A389170 a(n) = A057723(n) - A048146(n), where A048146 is sum of non-unitary divisors and A057723 is sum of positive divisors of n that are divisible by every prime that divides n.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 10, 13, 14, 15, 16, 17, 15, 19, 18, 21, 22, 23, 18, 25, 26, 27, 26, 29, 30, 31, 32, 33, 34, 35, 31, 37, 38, 39, 34, 41, 42, 43, 42, 42, 46, 47, 34, 49, 45, 51, 50, 53, 42, 55, 50, 57, 58, 59, 42, 61, 62, 60, 64, 65, 66, 67, 66, 69, 70, 71, 63, 73, 74, 70, 74, 77, 78, 79, 66, 81

Offset

1,2

Links

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

Formula

For any prime p and exponent e >= 0, a(p^e) = p^e. For any squarefree number k (A005117), a(k) = k, and for any other number k, a(k) < k.

a(n) = n - A389171(n).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(2) * (1/zeta(3) - 1 + Product_{primes p} (1 - 1/p^2 + 1/p^3)) = 0.95478985966058300203... . - Amiram Eldar, Oct 05 2025

Mathematica

f1[p_, e_] := (p^(e+1)-1)/(p-1) - 1;
f2[p_, e_] := (p^(e+1)-1)/(p-1);
f3[p_, e_] := p^e + 1;
a[1] = 1; a[n_] := Module[{f = FactorInteger[n]}, Times @@ f1 @@@ f - Times @@ f2 @@@ f + Times @@ f3 @@@ f]; Array[a, 100] (* Amiram Eldar, Oct 05 2025 *)

Prog

(PARI)
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
A048146(n) = (sigma(n)-A034448(n));
A057723(n) = { my(f=factor(n)~); prod(i=1, #f, sigma(f[1, i]^f[2, i])-1); };
A389170(n) = (A057723(n) - A048146(n));

Crossrefs

Cf. A005117, A034448, A048146, A057723, A389171.

Cf. A126706 (positions where a(k)<k), A303554 (where a(k)=k).

Sequence in context: A121758 A121759 A265310 * A291576 A355221 A180613

Adjacent sequences: A389167 A389168 A389169 * A389171 A389172 A389173

Keyword

nonn,easy

Author

Antti Karttunen, Oct 04 2025

Status

approved