A357983 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

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

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..10000

EXAMPLE

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.

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]

PROG

(PARI) \\ here b(n) is A064988.

b(n)=my(f=factor(n)); prod(i=1, #f~, my([p, e]=f[i, ]); prime(p)^e);

a(n)=my(f=factor(n)); prod(i=1, #f~, my([p, e]=f[i, ]); prime(b(primepi(p)))^e); \\ Andrew Howroyd, Nov 17 2025

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.

Sequence in context: A010078 A074639 A319525 * A338459 A343035 A002314

Adjacent sequences: A357980 A357981 A357982 * A357984 A357985 A357986

KEYWORD

nonn,mult

AUTHOR

Gus Wiseman, Oct 24 2022

STATUS

approved