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%I #21 Nov 05 2023 11:35:46
%S 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,
%T 6,29,16,14,13,15,12,31,17,22,12,37,10,41,14,18,19,43,16,35,15,26,22,
%U 47,18,21,20,34,23,53,12,59,29,30,32,33,14,61,26,38,15,67,24,71,31,30,34
%N Prime cascade: multiplicative with a(prime(m)^k) = prime(m-1) * prime(m)^(k-1).
%H Antti Karttunen, <a href="/A065769/b065769.txt">Table of n, a(n) for n = 1..16384</a>
%F a(A000040(n)) = A000040(n-1);
%F a(A000079(n)) = A000079(n-1);
%F a(A002110(n)) = A002110(n-1).
%F a(n) = A003557(n) * A064989(A007947(n)). - _Antti Karttunen_, Dec 31 2017
%F 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
%e a(63) = a(3^2*7^1) = a(3^2)*a(7^1) = (2*3^1)*(5*7^0) = 30.
%t 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 *)
%o (PARI)
%o A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); };
%o 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
%o (Scheme) (define (A065769 n) (* (A003557 n) (A064989 (A007947 n)))) ;; _Antti Karttunen_, Dec 31 2017
%Y Cf. A000040, A000079, A003557, A007947, A064989, A065770, A151799.
%K mult,nonn,easy
%O 1,3
%A _Henry Bottomley_, Nov 19 2001