# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a349170 Showing 1-1 of 1 %I A349170 #15 Nov 15 2021 13:02:54 %S A349170 1,5,7,19,11,35,15,65,37,55,23,133,27,75,77,211,35,185,39,209,105,115, %T A349170 47,455,91,135,175,285,59,385,63,665,161,175,165,703,75,195,189,715, %U A349170 83,525,87,437,407,235,95,1477,169,455,245,513,107,875,253,975,273,295,119,1463,123,315,555,2059,297,805,135,665 %N A349170 a(n) = Sum_{d|n} d * A003959(n/d), where A003959 is fully multiplicative with a(p) = (p+1). %C A349170 Dirichlet convolution of A003959 with the identity function, A000027. %C A349170 Dirichlet convolution of sigma (A000203) with A003968. %H A349170 Antti Karttunen, Table of n, a(n) for n = 1..20000 %F A349170 a(n) = Sum_{d|n} d * A003959(n/d). %F A349170 a(n) = Sum_{d|n} A349171(d). %F A349170 a(n) = Sum_{d|n} A000203(d) * A003968(n/d). %F A349170 a(n) = A038040(n) + A349140(n). %F A349170 For all n >= 1, a(n) >= A349129(n) >= A349130(n). %F A349170 Multiplicative with a(p^e) = (p+1)^(e+1) - p^(e+1). - _Amiram Eldar_, Nov 09 2021 %t A349170 f[p_, e_] := (p + 1)^(e + 1) - p^(e + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 09 2021 *) %o A349170 (PARI) %o A349170 A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); }; %o A349170 A349170(n) = sumdiv(n,d,d*A003959(n/d)); %Y A349170 Cf. A000027, A000203, A003959, A003968, A038040, A349129, A349130, A349140, A349171 (Möbius transform), A349172, A349173. %K A349170 nonn,mult %O A349170 1,2 %A A349170 _Antti Karttunen_, Nov 09 2021 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE