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

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

A166484 Prime sums of three Fermat numbers: primes of form 2^2^x + 2^2^y + 5.

Original entry on oeis.org

11, 13, 23, 37, 263, 277, 65543, 65557, 4295032837
Offset: 1

Views

Author

Jonathan Vos Post, Oct 14 2009, Oct 22 2009

Keywords

Comments

One can have a prime sum of two Fermat Primes, starting with 2 + 3 = 5.
Hence this current sequence is a proper subset of prime sums of a Fermat prime number of Fermat numbers, which in turn is a proper subset of prime sums of a Fermat number of Fermat numbers.
According to the Maple 9 primality test, the next term is larger than 10^300 if it exists. - R. J. Mathar, Oct 16 2009
At least one of the three Fermat numbers must be 3 because all Fermat numbers greater than 3 are equal to 2 (mod 3). Hence, the sum of three Fermat numbers greater than 3 is always a multiple of 3.
The next term, if it exists, has at least 1262612 digits. - Arkadiusz Wesolowski, Mar 06 2011

Examples

			a(1) = A000215(0) + A000215(0) + A000215(1) = 3 + 3 + 5 = 11, which is prime.
a(2) = A000215(0) + A000215(1) + A000215(1) = 3 + 5 + 5 = 13, which is prime.
a(3) = A000215(0) + A000215(0) + A000215(2) = 3 + 3 + 17 = 23, which is prime.
a(4) = A000215(0) + A000215(2) + A000215(2) = 3 + 17 + 17 = 37, which is prime.

Links

Crossrefs

Cf. A000040, A155877, A019434.

Programs

Formula

A155877 INTERSECTION A000040.
{p = (2^(2^a) + 1) + (2^(2^b) + 1) + (2^(2^c) + 1) for nonnegative integers a, b, c, such that p is prime}.

Extensions

a(9) from R. J. Mathar, Oct 16 2009
Definition improved by Arkadiusz Wesolowski, Feb 16 2011

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}}.
Showing 1-4 of 4 results.

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