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 1-4 of 4 results.

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

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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

A174057 Semi-sums (means) of a Fermat prime and a Mersenne prime.

Original entry on oeis.org

3, 4, 5, 6, 10, 12, 17, 18, 24, 65, 66, 72, 130, 132, 144, 192, 4097, 4098, 4104, 4224, 32770, 32772, 32784, 32832, 36864, 65537, 65538, 65544, 65664, 98304, 262145, 262146, 262152, 262272, 294912, 1073741825, 1073741826, 1073741832
Offset: 1

Views

Author

Jonathan Vos Post, Mar 06 2010

Keywords

Comments

The subsequence of prime semi-sums (means) of a Fermat prime and a Mersenne prime begins: 3, 5, 17, 65537 = (3 + 131071)/2. R. J. Mathar, on the remaining primes in the half sum, searched through all sums that can be created from the existing values of the two OEIS sequences, and that the next Fermat prime is known to be > 2^(2^32) + 1. So it is safe to say that the next prime > 65537 in the half sum (if it exists) is larger than 85070591730234615865843651857942085632, because adding that huge next Fermat prime would lead to even larger numbers. Of course one could easily boost that estimate by using the b-file of A000668.

Examples

			a(1) = 3 = half of first Mersenne prime + first Fermat prime = (3+3)/2.
a(2) = 4 = half of first Mersenne prime + 2nd Fermat prime = (3+5)/2.
a(3) = 5 = half of 2nd Mersenne prime + first Fermat prime = (7+3)/2.
a(4) = 6 = half of 2nd Mersenne prime + 2nd Fermat prime = (7+5)/2.
a(5) = 10 = half of 2nd Mersenne prime + 3rd Fermat prime = (3+17)/2.

Crossrefs

Cf. A000668, A171251-A171255, A155877 Sums of three Fermat numbers, A166484 Prime sums of three Fermat numbers, A174055, A174056.

Formula

{(A019434(i) + A000668(j))/2}. {(((2^p)-1) + (2^(2^k)+1))/2 = 2^(p-1) + 2^((2^k)-1) for p in A000043 and k in {0,1,2,3,4}}.

A269666 Prime sums of five Mersenne primes.

Original entry on oeis.org

19, 23, 31, 43, 47, 59, 71, 79, 83, 103, 107, 127, 131, 139, 151, 167, 179, 199, 223, 227, 251, 263, 271, 347, 419, 443, 8219, 8231, 8243, 8263, 8287, 8291, 8363, 8387, 8699, 16427, 16447, 16451, 16519, 16547, 16763, 24611, 32771, 131111, 131143, 131171
Offset: 1

Views

Author

Robert Israel, Mar 02 2016

Keywords

Comments

Primes of the form A000668(i_1) + ... + A000668(i_5), i_1 <= i_2 <= ... <= i_5.
There are 368 terms up to 10^1000, 13 more up to 10^1332, none between 10^1332 and 10^2916, and 9 between 10^2916 and 10^3000. Conjecture: the sequence is finite.

Examples

			a(1) = 3 + 3 + 3 + 3 + 7 = 19.
a(2) = 3 + 3 + 3 + 7 + 7 = 23.
a(3) = 3 + 7 + 7 + 7 + 7 = 31.

Links

Crossrefs

Cf. A000668, A174056.

Programs

  • Maple
    N:= 10^10: # 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(t -> (t <= N and isprime(t)), convert(
{seq(seq(seq(seq(seq(S[i]+S[j]+S[k]+S[l]+S[m],
  m=1..l),l=1..k),k=1..j),j=1..i),i=1..n-1)},list)));
  • Mathematica
    s = {3, 7, 31, 127, 8191, 131071, 524287} (* A000668 *); Take[Union@ Flatten@ Table[p = s[[a]] + s[[b]] + s[[c]] + s[[d]] + s[[e]]; If[ PrimeQ@ p, p, Sequence @@ {}], {e, 7}, {d, e}, {c, d}, {b, c}, {a, b}], 50] (* Robert G. Wilson v, Mar 02 2016 *)

A253682 Prime sums of three distinct Mersenne primes (A000668).

Original entry on oeis.org

41, 137, 2147483777, 162259895799233006081715459850241
Offset: 1

Views

Author

G. L. Honaker, Jr., Jan 08 2015

Keywords

Comments

a(1) has the exponents of the three distinct Mersenne primes of 1, 2 and 3; a(2) the exponents are 1, 2 and 4, a(3) the exponents are 1, 4 and 8 and a(4) the exponents are 1, 10 and 11. - Robert G. Wilson v, Jan 08 2015
a(5) > 10^30000, if it exists. - Jinyuan Wang, Jul 26 2021

Examples

			a(3) = 2147483777 because 2147483777 = (2^2-1) + (2^7-1) + (2^31-1).

Links

Crossrefs

Cf. A000040, A000043, A000668, A174056.

Programs

  • Mathematica
    exp={* the first 34 terms in A000043 *}; Do[ s = 2^exp[[p]] + 2^exp[[q]] + 2^exp[[r]] - 3; If[ PrimeQ@ s, Print[{p, q, r, s}]], {r, 3, 34}, {q, 2, r - 1}, {p, q - 1}] (* Robert G. Wilson v, Jan 08 2015 *)
Showing 1-4 of 4 results.

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