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.

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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

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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

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}}.

A279389 3 times Mersenne primes A000668.

Original entry on oeis.org

9, 21, 93, 381, 24573, 393213, 1572861, 6442450941, 6917529027641081853, 1856910058928070412348686333, 486777830487640090174734030864381, 510423550381407695195061911147652317181
Offset: 1

Views

Author

Omar E. Pol, Dec 20 2016

Keywords

Comments

Also sum of n-th Mersenne prime and the radical of n-th even perfect number.
The binary representation of a(n) has only two zeros, starting with "10" and ending with "01". The sequence begins: 1001, 10101, 1011101, 101111101, 101111111111101,...

Crossrefs

Cf. A000010, A000203, A000396, A000668, A051027, A147757.
Subsequence of A001748, and of A147758, and of A174055, and possibly of other sequences, see below:
Cf. A060482, A068156, A072087, A136252, A144482, A144856, A249780, A269633.

Formula

a(n) = 3*A000668(n) = A000668(n) + A139257(n).
a(n) = phi(M(n)) + sigma(sigma(M(n))) = A000010(A000668(n)) + A000203(A000203(A000668(n))) = A000010(A000668(n)) + A051027(A000668(n)).
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