A357983 Second MTF-transform of the primes (A000040). Replace prime(k) with prime(A064988(k)) in the prime factorization of n.

1, 2, 5, 4, 11, 10, 23, 8, 25, 22, 31, 20, 47, 46, 55, 16, 59, 50, 103, 44, 115, 62, 97, 40, 121, 94, 125, 92, 137, 110, 127, 32, 155, 118, 253, 100, 197, 206, 235, 88, 179, 230, 233, 124, 275, 194, 257, 80, 529, 242, 295, 188, 419, 250, 341, 184, 515, 274

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 shifting a number's prime indices along a function; see the Mathematica program.

Examples

			First, we have
- 4 = prime(1) * prime(1),
- A000040(1) = 2,
- A064988(4) = prime(2) * prime(2) = 9.
Similarly, A064988(3) = 5. Next,
- 35 = prime(3) * prime(4),
- A064988(3) = 5,
- A064988(4) = 9,
- a(35) = prime(5) * prime(9) = 253.
		

Crossrefs

Other multiplicative sequences: A003961, A357852, A064989, A357977, A357980.

Applying the transformation only once gives A064988.

The union is A076610 (numbers whose prime indices are themselves prime).

For partition numbers instead of primes we have A357979.

A000040 lists the primes.

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

Cf. A000720, A003964, A063834, 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[Prime]],100]