cp's OEIS Frontend

A325041 Heinz numbers of integer partitions whose product of parts is one greater than their sum.

%I A325041 #6 Mar 26 2019 21:06:30
%S A325041 1,15,42,54,100,132,312,560,720,816,1824,3520,4416,6272,8064,10368,
%T A325041 11136,16640,23808,38400,56832,78848,87040,101376,125952,264192,
%U A325041 389120,577536,745472,958464,1302528,1720320,1884160,1982464,2211840,2899968,5996544
%N A325041 Heinz numbers of integer partitions whose product of parts is one greater than their sum.
%C A325041 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose product of prime indices (A003963) is one more than their sum of prime indices (A056239).
%F A325041 A003963(a(n)) = A056239(a(n)) + 1.
%e A325041 The sequence of terms together with their prime indices begins:
%e A325041       1: {}
%e A325041      15: {2,3}
%e A325041      42: {1,2,4}
%e A325041      54: {1,2,2,2}
%e A325041     100: {1,1,3,3}
%e A325041     132: {1,1,2,5}
%e A325041     312: {1,1,1,2,6}
%e A325041     560: {1,1,1,1,3,4}
%e A325041     720: {1,1,1,1,2,2,3}
%e A325041     816: {1,1,1,1,2,7}
%e A325041    1824: {1,1,1,1,1,2,8}
%e A325041    3520: {1,1,1,1,1,1,3,5}
%e A325041    4416: {1,1,1,1,1,1,2,9}
%e A325041    6272: {1,1,1,1,1,1,1,4,4}
%e A325041    8064: {1,1,1,1,1,1,1,2,2,4}
%e A325041   10368: {1,1,1,1,1,1,1,2,2,2,2}
%e A325041   11136: {1,1,1,1,1,1,1,2,10}
%e A325041   16640: {1,1,1,1,1,1,1,1,3,6}
%e A325041   23808: {1,1,1,1,1,1,1,1,2,11}
%e A325041   38400: {1,1,1,1,1,1,1,1,1,2,3,3}
%t A325041 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A325041 Select[Range[10000],Times@@primeMS[#]==Total[primeMS[#]]+1&]
%Y A325041 Cf. A000720, A003963, A056239, A112798, A178503, A175508, A301987, A319000.
%Y A325041 Cf. A325032, A325033, A325036, A325037, A325038, A325042, A325044.
%K A325041 nonn
%O A325041 1,2
%A A325041 _Gus Wiseman_, Mar 25 2019