A346701 Heinz number of the odd bisection (odd-indexed parts) of the integer partition with Heinz number n.

1, 2, 3, 2, 5, 3, 7, 4, 3, 5, 11, 6, 13, 7, 5, 4, 17, 6, 19, 10, 7, 11, 23, 6, 5, 13, 9, 14, 29, 10, 31, 8, 11, 17, 7, 6, 37, 19, 13, 10, 41, 14, 43, 22, 15, 23, 47, 12, 7, 10, 17, 26, 53, 9, 11, 14, 19, 29, 59, 10, 61, 31, 21, 8, 13, 22, 67, 34, 23, 14, 71

Offset: 1

Author

Gus Wiseman, Aug 03 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 partition (2,2,2,1,1) has Heinz number 108 and odd bisection (2,2,1) with Heinz number 18, so a(108) = 18.
The partitions (3,2,2,1,1), (3,2,2,2,1), (3,3,2,1,1) have Heinz numbers 180, 270, 300 and all have odd bisection (3,2,1) with Heinz number 30, so a(180) = a(270) = a(300) = 30.
		

Crossrefs

Positions of last appearances are A000290 without the first term 0.

Positions of primes are A037143 (complement: A033942).

The even version is A329888.

Positions of first appearances are A342768.

The sum of prime indices of a(n) is A346699(n), non-reverse: A346697.

The non-reverse version is A346703.

The even non-reverse version is A346704.

A001221 counts distinct prime factors.

A001222 counts all prime factors.

A056239 adds up prime indices, row sums of A112798.

A103919 counts partitions by sum and alternating sum, reverse A344612.

A209281 (shifted) adds up the odd bisection of standard compositions.

A316524 gives the alternating sum of prime indices, reverse A344616.

A325534 counts separable partitions, ranked by A335433.

A325535 counts inseparable partitions, ranked by A335448.

A344606 counts alternating permutations of prime indices.

A344617 gives the sign of the alternating sum of prime indices.

A346700 gives the sum of the even bisection of reversed prime indices.

Cf. A025047, A027187, A027193, A053738, A106356, A277103, A341446, A344653, A345957, A345958, A345959, A346698, A346702.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Prime/@First/@Partition[Append[Reverse[primeMS[n]],0],2],{n,100}]

Formula

a(n) * A329888(n) = n.

A056239(a(n)) = A346699(n).