A322986 Number of distinct values obtained when the pi-based arithmetic derivative (A258851) is applied to the divisors of n.

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 5, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 8, 4, 7, 4, 4, 2, 12, 2, 4, 6, 7, 4, 8, 2, 6, 4, 8, 2, 11, 2, 4, 6, 6, 4, 8, 2, 9, 5, 4, 2, 11, 4, 4, 4, 8, 2, 12, 4, 6, 4, 4, 4, 12, 2, 6, 6, 9, 2, 7, 2, 8, 8

Offset: 1

Author

Antti Karttunen, Jan 04 2019

Keywords

• nonn

Examples

			Divisors of 28 are [1, 2, 4, 7, 14, 28]. When A258851 is applied to them, we get five distinct values: [0, 1, 4, 4, 15, 44] (because A258851(4) = A258851(7) = 4), thus a(28) = 5, one less than A000005(28)=6.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20160
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000005, A258851, A299701, A319686.

Differs from A000005 for the first time at n=28.

Programs

PARI

A258851(n) = n*sum(i=1, #n=factor(n)~, n[2, i]*primepi(n[1, i])/n[1, i]); \\ From A258851
A322986(n) = { my(m=Map(),s,k=0); fordiv(n,d,if(!mapisdefined(m,s=A258851(d)), mapput(m,s,s); k++)); (k); };
\\ Or maybe more efficiently as, after David A. Corneth's Oct 02 2018 program in A319686:
A322986(n) = { my(d = divisors(n)); for(i=1, #d, d[i] = A258851(d[i])); #Set(d); };

Formula

a(n) <= A000005(n).