n]) < N do od:
S:= {seq(numtheory:-mersenne([i]),i=1..n-1)}:
sort(select(isprime,convert(select(`<=`,{seq(seq(seq(s+t+u,s=S),t=S),u=S)},N),list))); # Robert Israel, Mar 02 2016
  • Mathematica
    Select[Total/@Tuples[Table[2^MersennePrimeExponent[n]-1,{n,20}],3],PrimeQ]//Union (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 22 2020 *)

Formula

A000668(i) + A000668(j) + A000668(k), with integers i,j,k not necessarily distinct. The supersequence of sums of three Mersenne primes is A174055.

Extensions

a(7)-a(8) from Donovan Johnson, Dec 22 2010
Showing 1-6 of 6 results.

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