A065769 Prime cascade: multiplicative with a(prime(m)^k) = prime(m-1) * prime(m)^(k-1).

1, 1, 2, 2, 3, 2, 5, 4, 6, 3, 7, 4, 11, 5, 6, 8, 13, 6, 17, 6, 10, 7, 19, 8, 15, 11, 18, 10, 23, 6, 29, 16, 14, 13, 15, 12, 31, 17, 22, 12, 37, 10, 41, 14, 18, 19, 43, 16, 35, 15, 26, 22, 47, 18, 21, 20, 34, 23, 53, 12, 59, 29, 30, 32, 33, 14, 61, 26, 38, 15, 67, 24, 71, 31, 30, 34

Offset: 1

Author

Henry Bottomley, Nov 19 2001

Keywords

• mult
• nonn
• easy

Examples

			a(63) = a(3^2*7^1) = a(3^2)*a(7^1) = (2*3^1)*(5*7^0) = 30.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A000040, A000079, A003557, A007947, A064989, A065770, A151799.

Programs

Mathematica

f[p_, e_] := If[p == 2, 1, NextPrime[p, -1]]*p^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 02 2023 *)

PARI

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); };
A065769(n) = { my(f=factor(n>>valuation(n,2))[, 1]~); (A003557(n) * factorback(vector(#f,i,precprime(f[i]-1)))); }; \\ Antti Karttunen, Dec 31 2017

Scheme

(define (A065769 n) (* (A003557 n) (A064989 (A007947 n)))) ;; Antti Karttunen, Dec 31 2017

Formula

a(A000040(n)) = A000040(n-1);

a(A000079(n)) = A000079(n-1);

a(A002110(n)) = A002110(n-1).

a(n) = A003557(n) * A064989(A007947(n)). - Antti Karttunen, Dec 31 2017

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - (p - q(p))/p^2) = 0.526221951..., where q(2) = 1, and q(p) = A151799(p) for an odd prime p. - Amiram Eldar, Nov 02 2023