A325043 Heinz numbers of integer partitions, with at least three parts, whose product of parts is one fewer than their sum.

18, 60, 168, 216, 400, 528, 1248, 2240, 2880, 3264, 7296, 14080, 17664, 25088, 32256, 41472, 44544, 66560, 95232, 153600, 227328, 315392, 348160, 405504, 503808, 1056768, 1556480, 2310144, 2981888, 3833856, 5210112, 6881280, 7536640, 7929856, 8847360, 11599872

Offset

1,1

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers with at least three prime factors (counted with multiplicity) whose product of prime indices (A003963) is one fewer than their sum of prime indices (A056239).

Links

Table of n, a(n) for n=1..36.

Formula

a(n) = 2 * A301988(n).

Example

The sequence of terms together with their prime indices begins:
     18: {1,2,2}
     60: {1,1,2,3}
    168: {1,1,1,2,4}
    216: {1,1,1,2,2,2}
    400: {1,1,1,1,3,3}
    528: {1,1,1,1,2,5}
   1248: {1,1,1,1,1,2,6}
   2240: {1,1,1,1,1,1,3,4}
   2880: {1,1,1,1,1,1,2,2,3}
   3264: {1,1,1,1,1,1,2,7}
   7296: {1,1,1,1,1,1,1,2,8}
  14080: {1,1,1,1,1,1,1,1,3,5}
  17664: {1,1,1,1,1,1,1,1,2,9}
  25088: {1,1,1,1,1,1,1,1,1,4,4}
  32256: {1,1,1,1,1,1,1,1,1,2,2,4}
  41472: {1,1,1,1,1,1,1,1,1,2,2,2,2}
  44544: {1,1,1,1,1,1,1,1,1,2,10}
  66560: {1,1,1,1,1,1,1,1,1,1,3,6}
  95232: {1,1,1,1,1,1,1,1,1,1,2,11}

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[10000], And[PrimeOmega[#]>2, Times@@primeMS[#]==Total[primeMS[#]]-1]&]

Crossrefs

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

Cf. A325032, A325033, A325036, A325037, A325038, A325041, A325042, A325044.

Sequence in context: A218617 A105521 A154563 * A338536 A090073 A327089

Adjacent sequences: A325040 A325041 A325042 * A325044 A325045 A325046

Keyword

nonn

Author

Gus Wiseman, Mar 25 2019

Extensions

More terms from Jinyuan Wang, Jun 27 2020

Status

approved