A326624 Heinz numbers of non-constant integer partitions whose geometric mean is an integer.

14, 42, 46, 57, 76, 106, 126, 161, 183, 185, 194, 196, 228, 230, 302, 371, 378, 393, 399, 412, 424, 454, 477, 515, 588, 622, 679, 684, 687, 722, 742, 781, 786, 838, 1057, 1064, 1077, 1082, 1115, 1134, 1150, 1157, 1159, 1219, 1244, 1272, 1322, 1563, 1589, 1654

Offset

1,1

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Links

Table of n, a(n) for n=1..50.

Wikipedia, Geometric mean

Example

The sequence of terms together with their prime indices begins:
    14: {1,4}
    42: {1,2,4}
    46: {1,9}
    57: {2,8}
    76: {1,1,8}
   106: {1,16}
   126: {1,2,2,4}
   161: {4,9}
   183: {2,18}
   185: {3,12}
   194: {1,25}
   196: {1,1,4,4}
   228: {1,1,2,8}
   230: {1,3,9}
   302: {1,36}
   371: {4,16}
   378: {1,2,2,2,4}
   393: {2,32}
   399: {2,4,8}
   412: {1,1,27}

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], !PrimePowerQ[#]&&IntegerQ[GeometricMean[primeMS[#]]]&]

Crossrefs

The case with prime powers is A326623.

Subsets whose geometric mean is an integer are A326027.

Cf. A001055, A067539, A078175, A102627, A316413, A326567/A326568, A326622, A326625.

Sequence in context: A214521 A064512 A118237 * A208360 A208359 A245629

Adjacent sequences: A326621 A326622 A326623 * A326625 A326626 A326627

Keyword

nonn

Author

Gus Wiseman, Jul 14 2019

Status

approved