cp's OEIS Frontend

A344998 a(n) = A342001(n) * A344753(n).

%I A344998 #20 Jan 30 2022 09:51:44
%S A344998 0,2,2,10,2,60,2,33,14,112,2,224,2,180,144,92,2,273,2,456,220,364,2,
%T A344998 660,22,480,66,768,2,2604,2,235,420,760,312,910,2,924,544,1394,2,4428,
%U A344998 2,1632,780,1300,2,1736,30,747,840,2184,2,1080,544,2392,1012,1984,2,8832,2,2244,1258,570,684,9516,2,3528,1404,8732
%N A344998 a(n) = A342001(n) * A344753(n).
%C A344998 From _Antti Karttunen_, Jan 30 2022: (Start)
%C A344998 In addition to 2's that occur on primes, there are also other duplicates, for example, a(39) = a(55) = 544, a(51) = a(91) = 840, a(65) = a(77) = 684, a(343) = a(6241) = 318, a(95) = a(119) = a(143) = 1200 and a(155) = a(203) = a(299) = a(323) = 2664. Note how, apart from 343 = 7^3 and 6241 = 79^2, the duplicate positions in above cases are all squarefree semiprimes, and how the sum of the two prime factors in those cases are equal. E.g. 95 = 5*19, 119 = 7*17, 143 = 11*13, with 5+19 = 7+17 = 11+13 = 24.
%C A344998 Indeed, for squarefree semiprimes pq, A342001(pq) = A003415(pq) = p+q and A344753(pq) = 2*A001065(pq) = 2*(1+p+q), and therefore the product A342001(pq) * A344753(pq) depends only on the sum of p+q.
%C A344998 (End)
%H A344998 Antti Karttunen, <a href="/A344998/b344998.txt">Table of n, a(n) for n = 1..20000</a>
%F A344998 a(n) = A342001(n) * A344753(n).
%F A344998 a(n) = A344999(n) + A345043(n).
%t A344998 A003415[n_] := If[n<2, 0, Module[{f = FactorInteger[n]}, If[PrimeQ[n], 1, Total[n*f[[All, 2]]/f[[All, 1]]]]]];
%t A344998 A003557[n_] := n*Times@@(1/FactorInteger[n][[All, 1]]);
%t A344998 A342001[n_] := A003415[n]/A003557[n];
%t A344998 A344753[n_] := Sum[d+If[SquareFreeQ[n/d], d, 0], {d, Most[Divisors[n]]}];
%t A344998 a[n_] := A342001[n]*A344753[n];
%t A344998 Array[a, 70] (* _Jean-François Alcover_, Jun 12 2021 *)
%o A344998 (PARI)
%o A344998 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A344998 A003557(n) = (n/factorback(factorint(n)[, 1]));
%o A344998 A342001(n) = (A003415(n) / A003557(n));
%o A344998 A344753(n) = sumdiv(n, d, (d<n)*(d+(issquarefree(n/d) * d)));
%o A344998 A344998(n) = (A342001(n) * A344753(n));
%Y A344998 Cf. A003415, A003557, A342001, A344753, A345043.
%Y A344998 Cf. A345003 [gives k for which a(k) = A344999(k)], A345004, A345005.
%K A344998 nonn
%O A344998 1,2
%A A344998 _Antti Karttunen_, Jun 05 2021