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A348507
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a(n) = A003959(n) - n, where A003959 is multiplicative with a(p^e) = (p+1)^e.
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14
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0, 1, 1, 5, 1, 6, 1, 19, 7, 8, 1, 24, 1, 10, 9, 65, 1, 30, 1, 34, 11, 14, 1, 84, 11, 16, 37, 44, 1, 42, 1, 211, 15, 20, 13, 108, 1, 22, 17, 122, 1, 54, 1, 64, 51, 26, 1, 276, 15, 58, 21, 74, 1, 138, 17, 160, 23, 32, 1, 156, 1, 34, 65, 665, 19, 78, 1, 94, 27, 74, 1, 360, 1, 40, 69, 104, 19, 90, 1, 406, 175, 44, 1, 204
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OFFSET
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1,4
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COMMENTS
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a(p*(n/p)) - (n/p) = (p+1)*a(n/p) holds for all prime divisors p of n, which can be seen by expanding the left hand side as (A003959(p*(n/p)) - (p*(n/p))) - (n/p) = (p+1)*A003959(n/p)-((p+1)*(n/p)) = (p+1)*(A003959(n/p)-(n/p)) = (p+1)*a(n/p). This implies that a(n) >= A003415(n) for all n. (See also comments in A348970). - Antti Karttunen, Nov 06 2021
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LINKS
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FORMULA
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(End)
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MATHEMATICA
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f[p_, e_] := (p + 1)^e; a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - n; Array[a, 100] (* Amiram Eldar, Oct 30 2021 *)
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PROG
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(PARI)
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
(PARI)
A020639(n) = if(1==n, n, (factor(n)[1, 1]));
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CROSSREFS
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Cf. A001065, A003415, A003959, A020639, A032742, A348029, A348508, A348929 [= gcd(n,a(n))], A348950, A348970.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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