A239205 -id:A239205 - cp's OEIS frontend

A239208 Numbers n such that sigma(n) divides the sum of the numbers x not coprime to n, with x<=n.

18, 135, 891, 4095, 10560, 13120, 14144, 21600, 23199, 74655, 144495, 192311, 404415, 4197375, 4612608, 5675775, 6664680, 9180800, 10953215, 11110400, 14381055, 18162144, 18420480, 18920000, 20765024, 25159680, 32058351, 41055200, 55889920, 65327104, 65982464

Offset: 1

Paolo P. Lava, Mar 12 2014

Numbers n such that sigma(n) | n/2(n+1-phi(n)).

			18/2*(19-phi(18)) = 117, sigma(18) = 39 and 117 / 39 = 3.

Giovanni Resta, Table of n, a(n) for n = 1..100

Cf. A000010, A000230, A238232, A239205.

with(numtheory); P:=proc(q) local a,n;
for n from 1 to q do a:=n/2*(n+1-phi(n)); if type(a/sigma(n),integer) then print(n);
fi; od; end: P(10^6);

a(14)-a(31) from Giovanni Resta, Mar 12 2014

A341933 a(n) = A023896(n) mod A000203(n).

Original entry on oeis.org

0, 1, 3, 4, 4, 6, 5, 1, 1, 2, 7, 24, 8, 18, 12, 2, 10, 15, 11, 38, 30, 2, 13, 36, 2, 30, 3, 0, 16, 48, 17, 4, 42, 2, 36, 34, 20, 42, 20, 50, 22, 60, 23, 20, 72, 2, 25, 12, 3, 35, 24, 36, 28, 6, 20, 72, 66, 2, 31, 144, 32, 66, 94, 8, 48, 84, 35, 80, 78, 120, 37, 84, 38, 78, 12, 108, 6, 96, 41, 164

Offset: 1

J. M. Bergot and Robert Israel, Feb 23 2021

nonn

a(k) is the sum of totatives of k modulo the sum of divisors of k.
If p is an odd prime, a(p) = (p+3)/2 and a(p^2) = (p-1)/2.
If p is a prime == 5 (mod 6), a(2*p) = 2.
If p is a prime == 1 (mod 6), a(2*p) = 2*p+4.
Are 2, 8 and 9 the only solutions to a(k) = 1?

			a(6) = 6 because the sum of totatives of 6 is 1+5 = 6, the sum of divisors of 6 is 1+2+3+6 = 12, and 6 mod 12 = 6.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000203, A023896, A239205, A308169.

f:= n -> n*numtheory:-phi(n)/2 mod numtheory:-sigma(n):
map(f, [$1..100]);

Array[Mod[# EulerPhi[#]/2 + Boole[# == 1]/2, DivisorSigma[1, #]] &, 80] (* Michael De Vlieger, Feb 23 2021 *)

cp's OEIS Frontend

A239208 Numbers n such that sigma(n) divides the sum of the numbers x not coprime to n, with x<=n.

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