cp's OEIS Frontend

A370820 Number of positive integers that are a divisor of some prime index of n.

%I A370820 #14 May 02 2024 09:47:15
%S A370820 0,1,2,1,2,2,3,1,2,2,2,2,4,3,3,1,2,2,4,2,3,2,3,2,2,4,2,3,4,3,2,1,3,2,
%T A370820 4,2,6,4,4,2,2,3,4,2,3,3,4,2,3,2,3,4,5,2,3,3,4,4,2,3,6,2,3,1,4,3,2,2,
%U A370820 4,4,6,2,4,6,3,4,4,4,4,2,2,2,2,3,3,4,4
%N A370820 Number of positive integers that are a divisor of some prime index of n.
%C A370820 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A370820 This sequence contains all nonnegative integers. In particular, a(prime(n)!) = n.
%e A370820 2045 has prime indices {3,80} with combined divisors {1,2,3,4,5,8,10,16,20,40,80}, so a(2045) = 11. In fact, 2045 is the least number with this property.
%t A370820 Table[Length[Union@@Divisors/@PrimePi/@First/@If[n==1,{},FactorInteger[n]]],{n,100}]
%o A370820 (PARI) a(n) = my(list=List(), f=factor(n)); for (i=1, #f~, fordiv(primepi(f[i,1]), d, listput(list, d))); #Set(list); \\ _Michel Marcus_, May 02 2024
%Y A370820 a(prime(n)) = A000005(n).
%Y A370820 Positions of ones are A000079 except for 1.
%Y A370820 a(n!) = A000720(n).
%Y A370820 a(prime(n)!) = a(prime(A005179(n))) = n.
%Y A370820 Counting prime factors instead of divisors gives A303975.
%Y A370820 Positions of 2's are A371127.
%Y A370820 Position of first appearance of n is A371131(n), sorted version A371181.
%Y A370820 RHS of A370802, A371128, A371130, A371165-A371170, A371177, A371178.
%Y A370820 A001221 counts distinct prime factors.
%Y A370820 A003963 gives product of prime indices.
%Y A370820 A027746 lists prime factors, A112798 indices, length A001222.
%Y A370820 A355731 counts choices of a divisor of each prime index, firsts A355732.
%Y A370820 A355741 counts choices of a prime factor of each prime index.
%Y A370820 Cf. A000792, A007416, A048249, A319899, A355737, A355739, A370348, A370808, A370809.
%K A370820 nonn
%O A370820 1,3
%A A370820 _Gus Wiseman_, Mar 15 2024