A322986 - OEIS

%I #13 Jan 04 2019 17:36:57

%S 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,

%T 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,

%U 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

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

%H Antti Karttunen, <a href="/A322986/b322986.txt">Table of n, a(n) for n = 1..20160</a>

%H Antti Karttunen, <a href="/A322986/a322986.txt">Data supplement: n, a(n) computed for n = 1..100000</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%F a(n) <= A000005(n).

%e 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.

%o (PARI)

%o A258851(n) = n*sum(i=1, #n=factor(n)~, n[2, i]*primepi(n[1, i])/n[1, i]); \\ From A258851

%o A322986(n) = { my(m=Map(),s,k=0); fordiv(n,d,if(!mapisdefined(m,s=A258851(d)), mapput(m,s,s); k++)); (k); };

%o \\ Or maybe more efficiently as, after _David A. Corneth_'s Oct 02 2018 program in A319686:

%o A322986(n) = { my(d = divisors(n)); for(i=1, #d, d[i] = A258851(d[i])); #Set(d); };

%Y Cf. A000005, A258851, A299701, A319686.

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

%K nonn

%O 1,2

%A _Antti Karttunen_, Jan 04 2019