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A325042 Heinz numbers of integer partitions whose product of parts is one fewer than their sum.

%I A325042 #5 Mar 26 2019 21:06:38
%S A325042 4,6,10,14,18,22,26,34,38,46,58,60,62,74,82,86,94,106,118,122,134,142,
%T A325042 146,158,166,168,178,194,202,206,214,216,218,226,254,262,274,278,298,
%U A325042 302,314,326,334,346,358,362,382,386,394,398,400,422,446,454,458,466
%N A325042 Heinz numbers of integer partitions whose product of parts is one fewer than their sum.
%C A325042 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 fewer than their sum of prime indices (A056239).
%F A325042 A003963(a(n)) = A056239(a(n)) - 1.
%F A325042 a(n) = 2 * A301987(n).
%e A325042 The sequence of terms together with their prime indices begins:
%e A325042     4: {1,1}
%e A325042     6: {1,2}
%e A325042    10: {1,3}
%e A325042    14: {1,4}
%e A325042    18: {1,2,2}
%e A325042    22: {1,5}
%e A325042    26: {1,6}
%e A325042    34: {1,7}
%e A325042    38: {1,8}
%e A325042    46: {1,9}
%e A325042    58: {1,10}
%e A325042    60: {1,1,2,3}
%e A325042    62: {1,11}
%e A325042    74: {1,12}
%e A325042    82: {1,13}
%e A325042    86: {1,14}
%e A325042    94: {1,15}
%e A325042   106: {1,16}
%e A325042   118: {1,17}
%e A325042   122: {1,18}
%t A325042 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A325042 Select[Range[1000],Times@@primeMS[#]==Total[primeMS[#]]-1&]
%Y A325042 Cf. A000720, A003963, A056239, A112798, A178503, A175508, A301987, A319000.
%Y A325042 Cf. A325032, A325033, A325036, A325037, A325038, A325041, A325044.
%K A325042 nonn
%O A325042 1,1
%A A325042 _Gus Wiseman_, Mar 25 2019