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%I #26 Dec 15 2024 12:58:39
%S 0,1,1,5,5,8,20,7,22
%N Position of first 0 in the n-th differences of the strict partition numbers A000009, or 0 if 0 does not appear.
%C Open problem: Do the 9th differences of the strict integer partition numbers contain a zero? If so, we must have a(9) > 10^5.
%C a(12) = 47. Conjecture: a(n) = 0 for n > 12. - _Chai Wah Wu_, Dec 15 2024
%e The 7th differences of A000009 are: 25, -16, 7, -6, 10, -9, 0, 10, ... so a(7) = 7.
%t Table[Position[Differences[PartitionsQ/@Range[0,100],k],0][[1,1]],{k,1,8}]
%o (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
%Y For primes we have A376678.
%Y For composites we have A377037.
%Y For squarefree numbers we have A377042.
%Y For nonsquarefree numbers we have A377050.
%Y For prime-powers we have A377055.
%Y Position of first zero in each row of A378622. See also:
%Y - A175804 is the version for partitions.
%Y - A293467 gives first column (up to sign).
%Y - A378970 gives row-sums.
%Y - A378971 gives row-sums of absolute value.
%Y A000009 counts strict integer partitions, differences A087897, A378972.
%Y A000041 counts integer partitions, differences A002865, A053445.
%Y Cf. A047966, A098859, A225486, A325244, A325282.
%Y Cf. A008284, A116608, A325242, A325268, A225485 or A325280.
%K nonn,more,new
%O 0,4
%A _Gus Wiseman_, Dec 12 2024