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A385720 Numbers k >= 1 such that k/A000005(k) + (k+1)/A000005(k+1) is an integer.

%I A385720 #20 Jul 14 2025 04:21:17
%S A385720 1,5,6,8,10,13,22,37,45,46,58,61,62,69,73,74,77,82,89,106,114,117,126,
%T A385720 146,149,150,154,157,166,167,178,186,193,197,198,206,221,226,233,237,
%U A385720 258,261,262,263,266,277,278,279,280,290,293,306,309,311,312,313
%N A385720 Numbers k >= 1 such that k/A000005(k) + (k+1)/A000005(k+1) is an integer.
%C A385720 k/A000005(k) + (k+1)/A000005(k+1) = (3*k + 1)/4 for k >= 5 from A256072.
%C A385720 k/A000005(k) + (k+1)/A000005(k+1) = (3*k + 2)/4 for k >= 6 from A077065.
%C A385720 k/A000005(k) + (k+1)/A000005(k+1) =< (3*k + 2)/4 for k >= 5.
%e A385720 For k = 6: 6/A000005(6) + 7/A000005(7) = 6/4 + 7/2 = 5, thus k = 6 is a term.
%t A385720 Position[Plus @@@ Partition[Table[n/DivisorSigma[0, n], {n, 1, 320}], 2, 1], _?IntegerQ] // Flatten (* _Amiram Eldar_, Jul 08 2025 *)
%o A385720 (PARI) isok(k) = denominator(k/numdiv(k) + (k+1)/numdiv(k+1)) == 1; \\ _Michel Marcus_, Jul 08 2025
%o A385720 (Python)
%o A385720 from itertools import count, islice
%o A385720 from sympy import divisor_count
%o A385720 def A385720_gen(startvalue=1): # generator of terms >= startvalue
%o A385720     m = max(startvalue,1)
%o A385720     a, b = divisor_count(m), divisor_count(m+1)
%o A385720     for k in count(m):
%o A385720         if not (k*b+(k+1)*a)%(a*b):
%o A385720             yield k
%o A385720         a, b = b, divisor_count(k+2)
%o A385720 A385720_list = list(islice(A385720_gen(),30)) # _Chai Wah Wu_, Jul 13 2025
%Y A385720 Cf. A000005, A005384, A005385, A077065, A256072.
%K A385720 nonn
%O A385720 1,2
%A A385720 _Ctibor O. Zizka_, Jul 07 2025