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 1-3 of 3 results.

A362640 Product of the larger primes, q, in the Goldbach partitions of 2n such that p + q = 2n, p <= q, and p,q prime (or 1 if no Goldbach partition of 2n exists).

Original entry on oeis.org

1, 2, 3, 5, 35, 7, 77, 143, 143, 221, 3553, 4199, 5681, 391, 7429, 551, 351509, 392863, 589, 24679, 765049, 47027, 1175921, 58642669, 2318087, 55883, 95041567, 84323, 2961799, 5037203051, 78647, 367569469, 14263488419, 2257, 403723843, 22531226387, 461671607, 761740327
Offset: 1

Views

Author

Wesley Ivan Hurt, Apr 28 2023

Keywords

Examples

			a(10) = 221; 2*10 = 20 has two Goldbach partitions, namely 17+3 and 13+7. The product of the larger parts of these partitions, is 17*13 = 221.

Links

Crossrefs

Cf. A010051, A045917, A337568 (product of all prime parts), A362641 (product of smaller primes p).

Programs

  • Mathematica
    Table[Product[(2 n - k)^((PrimePi[k] - PrimePi[k - 1]) (PrimePi[2 n - k] - PrimePi[2 n - k - 1])), {k, n}], {n, 40}]

Formula

a(n) = Product_{k=1..n} (2n - k)^(c(k)*c(2n - k)), where c is the prime characteristic (A010051).
a(n) = Product_{p+q = 2n, p<=q, and p,q prime} q.
a(n) = A337568(n) / A362641(n).

A362641 Product of the smaller primes, p, in the Goldbach partitions of 2n such that p + q = 2n, p <= q, and p,q prime (or 1 if no Goldbach partition of 2n exists).

Original entry on oeis.org

1, 2, 3, 3, 15, 5, 21, 15, 35, 21, 165, 385, 273, 55, 1001, 39, 2805, 7735, 133, 561, 13585, 273, 5865, 124355, 5187, 1265, 391391, 741, 27115, 19605131, 1767, 64515, 5766215, 217, 374187, 12212915, 313131, 170085, 142635185, 63973, 902451, 13147103255, 223041, 101065, 818183948197
Offset: 1

Views

Author

Wesley Ivan Hurt, Apr 28 2023

Keywords

Examples

			a(10) = 21; 2*10 = 20 has two Goldbach partitions, namely 17+3 and 13+7. The product of the smaller parts of these partitions, is 3*7 = 21.

Links

Crossrefs

Cf. A010051, A045917, A337568 (product of all prime parts), A362640 (product of the larger primes q).

Programs

  • Mathematica
    Table[Product[k^((PrimePi[k] - PrimePi[k - 1]) (PrimePi[2 n - k] - PrimePi[2 n - k - 1])), {k, n}], {n, 40}]

Formula

a(n) = Product_{k=1..n} k^(c(k)*c(2n - k)), where c is the prime characteristic (A010051).
a(n) = Product_{p+q = 2n, p<=q, and p,q prime} p.
a(n) = A337568(n) / A362640(n).

A350455 T(n,k) is the k-th semiprime whose sum of prime factors equals 2n, triangle T(n,k), n>=2, 1<=k<=A045917(n), read by rows.

Original entry on oeis.org

4, 9, 15, 21, 25, 35, 33, 49, 39, 55, 65, 77, 51, 91, 57, 85, 121, 95, 119, 143, 69, 133, 169, 115, 187, 161, 209, 221, 87, 247, 93, 145, 253, 289, 155, 203, 299, 323, 217, 361, 111, 319, 391, 185, 341, 377, 437, 123, 259, 403, 129, 205, 493, 529, 215, 287, 407
Offset: 2

Views

Author

Alois P. Heinz, Dec 31 2021

Keywords

Comments

Assuming Goldbach's conjecture, no row is empty.

Examples

			Triangle T(n,k) begins:
    4;
    9;
   15;
   21,  25;
   35     ;
   33,  49;
   39,  55;
   65,  77;
   51,  91;
   57,  85, 121;
   95, 119, 143;
   69, 133, 169;
  115, 187     ;
  161, 209, 221;
   87, 247     ;
   93, 145, 253, 289;
  155, 203, 299, 323;
  ...

Links

Crossrefs

Column k=1 gives A073046.
Last elements of rows give A102084.
Row sums give A228553.
Row products give A337568.
Row lengths give A045917.
Cf. A000040, A001358, A046315, A350419.

Programs

  • Maple
    T:= n-> seq(`if`(andmap(isprime, [h, 2*n-h]), h*(2*n-h), [][]), h=2..n):
seq(T(n), n=2..30);
Showing 1-3 of 3 results.

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

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