Revision History for A348507
(Underlined text is an addition;
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Showing entries 1-10
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#44 by Michael De Vlieger at Thu Nov 11 19:30:45 EST 2021
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#43 by Michel Marcus at Thu Nov 11 02:52:48 EST 2021
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#42 by Antti Karttunen at Wed Nov 10 17:16:28 EST 2021
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#41 by Antti Karttunen at Wed Nov 10 17:15:39 EST 2021
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| PROG
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A348507(n) = { my(s=0, m=1, spf); while(n>1, spf = A020639(n); n /= spf; s += m*n; m *= (1+spf)); (s); }; \\ (Compare this with my similar programs programgiven forin A003415 and forin A322582) - Antti Karttunen, Nov 06 2021
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#40 by Antti Karttunen at Wed Nov 10 17:14:43 EST 2021
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#39 by Antti Karttunen at Wed Nov 10 17:12:56 EST 2021
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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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| FORMULA
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For all n>= >= 1, A322582(n) <= A003415(n) <= a(n).
For n > 1, a(n) = a(A032742(n))*(1+A020639(n)) + A032742(n). [See the comments above, and compare this with Reinhard Zumkeller's May 09 2011 recursive formula for A003415 !]]
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#38 by Antti Karttunen at Wed Nov 10 16:56:40 EST 2021
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#37 by Antti Karttunen at Wed Nov 10 16:54:01 EST 2021
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| CROSSREFS
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Cf. A001065, A003415, A003959, A020639, A032742, A348029, A348508, A348929 [= gcd(n,a(n))], A348950, A348970, A348971 (Möbius transform)..
Cf. A348971 (Möbius transform), A349139, A349140, A349141, A349142, A349143 (other Dirichlet convolutions).
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#36 by Antti Karttunen at Wed Nov 10 16:50:44 EST 2021
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| CROSSREFS
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Cf. A001065, A003415, A003959, A020639, A032742, A348029, A348508, A348929 [= gcd(n,a(n))], A348950, A348970, A348971 (Möbius transform).
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#35 by Antti Karttunen at Sun Nov 07 06:42:31 EST 2021
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| CROSSREFS
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Cf. A001065, A003415, A003959, A020639, A032742, A348029, A348508, A348929 [= gcd(n,a(n))], A348970, A348971 (Möbius transform).
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