A079707 In prime factorization of n replace odd primes with their prime predecessor.

1, 2, 2, 4, 3, 4, 5, 8, 4, 6, 7, 8, 11, 10, 6, 16, 13, 8, 17, 12, 10, 14, 19, 16, 9, 22, 8, 20, 23, 12, 29, 32, 14, 26, 15, 16, 31, 34, 22, 24, 37, 20, 41, 28, 12, 38, 43, 32, 25, 18, 26, 44, 47, 16, 21, 40, 34, 46, 53, 24, 59, 58, 20, 64, 33, 28, 61, 52, 38, 30, 67, 32, 71, 62, 18

Offset: 1

Author

Reinhard Zumkeller, Jan 31 2003

Keywords

• nonn
• easy
• mult

Comments

Result after A061395(n)-1 iterations = A061142(n).

Links

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

Crossrefs

Cf. A001222, A005408, A000079, A003586, A003961, A061142, A061395, A064989, A151799.

Programs

Mathematica

f[p_, e_] := If[p == 2, 2, NextPrime[p, -1]]^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 29 2022 *)

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, f[i,1], precprime(f[i,1]-1))^f[i,2]);} \\ Amiram Eldar, Nov 29 2022

Formula

a(n) <= n; a(n) < n iff n > 1 is odd; a(n) = n iff n = 2^k.

A001222(a(n)) = A001222(n).

For 3-smooth numbers: a(2^i * 3^j) = 2^(i+j).

Multiplicative with 2->2 and prime(k)->prime(k-1), k>1.

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime > 2} ((p^2-p)/(p^2 - prevprime(p))) = 0.3310558934..., where prevprime is A151799. - Amiram Eldar, Nov 29 2022