cp's OEIS Frontend

A381635 Number of ways to partition the prime indices of n into constant blocks with distinct sums.

%I A381635 #5 Mar 09 2025 20:26:04
%S A381635 1,1,1,1,1,1,1,2,1,1,1,0,1,1,1,2,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,3,1,1,
%T A381635 1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,0,1,1,0,4,1,1,1,1,
%U A381635 1,1,1,2,1,1,1,1,1,1,1,1,2,1,1,0,1,1,1
%N A381635 Number of ways to partition the prime indices of n into constant blocks with distinct sums.
%C A381635 First differs from A381716 at a(1728) = 5, A381716(1728) = 4.
%C A381635 Also the number of factorizations on n into prime powers > 1 with distinct sums of prime indices (A056239).
%e A381635 The a(432) = 3 multiset partitions:
%e A381635   {{2,2,2},{1,1,1,1}}
%e A381635   {{1},{1,1,1},{2,2,2}}
%e A381635   {{1},{2},{2,2},{1,1,1}}
%e A381635 Note {{2},{2,2},{1,1,1,1}} is not included, as it does not have distinct block-sums.
%t A381635 hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
%t A381635 pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
%t A381635 Table[Length[Select[pfacs[n],UnsameQ@@hwt/@#&]],{n,100}]
%Y A381635 Without distinct block-sums we have A000688, after sums A381455 (upper), A381453 (lower).
%Y A381635 For distinct blocks instead of sums we have A050361, after sums A381715.
%Y A381635 For strict instead of constant we have A381633 (zeros A381806), after sums A381634.
%Y A381635 Positions of 0 are A381636.
%Y A381635 Taking block-sums (and sorting) gives A381716.
%Y A381635 Other multiset partitions of prime indices:
%Y A381635 More on multiset partitions into constant blocks: A006171, A279784, A295935.
%Y A381635 A001055 counts multiset partitions, see A317141 (upper), A300383 (lower).
%Y A381635 A003963 gives product of prime indices.
%Y A381635 A055396 gives least prime index, greatest A061395.
%Y A381635 A056239 adds up prime indices, row sums of A112798.
%Y A381635 A265947 counts refinement-ordered pairs of integer partitions.
%Y A381635 Cf. A000720, A001222, A002846, A005117, A213242, A213385, A293511, A299202, A300385, A317142, A381870.
%K A381635 nonn
%O A381635 1,8
%A A381635 _Gus Wiseman_, Mar 09 2025