1,2

From Antti Karttunen, Jan 30 2022: (Start)

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.

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. (End)

Antti Karttunen, Table of n, a(n) for n = 1..20000

a(n) = A344999(n) + A345043(n).

A003415[n_] := If[n<2, 0, Module[{f = FactorInteger[n]}, If[PrimeQ[n], 1, Total[n*f[[All, 2]]/f[[All, 1]]]]]];

A003557[n_] := n*Times@@(1/FactorInteger[n][[All, 1]]);

A342001[n_] := A003415[n]/A003557[n];

A344753[n_] := Sum[d+If[SquareFreeQ[n/d], d, 0], {d, Most[Divisors[n]]}];

a[n_] := A342001[n]*A344753[n];

Array[a, 70] (* Jean-François Alcover, Jun 12 2021 *)

(PARI)

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

A003557(n) = n/factorback(factorint(n)[, 1]);

A342001(n) = A003415(n) / A003557(n);

A344753(n) = sumdiv(n, d, (d<n)*(d+(issquarefree(n/d) * d)));

A344998(n) = A342001(n) * A344753(n);

Cf. A003415, A003557, A342001, A344753, A345043.

Cf. A345003 (gives k for which a(k) = A344999(k)), A345004, A345005.

Sequence in context: A319880 A133631 A137450 * A321415 A319692 A308693

Adjacent sequences: A344995 A344996 A344997 * A344999 A345000 A345001

Antti Karttunen, Jun 05 2021

approved