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

%I A079707 #11 Nov 29 2022 02:43:07
%S A079707 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,
%T A079707 23,12,29,32,14,26,15,16,31,34,22,24,37,20,41,28,12,38,43,32,25,18,26,
%U A079707 44,47,16,21,40,34,46,53,24,59,58,20,64,33,28,61,52,38,30,67,32,71,62,18
%N A079707 In prime factorization of n replace odd primes with their prime predecessor.
%C A079707 Result after A061395(n)-1 iterations = A061142(n).
%H A079707 Amiram Eldar, <a href="/A079707/b079707.txt">Table of n, a(n) for n = 1..10000</a>
%F A079707 a(n) <= n; a(n) < n iff n > 1 is odd; a(n) = n iff n = 2^k.
%F A079707 A001222(a(n)) = A001222(n).
%F A079707 For 3-smooth numbers: a(2^i * 3^j) = 2^(i+j).
%F A079707 Multiplicative with 2->2 and prime(k)->prime(k-1), k>1.
%F A079707 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
%t A079707 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 *)
%o A079707 (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
%Y A079707 Cf. A001222, A005408, A000079, A003586, A003961, A061142, A061395, A064989, A151799.
%K A079707 nonn,easy,mult
%O A079707 1,2
%A A079707 _Reinhard Zumkeller_, Jan 31 2003