A357977 Replace prime(k) with prime(A000041(k)) in the prime factorization of n.

1, 2, 3, 4, 5, 6, 11, 8, 9, 10, 17, 12, 31, 22, 15, 16, 47, 18, 79, 20, 33, 34, 113, 24, 25, 62, 27, 44, 181, 30, 263, 32, 51, 94, 55, 36, 389, 158, 93, 40, 547, 66, 761, 68, 45, 226, 1049, 48, 121, 50, 141, 124, 1453, 54, 85, 88, 237, 362, 1951, 60, 2659, 526

Offset: 1

Author

Gus Wiseman, Oct 23 2022

Keywords

• nonn
• mult

Comments

In the definition, taking A000041(k) instead of prime(A000041(k)) gives A299200.

Examples

			We have 35 = prime(3) * prime(4), so a(35) = prime(A000041(3)) * prime(A000041(4)) = prime(3) * prime(5) = 55.
		

Crossrefs

Applying the same transformation again gives A357979.

The strict version is A357978.

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

A000040 lists the primes.

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

Cf. A000041, A000720, A003964, A063834, A076610, A215366, A296150, A299201, A299202, 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[PartitionsP],100]

PARI

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