A191614 -id:A191614 - cp's OEIS frontend

A343068 Multiplicative with a(p^e) = e*a(p-1).

1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 2, 2, 2, 1, 2, 4, 4, 2, 2, 4, 1, 2, 2, 3, 4, 2, 3, 2, 2, 2, 2, 5, 2, 4, 2, 4, 4, 2, 2, 6, 6, 1, 1, 4, 4, 2, 2, 4, 2, 4, 4, 4, 4, 3, 4, 3, 2, 2, 2, 4, 4, 2, 2, 6, 4, 2, 2, 8, 2, 2, 2, 6, 6, 4, 4, 4, 2, 2, 2, 8, 4, 6, 6, 2, 8, 1

Offset: 1

José María Grau Ribas, Apr 04 2021

			a(2) = a(2-1) = 1; a(3) = a(3-1) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A279513, A191614.

a:= proc(n) option remember; `if`(n=1, 1,
      mul(i[2]*a(i[1]-1), i=ifactors(n)[2]))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 04 2021

a[1] = 1; a[p_,s_]:= a[p, s]=s a[p-1];
a[n_]:=a[n]= Module[{aux = FactorInteger[n]},Product[a[aux[[i, 1]],aux[[i, 2]]], {i, Length[aux]}]]; Table[a[n],{n,100}]

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = f[k,2]*a(f[k,1]-1); f[k,2] = 1); factorback(f); \\ Michel Marcus, Apr 23 2021

a(2^n) = n*a(2-1) = n.

A326304 Multiplicative with a(p^k) = a(p-1)^k + 1 for any k > 0 and any prime number p.

Original entry on oeis.org

1, 2, 3, 2, 3, 6, 7, 2, 5, 6, 7, 6, 7, 14, 9, 2, 3, 10, 11, 6, 21, 14, 15, 6, 5, 14, 9, 14, 15, 18, 19, 2, 21, 6, 21, 10, 11, 22, 21, 6, 7, 42, 43, 14, 15, 30, 31, 6, 37, 10, 9, 14, 15, 18, 21, 14, 33, 30, 31, 18, 19, 38, 35, 2, 21, 42, 43, 6, 45, 42, 43, 10

Offset: 1

Rémy Sigrist, Oct 17 2019

The sequence is well defined as computing a(p^k) involves terms of the form a(q) with q < p.

The fixed points are the divisors of 1806 = 2 * 3 * 7 * 43; they correspond to the first 16 terms of A191614.

			a(2) = a(1) + 1 = 1 + 1 = 2.
a(3) = a(2) + 1 = 2 + 1 = 3.
a(7) = a(6) + 1 = a(2)*a(3) + 1 = 2 * 3 + 1 = 7.
a(43) = a(42) + 1 = a(2)*a(3)*a(7) + 1 = 2*3*7 + 1 = 43.

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

Cf. A191614, A309243.

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

cp's OEIS Frontend

A343068 Multiplicative with a(p^e) = e*a(p-1).

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