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A065769
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Prime cascade: multiplicative with a(prime(m)^k) = prime(m-1) * prime(m)^(k-1).
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4
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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
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OFFSET
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1,3
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LINKS
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FORMULA
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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
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EXAMPLE
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a(63) = a(3^2*7^1) = a(3^2)*a(7^1) = (2*3^1)*(5*7^0) = 30.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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mult,nonn,easy
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AUTHOR
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STATUS
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approved
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