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A086665 Difference between sum of divisors of n and integer log of n, i.e., A000203(n) - A001414(n).

%I A086665 #17 Jul 06 2024 11:06:10
%S A086665 1,1,1,3,1,7,1,9,7,11,1,21,1,15,16,23,1,31,1,33,22,23,1,51,21,27,31,
%T A086665 45,1,62,1,53,34,35,36,81,1,39,40,79,1,84,1,69,67,47,1,113,43,81,52,
%U A086665 81,1,109,56,107,58,59,1,156,1,63,91,115,66,128,1,105,70,130,1,183,1,75,111,117,78,150,1,173,109,83
%N A086665 Difference between sum of divisors of n and integer log of n, i.e., A000203(n) - A001414(n).
%H A086665 Antti Karttunen, <a href="/A086665/b086665.txt">Table of n, a(n) for n = 1..65537</a>
%e A086665 a(10) = sigma(10) - (2 + 5) = 1 + 2 + 5 + 10 - 7 = 11.
%t A086665 sigma1 = Table[Plus @@ Divisors[n], {n, 100}]; sopfr = Prepend[ Array[ Plus @@ Map[ Times @@ #1 &, FactorInteger[ # ] ] &, 100, 2 ], 0 ]; Table[sigma1[[n]] - sopfr[[n]], {n, 100}] (* _Alonso del Arte_, Dec 01 2004 *)
%o A086665 (PARI) sodf(n)=local(x); x=factor(n); sum(i=1,length(x[,1]),sum(j=1,x[i,2],x[i,1])); for(i=1,60,print1(sigma(i)-sodf(i)","))
%o A086665 (PARI)
%o A086665 A001414(n) = ((n=factor(n))[, 1]~*n[, 2]);
%o A086665 A086665(n) = (sigma(n)-A001414(n)); \\ _Antti Karttunen_, Jul 06 2024
%Y A086665 Cf. A000203, A001414, A075255, A374126, A374127.
%K A086665 nonn
%O A086665 1,4
%A A086665 _Jon Perry_, Jul 27 2003
%E A086665 Data section extended up to a(82) by _Antti Karttunen_, Jul 06 2024