A348550 Heinz numbers of integer partitions whose length is 2/3 their sum, rounded down.

1, 3, 6, 9, 10, 18, 20, 36, 40, 54, 56, 60, 108, 112, 120, 216, 224, 240, 324, 336, 352, 360, 400, 648, 672, 704, 720, 800, 1296, 1344, 1408, 1440, 1600, 1664, 1944, 2016, 2112, 2160, 2240, 2400, 3328, 3888, 4032, 4224, 4320, 4480, 4800, 6656, 7776, 8064, 8448

Offset: 1

Author

Gus Wiseman, Nov 05 2021

Keywords

• nonn

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

Examples

			The terms and their prime indices begin:
    1: {}
    3: {2}
    6: {1,2}
    9: {2,2}
   10: {1,3}
   18: {1,2,2}
   20: {1,1,3}
   36: {1,1,2,2}
   40: {1,1,1,3}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
  108: {1,1,2,2,2}
  112: {1,1,1,1,4}
  120: {1,1,1,2,3}
  216: {1,1,1,2,2,2}
  224: {1,1,1,1,1,4}
  240: {1,1,1,1,2,3}
		

Crossrefs

The partitions with these as Heinz numbers are counted by A108711.

An adjoint version is A347452, counted by A119620.

The unrounded version is A348384, counted by A035377.

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. A001105, A028982, A028260, A119899, A316413, A346703, A346704.

Programs

Mathematica

Select[Range[1000],Floor[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]))); }
isA348550(n) = (bigomega(n)==floor((2/3)*A056239(n))); \\ Antti Karttunen, Nov 08 2021

Formula

A001222(a(n)) = floor(2*A056239(a(n))/3).