A225395 -id:A225395

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A328880

If n = Product (p_j^k_j) then a(n) = Product (a(pi(p_j)) + 1), where pi = A000720, with a(1) = 1.

+10
3

1, 2, 3, 2, 4, 6, 3, 2, 3, 8, 5, 6, 7, 6, 12, 2, 4, 6, 3, 8, 9, 10, 4, 6, 4, 14, 3, 6, 9, 24, 6, 2, 15, 8, 12, 6, 7, 6, 21, 8, 8, 18, 7, 10, 12, 8, 13, 6, 3, 8, 12, 14, 3, 6, 20, 6, 9, 18, 5, 24, 7, 12, 9, 2, 28, 30, 4, 8, 12, 24, 9, 6, 10, 14, 12, 6, 15, 42, 11, 8

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

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..80.

Index entries for sequences computed from indices in prime factorization

FORMULA

a(n) = a(prime(n)) - 1.

EXAMPLE

a(36) = 6 because 36 = 2^2 * 3^2 = prime(1)^2 * prime(2)^2 and (a(1) + 1) * (a(2) + 1) = (1 + 1) * (2 + 1) = 6.

MATHEMATICA

a[1] = 1; a[n_] := Times @@ (a[PrimePi[#[[1]]]] + 1 & /@ FactorInteger[n]); Table[a[n], {n, 1, 80}]

PROG

(PARI) a(n)={my(f=factor(n)[, 1]); prod(i=1, #f, 1 + a(primepi(f[i])))} \\ Andrew Howroyd, Oct 29 2019

CROSSREFS

Cf. A000720, A048250, A156061, A184160, A225395, A282446, A328879.

KEYWORD

nonn,mult

AUTHOR

Ilya Gutkovskiy, Oct 29 2019

STATUS

approved

A301315

Multiplicative with a(p^e) = prime(p) ^ a(e) (where prime(k) denotes the k-th prime number).

+10
1

1, 3, 5, 27, 11, 15, 17, 243, 125, 33, 31, 135, 41, 51, 55, 7625597484987, 59, 375, 67, 297, 85, 93, 83, 1215, 1331, 123, 3125, 459, 109, 165, 127, 177147, 155, 177, 187, 3375, 157, 201, 205, 2673, 179, 255, 191, 837, 1375, 249, 211, 38127987424935, 4913, 3993

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

OFFSET

1,2

COMMENTS

This sequence is a recursive version of A064988.

This sequence is injective (all terms are distinct).

LINKS

Rémy Sigrist, Table of n, a(n) for n = 1..10000

FORMULA

A225395(a(n)) = n.

A279690(a(n)) = A279690(n).

MATHEMATICA

Fold[Function[{a, n}, Append[a, Times @@ Map[Prime[#1]^a[[#2]] & @@ # &, FactorInteger@ n]]], {1}, Range[2, 50]] (* Michael De Vlieger, Mar 19 2018 *)

PROG

(PARI) a(n) = my (f=factor(n)); prod (i=1, #f~, prime(f[i, 1])^a(f[i, 2]))