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

1,2

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.

Links

Table of n, a(n) for n=1..51.

Formula

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

Example

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}

Mathematica

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

Prog

(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

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.

Sequence in context: A085780 A166047 A310141 * A331841 A223999 A107084

Adjacent sequences: A348547 A348548 A348549 * A348551 A348552 A348553

Keyword

nonn

Author

Gus Wiseman, Nov 05 2021

Status

approved