|
%I #18 Jan 08 2024 01:35:29
%S 1,6,12,24,36,60,72,120,144,216,288,360,432,960,720,864,1296,1440,
%T 1728,2160,2592,3456,7560,4320,5184,7776,10800,8640,10368,12960,15552,
%U 17280,20736,40320,25920,31104,41472,60480,64800,51840,62208,77760,93312
%N Least positive integer such that the number of divisors having two distinct prime factors is n.
%C Does this contain every power of six, namely 1, 6, 36, 216, 1296, 7776, ...?
%C Yes, every power of six is a term, since 6^k = 2^k * 3^k is the least positive integer having n = tau(6^k) - (2k+1) divisors with two distinct prime factors. - _Ivan N. Ianakiev_, Nov 11 2023
%H Amiram Eldar, <a href="/A367099/b367099.txt">Table of n, a(n) for n = 0..4469</a>
%e The divisors of 60 having two distinct prime factors are: 6, 10, 12, 15, 20. Since 60 is the first number having five such divisors, we have a(5) = 60.
%e The terms together with their prime indices begin:
%e 1: {}
%e 6: {1,2}
%e 12: {1,1,2}
%e 24: {1,1,1,2}
%e 36: {1,1,2,2}
%e 60: {1,1,2,3}
%e 72: {1,1,1,2,2}
%e 120: {1,1,1,2,3}
%e 144: {1,1,1,1,2,2}
%e 216: {1,1,1,2,2,2}
%e 288: {1,1,1,1,1,2,2}
%e 360: {1,1,1,2,2,3}
%e 432: {1,1,1,1,2,2,2}
%e 960: {1,1,1,1,1,1,2,3}
%e 720: {1,1,1,1,2,2,3}
%e 864: {1,1,1,1,1,2,2,2}
%t nn=1000;
%t w=Table[Length[Select[Divisors[n],PrimeNu[#]==2&]],{n,nn}];
%t spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];
%t Table[Position[w,k][[1,1]],{k,0,spnm[w]}]
%o (PARI) a(n) = my(k=1); while (sumdiv(k, d, omega(d)==2) != n, k++); k; \\ _Michel Marcus_, Nov 11 2023
%Y The version for all divisors is A005179 (firsts of A000005).
%Y For all prime factors (A001222) we have A220264, firsts of A086971.
%Y Positions of first appearances in A367098 (counts divisors in A007774).
%Y A000961 lists prime powers, complement A024619.
%Y A001221 counts distinct prime factors.
%Y A001358 lists semiprimes, squarefree A006881, complement A100959.
%Y A367096 lists semiprime divisors, sum A076290.
%Y Cf. A001248, A054753, A056170, A079275, A146289, A366740, A367093.
%K nonn
%O 0,2
%A _Gus Wiseman_, Nov 09 2023
|
