A326066 - OEIS

%I #17 Dec 21 2024 11:09:51

%S 0,2,3,4,5,8,7,8,9,12,11,16,13,16,18,16,17,26,19,24,24,24,23,32,25,28,

%T 27,32,29,48,31,32,36,36,40,52,37,40,42,48,41,64,43,48,54,48,47,64,49,

%U 62,54,56,53,80,60,64,60,60,59,96,61,64,72,64,70,96,67,72,72,96,71,104,73,76,93,80,84,112,79,96,81,84,83

%N a(n) = sigma(n) - sigma(A032742(n)), where A032742 gives the largest proper divisor of n.

%H Antti Karttunen, <a href="/A326066/b326066.txt">Table of n, a(n) for n = 1..20000</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.

%F a(n) = A000203(n) - A326065(n) = A000203(n) - A000203(A032742(n)).

%F a(1) = 0; for n > 1, if n is of the form p^k (p prime and exponent k >= 1), then a(n) = n, otherwise a(n) > n.

%F For terms in A247180, i.e., when n = A020639(n) * A032742(n), with the smallest prime factor A020639(n) unitary, a(n) = A020639(n) * A326065(n).

%F Sum_{k=1..n} a(k) ~ (zeta(2)/2) * (1 - c) * n^2, where c is defined in the corresponding formula in A326065. . - _Amiram Eldar_, Dec 21 2024

%t Join[{0},Table[DivisorSigma[1,n]-DivisorSigma[1,Divisors[n][[-2]]],{n,2,100}]] (* _Harvey P. Dale_, Jan 12 2022 *)

%o (PARI)

%o A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));

%o A326065(n) = sigma(A032742(n));

%o A326066(n) = (sigma(n) - sigma(A032742(n)));

%Y Cf. A000203, A013661, A020639, A032742, A246655 (positions of fixed points), A247180, A326065, A326067, A326135, A326136.

%K nonn

%O 1,2

%A _Antti Karttunen_, Jun 06 2019