|
%I #20 Sep 11 2018 17:02:28
%S 1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,4,0,
%T 10,0,0,0,4,0,0,2,0,0,2,0,0,0,0,0,6,0,0,0,6,0,6,0,0,2,0,0,2,0,18,4,0,
%U 0,8,10,0,0,0,0,2,0,20,4,0,0,0,0,0,2,24,0,10,0,0,2,10,0,10,0,12,0,0,0,4,0,0,6,0,0,26
%N Number of terms in n-th cyclotomic polynomial minus largest prime factor of n; a(1)=1 by convention.
%C When (as at n=105) coefficients are not equal 1 or -1 then terms in C[n,x] are counted with multiplicity. - This is the comment by the original author. However, the claim contradicts the given formula, as A051664 counts each nonzero coefficient just once, regardless of its value. For the version summing the absolute values of the coefficients (thus "with multiplicity"), see A318886. - _Antti Karttunen_, Sep 10 2018
%H Antti Karttunen, <a href="/A070536/b070536.txt">Table of n, a(n) for n = 1..65537</a>
%F a(n) = A051664(n) - A006530(n).
%e n=21: Cyclotomic[21,x]=1-x+x^3-x^4+x^6-x^8+x^9-x^11+x^12 has 9 terms while largest prime factor of 21 is 7
%t Array[Length@ Cyclotomic[#, x] - FactorInteger[#][[-1, 1]] &, 105] (* _Michael De Vlieger_, Sep 10 2018 *)
%o (PARI)
%o A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1); \\ From A006530.
%o A051664(n) = length(select(x->x!=0, Vec(polcyclo(n)))); \\ From A051664
%o A070536(n) = (A051664(n) - A006530(n)); \\ _Antti Karttunen_, Sep 10 2018
%Y Cf. A006530, A051664, A070537, A070776.
%Y Differs from A318886 for the first time at n=105, where a(105) = 26, while A318886(105) = 28.
%K nonn
%O 1,15
%A _Labos Elemer_, May 03 2002
%E Data section extended to 105 terms by _Antti Karttunen_, Sep 10 2018
|
