A357979 Second MTF-transform of A000041. Replace prime(k) with prime(A357977(k)) in the prime factorization of n.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 31, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 59, 32, 33, 62, 35, 36, 37, 38, 39, 40, 127, 42, 79, 44, 45, 46, 47, 48, 49, 50, 93, 52, 53, 54, 55, 56, 57, 58, 211, 60, 61, 118, 63, 64, 65, 66

Offset: 1

Author

Gus Wiseman, Oct 24 2022

Keywords

• nonn
• mult

Comments

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. We define the MTF-transform as applying a function horizontally along a number's prime indices; see the Mathematica program.

Examples

			We have:
- 51 = prime(2) * prime(7),
- A357977(2) = 2,
- A357977(7) = 11,
- a(51) = prime(2) * prime(11) = 93.
		

Crossrefs

Other multiplicative sequences: A003961, A357852, A064988, A064989, A357980.

Applying the transformation only once gives A357977, strict A357978.

For primes instead of partition numbers we have A357983.

A000040 lists the primes.

A056239 adds up prime indices, row-sums of A112798.

Cf. A000041, A000720, A003964, A063834, A076610, A215366, A296150, A299201, A299202, A357975.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mtf[f_][n_]:=Product[If[f[i]==0,1,Prime[f[i]]],{i,primeMS[n]}];
Array[mtf[mtf[PartitionsP]],100]