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

1,2

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).

Links

Table of n, a(n) for n=1..37.

Formula

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

Example

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}

Mathematica

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

Crossrefs

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

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

Sequence in context: A371050 A070007 A380217 * A154267 A173351 A051867

Adjacent sequences: A325038 A325039 A325040 * A325042 A325043 A325044

Keyword

nonn

Author

Gus Wiseman, Mar 25 2019

Status

approved