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Revision History for A349133

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A349133

Dirichlet convolution of A003415 with A003958, where A003415 is the arithmetic derivative and A003958 is fully multiplicative with a(p) = (p-1).
(history; published version)

#11 by Michael De Vlieger at Sun Nov 14 17:44:34 EST 2021

STATUS

reviewed

approved

#10 by Michel Marcus at Sun Nov 14 07:33:10 EST 2021

STATUS

proposed

reviewed

#9 by Antti Karttunen at Sun Nov 14 06:59:05 EST 2021

STATUS

editing

proposed

#8 by Antti Karttunen at Sun Nov 14 06:17:43 EST 2021

LINKS

Antti Karttunen, <a href="/A349133/b349133.txt">Table of n, a(n) for n = 1..20000</a>

STATUS

approved

editing

#7 by Susanna Cuyler at Tue Nov 09 18:40:44 EST 2021

STATUS

proposed

approved

#6 by Amiram Eldar at Tue Nov 09 13:24:45 EST 2021

STATUS

editing

proposed

#5 by Amiram Eldar at Tue Nov 09 13:24:40 EST 2021

MATHEMATICA

f1[p_, e_] := e/p; f2[p_, e_] := (p - 1)^e; a1[1] = 0; a1[n_] := n*Plus @@ (f1 @@@ FactorInteger[n]); a2[1] = 1; a2[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := DivisorSum[n, a1[#] * a2[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 09 2021 *)

STATUS

proposed

editing

#4 by Antti Karttunen at Tue Nov 09 13:05:25 EST 2021

STATUS

editing

proposed

#3 by Antti Karttunen at Tue Nov 09 02:13:32 EST 2021

CROSSREFS

Cf. A003415, A003958, A349129, A349130, A349131, A349132, A349173.

#2 by Antti Karttunen at Tue Nov 09 02:03:09 EST 2021

NAME

allocated for Antti KarttunenDirichlet convolution of A003415 with A003958, where A003415 is the arithmetic derivative and A003958 is fully multiplicative with a(p) = (p-1).

DATA

0, 1, 1, 5, 1, 8, 1, 17, 8, 12, 1, 32, 1, 16, 14, 49, 1, 43, 1, 52, 18, 24, 1, 100, 14, 28, 43, 72, 1, 87, 1, 129, 26, 36, 22, 151, 1, 40, 30, 168, 1, 119, 1, 112, 91, 48, 1, 276, 20, 103, 38, 132, 1, 194, 30, 236, 42, 60, 1, 323, 1, 64, 123, 321, 34, 183, 1, 172, 50, 183, 1, 443, 1, 76, 131, 192, 34, 215, 1, 472

OFFSET

1,4

FORMULA

a(n) = Sum_{d|n} A003415(d) * A003958(n/d).

For all n >= 1, a(n) <= A349173(n).

PROG

(PARI)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };

A349133(n) = sumdiv(n, d, A003415(d)*A003958(n/d));

CROSSREFS

Cf. A003415, A003958, A349129, A349130, A349131, A349132.

KEYWORD

allocated

nonn

AUTHOR

Antti Karttunen, Nov 09 2021

STATUS

approved

editing