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

4, 6, 10, 14, 18, 22, 26, 34, 38, 46, 58, 60, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 168, 178, 194, 202, 206, 214, 216, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 400, 422, 446, 454, 458, 466

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 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..56.

Formula

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

a(n) = 2 * A301987(n).

Example

The sequence of terms together with their prime indices begins:
    4: {1,1}
    6: {1,2}
   10: {1,3}
   14: {1,4}
   18: {1,2,2}
   22: {1,5}
   26: {1,6}
   34: {1,7}
   38: {1,8}
   46: {1,9}
   58: {1,10}
   60: {1,1,2,3}
   62: {1,11}
   74: {1,12}
   82: {1,13}
   86: {1,14}
   94: {1,15}
  106: {1,16}
  118: {1,17}
  122: {1,18}

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[1000], Times@@primeMS[#]==Total[primeMS[#]]-1&]

Crossrefs

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

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

Sequence in context: A310584 A165779 A287375 * A137860 A184335 A243428

Adjacent sequences: A325039 A325040 A325041 * A325043 A325044 A325045

Keyword

nonn

Author

Gus Wiseman, Mar 25 2019

Status

approved