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

1, 15, 42, 54, 100, 132, 312, 560, 720, 816, 1824, 3520, 4416, 6272, 8064, 10368, 11136, 16640, 23808, 38400, 56832, 78848, 87040, 101376, 125952, 264192, 389120, 577536, 745472, 958464, 1302528, 1720320, 1884160, 1982464, 2211840, 2899968, 5996544

Offset: 1

Author

Gus Wiseman, Mar 25 2019

Keywords

• nonn

Comments

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).

Examples

			The sequence of terms together with their prime indices begins:
      1: {}
     15: {2,3}
     42: {1,2,4}
     54: {1,2,2,2}
    100: {1,1,3,3}
    132: {1,1,2,5}
    312: {1,1,1,2,6}
    560: {1,1,1,1,3,4}
    720: {1,1,1,1,2,2,3}
    816: {1,1,1,1,2,7}
   1824: {1,1,1,1,1,2,8}
   3520: {1,1,1,1,1,1,3,5}
   4416: {1,1,1,1,1,1,2,9}
   6272: {1,1,1,1,1,1,1,4,4}
   8064: {1,1,1,1,1,1,1,2,2,4}
  10368: {1,1,1,1,1,1,1,2,2,2,2}
  11136: {1,1,1,1,1,1,1,2,10}
  16640: {1,1,1,1,1,1,1,1,3,6}
  23808: {1,1,1,1,1,1,1,1,2,11}
  38400: {1,1,1,1,1,1,1,1,1,2,3,3}
		

Crossrefs

Cf. A000720, A003963, A056239, A112798, A178503, A175508, A301987, A319000.

Cf. A325032, A325033, A325036, A325037, A325038, A325042, A325044.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[10000],Times@@primeMS[#]==Total[primeMS[#]]+1&]

Formula

A003963(a(n)) = A056239(a(n)) + 1.