A228553 -id:A228553 - cp's OEIS frontend

A339307 Numbers k such that A228553(k) is prime.

11, 20, 24, 27, 32, 35, 37, 38, 68, 74, 78, 79, 82, 96, 98, 101, 103, 122, 137, 149, 161, 188, 209, 218, 239, 249, 258, 276, 293, 321, 325, 333, 363, 365, 372, 433, 445, 449, 499, 535, 567, 568, 580, 588, 594, 595, 622, 644, 656, 663, 666, 682, 688, 693, 721, 742, 754, 756, 760, 796, 814, 845

Offset: 1

J. M. Bergot and Robert Israel, Nov 29 2020

nonn

			a(3) = 24 is in the sequence because A228553(24) = 1987 is prime.

Robert Israel, Table of n, a(n) for n = 1..5000

Cf. A228553, A339309.

f:= proc(n)
  local S;
  S:= select(t -> isprime(t) and isprime(2*n-t), [seq(i,i=3..n,2)]);
  add(t*(2*n-t),t=S)
end proc:
f(2):= 4:
select(isprime@f, [$1..1000]);

A339309 Numbers k such that A228553(k) is twice a prime.

Original entry on oeis.org

2, 5, 7, 8, 9, 10, 14, 16, 34, 53, 55, 83, 91, 140, 172, 180, 182, 187, 189, 242, 246, 254, 265, 288, 361, 398, 404, 421, 435, 490, 508, 559, 574, 623, 706, 727, 736, 747, 770, 773, 777, 984, 985, 1065, 1093, 1106, 1125, 1187, 1245, 1256, 1287, 1341, 1346, 1361, 1372, 1381, 1449, 1467, 1573, 1596

Offset: 1

J. M. Bergot and Robert Israel, Nov 29 2020

nonn

			a(3) = 7 is a term because A228553(7) = 82 = 2*41 and 41 is prime.

Robert Israel, Table of n, a(n) for n = 1..3016

Cf. A228553, A339307.

f:= proc(n)
  local S;
  S:= select(t -> isprime(t) and isprime(2*n-t), [seq(i, i=3..n, 2)]);
  add(t*(2*n-t), t=S)
end proc:
f(2):= 4:
filter:= proc(t) local v; v:= f(t); v::even and isprime(v) end proc:
select(filter, [$1..2000]);

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

Alois P. Heinz, Dec 31 2021

Assuming Goldbach's conjecture, no row is empty.

			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;
  ...

Alois P. Heinz, Rows n = 2..1000, flattened
Wikipedia, Goldbach's conjecture

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.

T:= n-> seq(`if`(andmap(isprime, [h, 2*n-h]), h*(2*n-h), [][]), h=2..n):
seq(T(n), n=2..30);

cp's OEIS Frontend

A339307 Numbers k such that A228553(k) is prime.

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A339309 Numbers k such that A228553(k) is twice a prime.

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Maple

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.

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Maple