A371285 Heinz number of the multiset union of the divisor sets of each prime index of n.

1, 2, 6, 4, 10, 12, 42, 8, 36, 20, 22, 24, 390, 84, 60, 16, 34, 72, 798, 40, 252, 44, 230, 48, 100, 780, 216, 168, 1914, 120, 62, 32, 132, 68, 420, 144, 101010, 1596, 2340, 80, 82, 504, 4386, 88, 360, 460, 5170, 96, 1764, 200, 204, 1560, 42294, 432, 220, 336

Offset: 1

Author

Gus Wiseman, Mar 21 2024

Keywords

• nonn
• mult

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.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The prime indices of 105 are {2,3,4}, with divisor sets {{1,2},{1,3},{1,2,4}}, with multiset union {1,1,1,2,2,3,4}, with Heinz number 2520, so a(105) = 2520.
The terms together with their prime indices begin:
          1: {}
          2: {1}
          6: {1,2}
          4: {1,1}
         10: {1,3}
         12: {1,1,2}
         42: {1,2,4}
          8: {1,1,1}
         36: {1,1,2,2}
         20: {1,1,3}
         22: {1,5}
         24: {1,1,1,2}
        390: {1,2,3,6}
         84: {1,1,2,4}
         60: {1,1,2,3}
         16: {1,1,1,1}
         34: {1,7}
         72: {1,1,1,2,2}
		

Crossrefs

Product of A275700 applied to each prime index.

The squarefree case is also A275700.

The sorted version is A371286.

A000005 counts divisors.

A001221 counts distinct prime factors.

A027746 lists prime factors, A112798 indices, length A001222.

A355731 counts choices of a divisor of each prime index, firsts A355732.

A355741 counts choices of a prime factor of each prime index.

Cf. A000720, A003963, A005179, A007416, A034729, A054973, A056239, A321898, A370820, A371165, A371181, A371284, A371288.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Prime/@Join@@Divisors/@prix[n],{n,100}]

Formula

If n = prime(x_1)*...*prime(x_k) then a(n) = A275700(x_1)*...*A275700(x_k).