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Numbers k such that 3k and 4k have the same number of divisors.
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%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