A119790 a(n) is the sum of the positive integers each of which is <= n and is divisible by exactly one prime dividing n (but is coprime to every other prime dividing n). (a(1) = 1).

1, 2, 3, 6, 5, 9, 7, 20, 18, 25, 11, 36, 13, 49, 45, 72, 17, 81, 19, 100, 84, 121, 23, 144, 75, 169, 135, 196, 29, 210, 31, 272, 198, 289, 175, 324, 37, 361, 273, 400, 41, 420, 43, 484, 405, 529, 47, 576, 196, 625, 459, 676, 53, 729, 385, 784, 570, 841, 59, 840, 61, 961

Offset

1,2

Comments

a(n) is divisible by A026741(n). - Robert Israel, Oct 01 2017

The number of these integers is A116512(n), for n >= 2. - Amiram Eldar, Nov 20 2025

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Formula

From Amiram Eldar, Nov 20 2025: (Start)

a(n) = Sum_{k=1..n, gcd(k,n) is in A246655} k = Sum_{k=1..n} A069513(gcd(k,n)) * k, for n >= 2.

a(n) = (n + f(n)) * A116512(n) / 2, where f(n) = p if n = p^k is a prime power (where p is prime) and f(n) = 0 otherwise, for n >= 2.

Dirichlet g.f.: 1 + (zeta(s-2)/zeta(s-1) + 1) * A(s-1) / 2, where A(s) = Sum_{n>=1} A069513(n)/n^s = Sum_{p prime} 1/(p^s-1).

Sum_{k=1..n} a(k) ~ c * n^3 / 6, where c = Sum_{p prime} (1/(p^2-1)) / zeta(2) = A154945 / A013661 = 0.3353893075569103736099... . (End)

Example

12 is divisible by 2 and 3. The positive integers which are <= 12 and which are divisible by 2 or 3 but not by both 2 and 3 are: 2, 3, 4, 8, 9, 10. a(12) = the sum of these integers, which is 36.

Maple

f:= proc(n) local P;
  P:= convert(numtheory:-factorset(n), list);
  convert(select(k -> nops(select(p->k mod p = 0, P))=1, [$2..n]), `+`)
end proc:
1, seq(f(n), n=2..100); # Robert Israel, Oct 01 2017

Mathematica

Table[Total@ Select[Range@ n, Function[k, Total@ Boole@ Map[Divisible[k, #] &, FactorInteger[n][[All, 1]]] == 1]], {n, 62}] (* Michael De Vlieger, Oct 01 2017 *)
a[n_] := Module[{f = FactorInteger[n], p}, p = f[[;; , 1]]; (n + If[Length[p] == 1, p[[1]], 0]) * n * Times @@ (1 - 1/p) * Total[1/(p - 1)] / 2]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Nov 20 2025 *)

Prog

(PARI) a(n) = if(n == 1, 1 , my(f = factor(n), p = f[, 1]); (n + if(#p == 1, p[1], 0)) * n * prod(i = 1, #p, 1 - 1/p[i]) * sum(i = 1, #p, 1/(p[i] - 1)) / 2); \\ Amiram Eldar, Nov 20 2025

Crossrefs

Cf. A013661, A026741, A069513, A116512, A119794, A120499, A154945.

Sequence in context: A070038 A328638 A327420 * A272214 A319785 A080029

Adjacent sequences: A119787 A119788 A119789 * A119791 A119792 A119793

Keyword

nonn,easy

Author

Leroy Quet, Jul 30 2006

Extensions

Corrected and extended by Joshua Zucker, Aug 12 2006

Status

approved