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

1,2

Comments

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

Links

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

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A171580 A246796 A177235 * A233511 A205793 A178431

Adjacent sequences: A079704 A079705 A079706 * A079708 A079709 A079710

Keyword

nonn,easy,mult

Author

Reinhard Zumkeller, Jan 31 2003

Status

approved