A377285 Position of first 0 in the n-th differences of the strict partition numbers A000009, or 0 if 0 does not appear.

0, 1, 1, 5, 5, 8, 20, 7, 22

Offset: 0

Author

Gus Wiseman, Dec 12 2024

Keywords

• nonn
• more

Comments

Open problem: Do the 9th differences of the strict integer partition numbers contain a zero? If so, we must have a(9) > 10^5.

a(12) = 47. Conjecture: a(n) = 0 for n > 12. - Chai Wah Wu, Dec 15 2024

Examples

			The 7th differences of A000009 are: 25, -16, 7, -6, 10, -9, 0, 10, ... so a(7) = 7.
		

Crossrefs

For primes we have A376678.

For composites we have A377037.

For squarefree numbers we have A377042.

For nonsquarefree numbers we have A377050.

For prime-powers we have A377055.

Position of first zero in each row of A378622. See also:

- A175804 is the version for partitions.

- A293467 gives first column (up to sign).

- A378970 gives row-sums.

- A378971 gives row-sums of absolute value.

A000009 counts strict integer partitions, differences A087897, A378972.

A000041 counts integer partitions, differences A002865, A053445.

Cf. A047966, A098859, A225486, A325244, A325282.

Cf. A008284, A116608, A325242, A325268, A225485 or A325280.

Programs

Mathematica

Table[Position[Differences[PartitionsQ/@Range[0,100],k],0][[1,1]],{k,1,8}]

PARI

a(n, nn=100) = my(q='q+O('q^nn), v=Vec(eta(q^2)/eta(q))); for (i=1, n, my(w=vector(#v-1, k, v[k+1]-v[k])); v = w;); my(vz=select(x->x==0, v, 1)); if (#vz, vz[1]); \\ Michel Marcus, Dec 15 2024