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

1,2

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

Formula

a(n) <= A000005(n).

Example

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.

Prog

(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); };

Crossrefs

Cf. A000005, A258851, A299701, A319686.

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

Sequence in context: A325560 A318412 A365208 * A335519 A167447 A134687

Adjacent sequences: A322983 A322984 A322985 * A322987 A322988 A322989

Keyword

nonn

Author

Antti Karttunen, Jan 04 2019

Status

approved