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.

A001172 Smallest even number that is an unordered sum of two odd primes in exactly n ways.

Original entry on oeis.org

0, 6, 10, 22, 34, 48, 60, 78, 84, 90, 114, 144, 120, 168, 180, 234, 246, 288, 240, 210, 324, 300, 360, 474, 330, 528, 576, 390, 462, 480, 420, 570, 510, 672, 792, 756, 876, 714, 798, 690, 1038, 630, 1008, 930, 780, 960, 870, 924, 900, 1134, 1434, 840, 990, 1302
Offset: 0

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Author

N. J. A. Sloane, Eric Wolman, Dec 17 1969

Keywords

Comments

a(n) = A023036(n) for all n > 1.

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000954, A000974, A002375, A005843, A023036.
Divided by 2: A258713.

Programs

  • Mathematica
    nn = 55; a = Table[0, {nn}]; n = 6; While[Times @@ a == 0, c = 0; k = 3; While[k <= n/2, If[PrimeQ[k] && PrimeQ[n - k], c++]; k++]; If[c <= nn && a[[c]] == 0, a[[c]] = n]; n = n + 2]; Prepend[a, 0]

Extensions

a(0) corrected by Zak Seidov, Sep 30 2011

A000954 Conjecturally largest even integer which is an unordered sum of two primes in exactly n ways.

Original entry on oeis.org

2, 12, 68, 128, 152, 188, 332, 398, 368, 488, 632, 692, 626, 992, 878, 908, 1112, 998, 1412, 1202, 1448, 1718, 1532, 1604, 1682, 2048, 2252, 2078, 2672, 2642, 2456, 2936, 2504, 2588, 2978, 3092, 3032, 3218, 3272, 3296, 3632, 3548, 3754, 4022, 4058, 4412
Offset: 0

Views

Author

Bill Gosper

Keywords

Comments

The Goldbach conjecture is that every even number is the sum of two primes.

Examples

			2 is largest even integer which is the sum of two primes in 0 ways, 12 is largest even integer which is the unordered sum of two primes in 1 way (5+7), etc.

Links

Crossrefs

Cf. A045917, A023036, A000974, A001172, A002375.

Programs

  • Mathematica
    f[n_] := Block[{c = 0, k = 3}, While[k <= n/2, If[PrimeQ[k] && PrimeQ[n - k], c++ ]; k++ ]; c]; a = Table[0, {50}]; a[[1]] = 2; a[[2]] = 4; Do[m = n; b = f[n]; If[b < 100, a[[b + 1]] = n], {n, 6, 20000, 2}] (* Robert G. Wilson v, Dec 20 2003 *)

A229492 Conjectured number of positive even numbers having exactly n representations as the unordered sum of two odd primes.

Original entry on oeis.org

2, 3, 9, 11, 11, 16, 16, 18, 20, 23, 16, 29, 16, 25, 27, 23, 22, 25, 35, 29, 26, 25, 27, 27, 27, 33, 28, 44, 35, 21, 29, 35, 38, 33, 39, 37, 34, 35, 31, 31, 28, 41, 37, 32, 44, 35, 37, 41, 44, 33, 37, 32, 47, 39, 43, 47, 33, 37, 48, 41, 37, 48, 34, 35, 47, 36, 29
Offset: 0

Views

Author

T. D. Noe, Oct 31 2013

Keywords

Comments

This sequence is closely related to the Goldbach conjecture, which is equivalent to a(0) = 2; that is, there are no numbers other than 2 and 4 that have no representations as the sum of two odd primes. This is sequence A000974, except for the first two terms.

Examples

			There are 3 even numbers with one representation: 6 = 3+3, 8 = 3+5, and 12 = 5+7. Hence, a(1) = 3.

Crossrefs

Cf. A000974 (using both even and odd primes).
Cf. A002375 (number of representations of 2n as the sum of two odd primes).

A352596 Conjecturally the number of positive even integers that can be expressed as the sum of two distinct primes in exactly n ways.

Original entry on oeis.org

3, 5, 9, 12, 12, 16, 19, 13, 24, 19, 21, 25, 15, 29, 28, 16, 31, 22, 34, 32, 20, 29, 26, 24, 28, 36, 34, 35, 37, 22, 29, 37, 36, 34, 39, 32, 39, 35, 28, 31, 28
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 22 2022

Keywords

Crossrefs

Cf. A000974, A038609, A061357, A077914, A077969, A078299, A080854, A080862, A117929, A229492, A352305.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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