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A295879
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Multiplicative with a(p) = 1, a(p^e) = prime(e-1) if e > 1.
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4
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1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 5, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 7, 1, 2, 2, 4, 1, 1, 1, 3, 1, 1, 1, 6, 1, 1, 1, 5, 1, 1, 1, 2, 2, 1, 1, 3, 2, 1, 1, 2, 3, 2, 1, 13
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OFFSET
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1,4
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COMMENTS
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This sequence can be used as a filter. It matches at least to the following sequences related to the counting of various non-unitary prime divisors:
For all i, j:
An encoding of the prime signature of A057521(n), the powerful part of n. - Peter Munn, Apr 06 2024
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LINKS
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FORMULA
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a(1) = 1; for n>1, if n = Product prime(i)^e(i), then a(n) = Product A008578(e(i)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/p^2 + Sum_{k>=1} (prime(k+1)-prime(k))/p^(k+2)) = 2.208... . - Amiram Eldar, Nov 18 2022
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MATHEMATICA
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Array[Apply[Times, FactorInteger[#] /. {p_, e_} /; p > 0 :> Which[p == 1, 1, e == 1, 1, True, Prime[e - 1]]] &, 128] (* Michael De Vlieger, Nov 29 2017 *)
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PROG
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(Scheme, with memoization-macro definec)
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, 1, prime(f[i, 2]-1))); } \\ Amiram Eldar, Nov 18 2022
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CROSSREFS
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Differs from A000688 for the first time at n=128, where a(128) = 13, while A000688(128) = 15.
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KEYWORD
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nonn,mult,easy,changed
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AUTHOR
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STATUS
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approved
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