A318876 - OEIS

%I #19 Jul 08 2024 11:57:16

%S 1,1,4,1,6,4,8,1,13,6,12,4,14,8,24,1,18,13,20,6,32,12,24,4,31,14,40,8,

%T 30,24,32,1,48,18,48,13,38,20,56,6,42,32,44,12,78,24,48,4,57,31,72,14,

%U 54,40,72,8,80,30,60,24,62,32,104,1,84,48,68,18,96,48,72,13,74,38,124,20,96,56,80,6,121,42,84,32,108,44,120,12,90,78,112,24,128,48,120,4,98,57,156,31,102,72,104,14,87

%N Sum of divisors d of n for which 2*phi(d) > d.

%C Not multiplicative as a(3) = 4, a(5) = 6 and a(7) = 8, but a(105) = 87, not a(3)*a(5)*a(7) = 4*6*8 = 192 = A000593(105).

%H Antti Karttunen, <a href="/A318876/b318876.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>

%F a(n) = Sum_{d|n} [2*phi(d) > d]*d, where [ ] are the Iverson brackets.

%F For all n >= 1, a(n) + A318877(n) + 2*(A006519(n)-1) = A000203(n).

%e n = 105 has divisors [1, 3, 5, 7, 15, 21, 35, 105]. When A083254 is applied to them, all others except the last one result a positive number, thus a(105) = 1+3+5+7+15+21+35 = 87.

%t A318876[n_] := DivisorSum[n,  # &, 2*EulerPhi[#] > # &];

%t Array[A318876, 100] (* _Paolo Xausa_, Jul 08 2024 *)

%o (PARI) A318876(n) = sumdiv(n,d,((2*eulerphi(d))>d)*d);

%Y Cf. A000010, A000203, A006519, A083254, A318874, A318877, A318878.

%Y Cf. also A187793.

%Y Differs from A000593 for the first time at n=105, where a(105) = 87, while A000593(105) = 192.

%K nonn

%O 1,3

%A _Antti Karttunen_, Sep 05 2018