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

1,2

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.

Links

Table of n, a(n) for n=1..66.

Example

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

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]

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.

Sequence in context: A194967 A023770 A371864 * A023797 A032951 A288139

Adjacent sequences: A357976 A357977 A357978 * A357980 A357981 A357982

Keyword

nonn,mult

Author

Gus Wiseman, Oct 24 2022

Status

approved