A357980 Replace prime(k) with prime(A000720(k)) in the prime factorization of n, assuming prime(0) = 1.

1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 6, 1, 7, 4, 7, 3, 6, 5, 7, 2, 9, 5, 8, 3, 7, 6, 11, 1, 10, 7, 9, 4, 11, 7, 10, 3, 13, 6, 13, 5, 12, 7, 13, 2, 9, 9, 14, 5, 13, 8, 15, 3, 14, 7, 17, 6, 17, 11, 12, 1, 15, 10, 19, 7, 14, 9, 19, 4, 19, 11, 18, 7, 15, 10

Offset: 1

Author

Gus Wiseman, Oct 24 2022

Keywords

• nonn
• mult

Comments

In the definition, taking A000720(k) in place of prime(A000720(k)) gives A357984.

Examples

			We have 90 = prime(1) * prime(2)^2 * prime(3), so a(90) = prime(0) * prime(1)^2 * prime(2) = 12.
		

Crossrefs

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

The version for p instead of pi is A357977, strict A357978.

The triangular version is A357984.

A000040 lists the prime numbers.

A000720 = PrimePi.

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

Cf. A063834, A066207, A076610, A215366, A296150, A357975, A357983.

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[PrimePi],100]

PARI

myprime(n) = if (n==0, 1, prime(n));
a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = myprime(primepi(primepi(f[k,1])))); factorback(f); \\ Michel Marcus, Oct 25 2022