A358010 -id:A358010 - cp's OEIS frontend

A358011 Number of partitions of n into at most 6 distinct prime parts.

1, 0, 1, 1, 0, 2, 0, 2, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 4, 3, 4, 4, 4, 5, 5, 5, 6, 5, 6, 7, 6, 9, 7, 9, 9, 9, 11, 11, 11, 13, 12, 14, 15, 15, 17, 16, 18, 19, 20, 21, 23, 22, 25, 26, 27, 30, 29, 32, 31, 35, 36, 39, 40, 42, 42, 45, 49, 50, 52, 55, 53, 61, 61, 67, 67, 70, 70, 77, 77, 86, 84

Offset: 0

Ilya Gutkovskiy, Oct 24 2022

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 501 terms from Robert Israel)

Cf. A000040, A000586, A219180, A219200, A223893, A347550, A347588, A347610, A347743, A358009, A358010.

P:= select(isprime,[2,seq(i,i=3..100,2)]):
G:= mul(1+t*x^p, p=P):
f:= proc(n) local i,S;
   S:= coeff(G,x,n);
   add(coeff(S,t,i),i=0..6)
end proc;
map(f, [$0..100]); # Robert Israel, May 14 2025

a(n) = Sum_{k=0..6} A219180(n,k). - Alois P. Heinz, May 14 2025

A358009 Number of partitions of n into at most 4 distinct prime parts.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 0, 2, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 4, 3, 4, 4, 4, 5, 5, 5, 6, 5, 5, 7, 5, 9, 7, 9, 7, 9, 9, 11, 9, 12, 8, 13, 11, 14, 13, 13, 12, 16, 14, 18, 17, 16, 17, 20, 17, 23, 19, 21, 19, 24, 23, 28, 24, 26, 25, 26, 30, 30, 29, 29, 29, 32, 36, 37, 36, 32, 38, 35, 43, 41, 43, 20

Offset: 0

Ilya Gutkovskiy, Oct 24 2022

nonn

Cf. A000040, A000586, A219180, A219198, A223893, A347550, A347578, A347586, A347741, A358010, A358011.

A363342 Array read by descending antidiagonals. A(n,k), n > 1 and k > 0, is the least m such that the number of partitions of m into n distinct prime parts is exactly k, or -1 if no such number exists.

Original entry on oeis.org

5, 16, 10, 24, 18, 17, 36, 26, 23, 28, 48, 31, 29, 34, 41, 60, 35, 33, -1, 47, 58, 78, 39, 37, 40, 70, 64, 77, 84, 80, 41, 55, 53, 72, 87, 100

Offset: 2

Jean-Marc Rebert, May 28 2023

			A(2, 1) = 5 = 2 + 3, because 5 is the least number for which there exists exactly one partition into 2 distinct primes.
A(2, 2) = 16 = 3 + 13 = 5 + 11, because 16 is the least number for which there exist exactly 2 partitions into 2 distinct primes.
Array begins:
  2:   5, 16, 24, 36, 48, 60, 78, 84, ...
  3:  10, 18, 26, 31, 35, 39, 80, ...
  4:  17, 23, 29, 33, 37, 41, ...
  5:  28, 34, -1, 40, 55, ...
  6:  41, 47, 70, 53, ...
  7:  58, 64, 72, ...
  8:  77, 87, ...
  9: 100, ...

Cf. A000586, A000607, A358010.

cp's OEIS Frontend

A358011 Number of partitions of n into at most 6 distinct prime parts.

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A358009 Number of partitions of n into at most 4 distinct prime parts.

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A363342 Array read by descending antidiagonals. A(n,k), n > 1 and k > 0, is the least m such that the number of partitions of m into n distinct prime parts is exactly k, or -1 if no such number exists.

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