# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a321455 Showing 1-1 of 1 %I A321455 #14 Jul 14 2019 06:30:12 %S A321455 1,1,1,2,1,1,1,2,2,1,1,2,1,1,1,3,1,1,1,1,1,1,1,1,2,1,2,1,1,2,1,2,1,1, %T A321455 1,3,1,1,1,2,1,1,1,1,1,1,1,3,2,1,1,1,1,1,1,1,1,1,1,1,1,1,2,4,1,1,1,1, %U A321455 1,2,1,1,1,1,1,1,1,1,1,1,3,1,1,2,1,1,1 %N A321455 Number of ways to factor n into factors > 1 all having the same sum of prime indices. %C A321455 Also the number of multiset partitions of the multiset of prime indices of n with equal block-sums. %C A321455 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The sum of prime indices of n is A056239(n). %H A321455 Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict. %e A321455 The a(1440) = 6 factorizations into factors all having the same sum of prime indices: %e A321455 (10*12*12) %e A321455 (5*6*6*8) %e A321455 (9*10*16) %e A321455 (30*48) %e A321455 (36*40) %e A321455 (1440) %e A321455 The a(900) = 5 multiset partitions with equal block-sums: %e A321455 {{1,1,2,2,3,3}} %e A321455 {{3,3},{1,1,2,2}} %e A321455 {{1,2,3},{1,2,3}} %e A321455 {{1,3},{1,3},{2,2}} %e A321455 {{3},{3},{1,2},{1,2}} %t A321455 hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]]; %t A321455 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]]; %t A321455 Table[Length[Select[facs[n],SameQ@@hwt/@#&]],{n,100}] %Y A321455 Positions of 1's are A321453. Positions of terms > 1 are A321454. %Y A321455 Cf. A001055, A035470, A056239, A279787, A305551, A321469, A322794, A326515, A326516, A326518, A326534. %K A321455 nonn %O A321455 1,4 %A A321455 _Gus Wiseman_, Nov 10 2018 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE