A348384 Heinz numbers of integer partitions whose length is 2/3 their sum.

1, 6, 36, 40, 216, 224, 240, 1296, 1344, 1408, 1440, 1600, 6656, 7776, 8064, 8448, 8640, 8960, 9600, 34816, 39936, 46656, 48384, 50176, 50688, 51840, 53760, 56320, 57600, 64000, 155648, 208896, 239616, 266240, 279936, 290304, 301056, 304128, 311040, 315392

Offset: 1

Author

Gus Wiseman, Nov 13 2021

Keywords

• nonn

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers whose sum of prime indices is 3/2 their number. Counting the partitions with these Heinz numbers gives A035377(n) = A000041(n/3) if n is a multiple of 3, otherwise 0.

Examples

			The terms and their prime indices begin:
     1: {}
     6: {1,2}
    36: {1,1,2,2}
    40: {1,1,1,3}
   216: {1,1,1,2,2,2}
   224: {1,1,1,1,1,4}
   240: {1,1,1,1,2,3}
  1296: {1,1,1,1,2,2,2,2}
  1344: {1,1,1,1,1,1,2,4}
  1408: {1,1,1,1,1,1,1,5}
  1440: {1,1,1,1,1,2,2,3}
  1600: {1,1,1,1,1,1,3,3}
  6656: {1,1,1,1,1,1,1,1,1,6}
  7776: {1,1,1,1,1,2,2,2,2,2}
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000

Crossrefs

These partitions are counted by A035377.

Rounding down gives A348550 or A347452, counted by A108711 or A119620.

A000041 counts integer partitions.

A001222 counts prime factors with multiplicity.

A056239 adds up prime indices, row sums of A112798.

A316524 gives the alternating sum of prime indices (reverse: A344616).

A344606 counts alternating permutations of prime factors.

Cf. A000070, A000097, A028260, A028982, A032766, A236914, A316413, A347457, A348551.

Programs

Mathematica

Select[Range[1000],2*Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]==3*PrimeOmega[#]&]

PARI

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
isA348384(n) = (A056239(n)==(3/2)*bigomega(n)); \\ Antti Karttunen, Nov 22 2021

Formula

The sequence contains n iff A056239(n) = 3*A001222(n)/2. Here, A056239 adds up prime indices, while A001222 counts them with multiplicity.

Intersection of A028260 and A347452.