A370348 - OEIS

%I #16 Mar 22 2024 16:28:12

%S 4,8,12,16,18,20,24,27,32,36,40,44,48,50,54,56,60,64,68,72,80,81,84,

%T 88,90,96,100,108,112,120,124,125,126,128,132,135,136,144,150,160,162,

%U 164,168,176,180,184,189,192,196,198,200,204,208,216,220,224,225,236,240,242,243,248,250,252,256

%N Numbers k such that there are fewer divisors of prime indices of k than there are prime indices of k.

%C No multiple of a term is a term of A368110.

%H Robert Israel, <a href="/A370348/b370348.txt">Table of n, a(n) for n = 1..10000</a>

%e a(5) = 18 is a term because the prime indices of 18 = 2 * 3^2 are 1,2,2, and there are 3 of these but only 2 divisors of prime indices, namely 1 and 2.

%p filter:= proc(n) uses numtheory; local F,D,t;

%p    F:= map(t -> [pi(t[1]),t[2]], ifactors(n)[2]);

%p    D:= `union`(seq(divisors(t[1]), t = F));

%p    nops(D) < add(t[2], t = F)

%p end proc:

%p select(filter, [$1..300]);

%t filter[n_] := Module[{F, d},

%t     F = {PrimePi[#[[1]]], #[[2]]}& /@ FactorInteger[n];

%t     d = Union[Flatten[Divisors /@ F[[All, 1]]]];

%t     Length[d] < Total[F[[All, 2]]]];

%t Select[Range[300], filter] (* _Jean-François Alcover_, Mar 08 2024, after Maple code *)

%Y The LHS is A370820, firsts A371131.

%Y The version for equality is A370802, counted by A371130, strict A371128.

%Y For submultisets instead of parts on the RHS we get A371167.

%Y The opposite version is A371168, counted by A371173.

%Y The weak version is A371169.

%Y The complement is A371170.

%Y Partitions of this type are counted by A371171.

%Y A000005 counts divisors.

%Y A001221 counts distinct prime factors.

%Y A027746 lists prime factors, indices A112798, length A001222.

%Y A355731 counts choices of a divisor of each prime index, firsts A355732.

%Y Cf. A003963, A239312, A303975, A319899, A355529, A355739, A370808, A371127.

%K nonn

%O 1,1

%A _Robert Israel_, Feb 15 2024