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Terms of A097703 that are not of the form 3*k + 1.
5

%I #16 Aug 31 2024 04:27:04

%S 12,24,60,62,84,87,122,137,144,162,171,180,212,237,264,269,287,302,

%T 312,318,362,387,416,420,422,423,437,462,465,480,512,537,563,587,591,

%U 612,662,665,684,687,710,722,737,759,762,786,812,837,840,857,887,902

%N Terms of A097703 that are not of the form 3*k + 1.

%C Conjecture: "most" of the terms also belong to [(A067778-1)/2]. Exceptions are {302, 2117, ...} (A098241). In other words, most terms satisfy: GCD(2*k+1, numerator(B(4*k+2))) is not squarefree, with B(n) the Bernoulli numbers.

%H Amiram Eldar, <a href="/A097704/b097704.txt">Table of n, a(n) for n = 1..10000</a>

%t usigma[n_] := Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[ #, n/# ] == 1 &]]; Complement[ Range[1017], Table[3k - 2, {k, 340}], (Select[ Range[220000], DivisorSigma[1, # ] == 2usigma[ # ] &] - 108)/216] (* _Robert G. Wilson v_, Aug 28 2004 *)

%o (PARI) is(k) = if(k % 3 == 1, 0, my(f = factor(216*k + 108)); sigma(f) != 2 * prod(i = 1, #f~, 1 + f[i,1]^f[i,2])); \\ _Amiram Eldar_, Aug 31 2024

%Y Intersection of A007494 and A097703.

%Y Cf. A013929, A027641, A027642, A067778, A098241.

%K nonn

%O 1,1

%A _Ralf Stephan_, Aug 26 2004