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Odd numbers k for which A003415(sigma(k^2))-(k^2) is strictly positive and even. Here A003415 is the arithmetic derivative.
4

%I #14 Sep 22 2021 09:25:35

%S 201,231,237,259,273,315,333,399,429,455,483,525,555,585,627,651,665,

%T 741,763,855,903,975,1057,1071,1085,1113,1209,1235,1351,1395,1407,

%U 1505,1533,1635,1659,1677,1767,1785,1935,2037,2079,2163,2211,2265,2317,2331,2345,2451,2457,2479,2541,2555,2583,2607,2611,2613

%N Odd numbers k for which A003415(sigma(k^2))-(k^2) is strictly positive and even. Here A003415 is the arithmetic derivative.

%C A square root of any hypothetical odd term x in A005820 (triperfect numbers) would be a member of this sequence, because such x should be a term of A342923 [Numbers x such that A342925(x)-x = 3*A003415(x)], and as the right hand side would then certainly be even (A235992 contains all odd squares), the left hand side should also be even. See also comments in A347870 and in A347391.

%t ad[1] = 0; ad[n_] := n * Total@(Last[#]/First[#]& /@ FactorInteger[n]); Select[Range[1, 3000, 2], (d = ad[DivisorSigma[1, #^2]] - #^2) > 0 && EvenQ[d] &] (* _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 isA347887(n) = if(!(n%2),0,my(u=(A003415(sigma(n^2))-(n^2))); ((u>0)&&!(u%2)));

%Y Cf. A000203, A003415, A005820, A097023, A235991, A235992, A342923, A342925, A342926, A347383, A347391.

%Y Subsequence of A347881 and of A347885. The intersection with A347882 gives A347888.

%K nonn

%O 1,1

%A _Antti Karttunen_, Sep 19 2021