A223893 -id:A223893 - cp's OEIS frontend

A347550 Number of partitions of n into at most 2 distinct prime parts.

1, 0, 1, 1, 0, 2, 0, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 3, 1, 2, 0, 2, 1, 3, 2, 2, 1, 3, 0, 4, 1, 1, 1, 3, 1, 4, 2, 3, 1, 3, 1, 5, 1, 4, 0, 3, 1, 5, 1, 3, 0, 3, 1, 6, 2, 2, 1, 5, 0, 6, 1, 2, 1, 5, 1, 6, 2, 4, 1, 5, 0, 7, 1, 4, 1, 4, 1, 8, 1, 4

Offset: 0

Ilya Gutkovskiy, Sep 08 2021

nonn

Cf. A000040, A000586, A117929, A219180, A223893.

a(n) = Sum_{k=0..2} A219180(n,k). - Alois P. Heinz, Sep 08 2021

A358010 Number of partitions of n into at most 5 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, 6, 7, 6, 9, 7, 9, 9, 9, 11, 11, 11, 13, 12, 13, 15, 15, 17, 15, 18, 17, 20, 20, 23, 20, 25, 22, 27, 28, 28, 27, 30, 29, 36, 34, 38, 36, 41, 35, 48, 41, 48, 44, 50, 46, 58, 53, 61, 54, 64, 55, 72, 66, 74

Offset: 0

Ilya Gutkovskiy, Oct 24 2022

nonn

Cf. A000040, A000586, A219180, A219199, A223893, A347550, A347587, A347609, A347742, A358009, A358011.

A358011 Number of partitions of n into at most 6 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, 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.

A269329 Number of partitions of a positive integer n into two distinct primes such that for even n, it is of the form n = p + q and for odd n, it is of the form n = 2p' + q'.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 4, 2, 2, 2, 2, 2, 3, 3, 3, 2, 4, 2, 4, 3, 2, 2, 4, 3, 3, 4, 4, 1, 3, 3, 5, 4, 4, 3, 6, 3, 4, 5, 4, 4, 6, 3, 5, 5, 3, 3, 6, 3, 3, 6, 3, 2, 7, 5, 7, 6, 5, 2, 6, 5, 4, 6, 4, 4, 8, 5, 7, 7, 5, 4, 8, 4, 4, 8, 7, 4, 7, 4, 7, 9, 4, 4, 10, 4, 5, 7, 6, 3, 9

Offset: 1

Frank M Jackson and M. B. Rees, Feb 23 2016

nonn

This sequence combines Levy's conjecture for odd positive numbers with the Goldbach conjecture for even positive numbers and strenghtens both by restricting the prime pairs to be distinct. I.e., every positive integer n > 6 is the sum of two distinct primes p and q such that for n even, it is of the form n = p + q and for n odd, it is of the form n = 2p' + q'.

			a(23)=3. Hence there are 3 partitions (as defined above) of the odd integer 23, namely 19+2+2, 17+3+3 and 13+5+5. a(24)=3. Hence there are 3 partitions of the even integer 24, namely 19+5, 17+7 and 13+11.

Cf. A002375, A061357, A194830, A223893, A225522.

parts[n_, a_, b_] := Select[IntegerPartitions[n, {a+b}, Prime@Range[PrimePi[n]]], Length[Union@#]==2&&MemberQ[Values@Counts@#, a] &]; lst1=Table[Length@parts[2n-1, 1, 2], {n, 1, 200}]; lst2=Table[Length@parts[2n, 1, 1], {n, 1, 200}]; Riffle[lst1, lst2]

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A347550 Number of partitions of n into at most 2 distinct prime parts.

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A358010 Number of partitions of n into at most 5 distinct prime parts.

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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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A269329 Number of partitions of a positive integer n into two distinct primes such that for even n, it is of the form n = p + q and for odd n, it is of the form n = 2p' + q'.

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