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Numbers k such that sigma(k) > 2*k and A003415(sigma(k)) < k, where A003415 is the arithmetic derivative, and sigma is the sum of divisors function.
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%I #12 Sep 19 2021 22:02:46

%S 18,36,100,144,324,400,576,784,900,1296,1458,1600,1936,2304,2500,2704,

%T 2916,3136,3600,4624,5184,5202,5776,6400,7744,8464,9216,9604,10000,

%U 10404,10816,11664,12100,13122,13456,14400,15376,17424,18496,19044,23104,25600,26244,28900,30258,30276,30976,32400,33856,36864,38416

%N Numbers k such that sigma(k) > 2*k and A003415(sigma(k)) < k, where A003415 is the arithmetic derivative, and sigma is the sum of divisors function.

%C Numbers k such that A033880(k) is positive but A342926(k) is negative.

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

%t ad[1] = 0; ad[n_] := n * Total@(Last[#]/First[#]& /@ FactorInteger[n]); Select[Range[1, 40000], DivisorSigma[1, #] > 2*# && ad[DivisorSigma[1, #]] < # &] (* _Amiram Eldar_, Sep 19 2021 *)

%o (PARI)

%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

%o isA347889(n) = ((A003415(sigma(n))<n)&&(sigma(n)>(2*n)));

%Y Intersection of A005101 and A343216. Subsequence A347890 gives the odd terms.

%Y Cf. A000203, A003415, A033880, A342926.

%K nonn

%O 1,1

%A _Antti Karttunen_, Sep 19 2021