Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #32 Feb 03 2024 10:16:55
%S 2,10,14,22,24,26,34,38,46,50,58,62,70,74,82,86,94,98,106,110,118,120,
%T 122,130,134,142,146,154,158,166,168,170,178,182,190,194,202,206,214,
%U 218,226,230,238,242,250,254,262,264,266,274,278,286,288,290,298,302,310,312,314
%N Numbers k such that 3k and 4k have the same number of divisors.
%C Includes all numbers whose prime factorization has one 2 and no 3's, or three 2's and one 3.
%C Numbers k such that v_2(k) - 2*v_3(k) = 1, where v_p(k) is the p-adic valuation of k. - _Amiram Eldar_, Dec 12 2021
%C Numbers of the form 2 * 12^k * A007310(m) for k >= 0 and m >= 1. - _David A. Corneth_, Dec 12 2021
%C The asymptotic density of this sequence is 2/11. - _Amiram Eldar_, Feb 02 2024
%H Winston de Greef, <a href="/A350059/b350059.txt">Table of n, a(n) for n = 1..10000</a>
%e 30 is not in the sequence: 30*3=90 has 12 divisors, but 30*4=120 has 16 divisors.
%t Select[Range[300], IntegerExponent[#, 2] - 2 * IntegerExponent[#, 3] == 1 &] (* _Amiram Eldar_, Dec 12 2021 *)
%o (PARI) isok(k) = numdiv(3*k) == numdiv(4*k); \\ _Michel Marcus_, Dec 12 2021
%o (PARI) isok(n) = valuation(n,2) - 2 * valuation(n, 3) == 1; \\ _Amiram Eldar_, Feb 02 2024
%o (Python)
%o from sympy import multiplicity as v
%o def ok(n): return v(2, n) - 2*v(3, n) == 1
%o print([k for k in range(1, 315) if ok(k)]) # _Michael S. Branicky_, Feb 02 2024
%Y Cf. A000005, A007310, A007814, A007949.
%K nonn,easy
%O 1,1
%A _J. Lowell_, Dec 11 2021
%E More terms from _Michel Marcus_, Dec 12 2021