A347452 Heinz numbers of integer partitions whose sum is 3/2 their length, rounded down.

1, 2, 6, 12, 36, 40, 72, 80, 216, 224, 240, 432, 448, 480, 1296, 1344, 1408, 1440, 1600, 2592, 2688, 2816, 2880, 3200, 6656, 7776, 8064, 8448, 8640, 8960, 9600, 13312, 15552, 16128, 16896, 17280, 17920, 19200, 34816, 39936, 46656, 48384, 50176, 50688, 51840

Offset

1,2

Comments

Also numbers whose sum of prime indices is 3/2 their number, rounded down, where a prime index of n is a number m such that prime(m) divides n.

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.

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

Counting the partitions with these Heinz numbers gives A119620 with zeros interspersed every three terms.

Links

Table of n, a(n) for n=1..45.

Example

The initial terms and their prime indices:
      1: {}
      2: {1}
      6: {1,2}
     12: {1,1,2}
     36: {1,1,2,2}
     40: {1,1,1,3}
     72: {1,1,1,2,2}
     80: {1,1,1,1,3}
    216: {1,1,1,2,2,2}
    224: {1,1,1,1,1,4}
    240: {1,1,1,1,2,3}
    432: {1,1,1,1,2,2,2}
    448: {1,1,1,1,1,1,4}
    480: {1,1,1,1,1,2,3}
   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}

Mathematica

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

Crossrefs

Counting terms by Heinz weight (in A032766) gives A119620.

An adjoint version is A348550, counted by A108711.

A000041 counts 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 wiggly permutations of prime factors.

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

Sequence in context: A204512 A099576 A303479 * A307015 A026549 A120766

Adjacent sequences: A347449 A347450 A347451 * A347453 A347454 A347455

Keyword

nonn

Author

Gus Wiseman, Oct 28 2021

Status

approved