cp's OEIS Frontend

A346608 Indices k such that A047994(k) != A344005(k).

%I A346608 #37 Dec 24 2024 07:17:59
%S A346608 12,15,20,21,24,28,30,33,35,36,39,40,42,44,45,48,51,52,55,56,57,60,63,
%T A346608 65,66,68,69,70,72,75,76,77,78,80,84,85,87,88,90,91,92,93,95,96,99,
%U A346608 100,102,104,105,108,110,111,112,114,115,116,117,119,120,123,124,126,129,130,132,133,135,136,138,140
%N A346608 Indices k such that A047994(k) != A344005(k).
%C A346608 Conjectures: (i) For all k in this sequence, A047994(k) >= A344005(k).
%C A346608 (ii) Equals composite numbers with {18, 2*p (p prime), p^i (p prime, i >= 2)} deleted.
%C A346608 The second conjecture asserts that this is equal to A265128 with {0, 1, 18} deleted.
%C A346608 I believe I have a proof of both conjectures, although I have not yet written out all the details.
%C A346608 Numbers k that are in A265128, but do not appear here are: 1, 18, 50, 54, 98, 162, 242, 250, 338, 486, 578, 686, ... probably given by {1} UNION A354929. Hence conjecture: the sequence consists of numbers that are neither a power of prime, or 2 * power of prime. - _Antti Karttunen_, Jun 14 2022
%C A346608 Is this the set of all k such that Phi_k(-1) = Phi_k(0) = Phi_k(1) where Phi_k denotes the k-th cyclotomic polynomial? - _Jeppe Stig Nielsen_, Jun 26 2023
%H A346608 N. J. A. Sloane, <a href="/A346608/b346608.txt">Table of n, a(n) for n = 1..8020</a>
%o A346608 (PARI)
%o A346608 A047994(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^f[2, i])-1); };
%o A346608 A344005(n) = for(m=1, oo, if((m*(m+1))%n==0, return(m))); \\ From A344005
%o A346608 A354924(n) = (A047994(n)==A344005(n));
%o A346608 isA346608(n) = !A354924(n); \\ _Antti Karttunen_, Jun 13 2022
%o A346608 (PARI) isA346608conjectured(n) = ((n>1) && !isprimepower(n) && ((n%2) || !isprimepower(n/2)));
%Y A346608 Cf. A047994, A265128, A344005, A345992, A354928 (complement).
%Y A346608 Positions of nonzeros in A346607. Positions of zeros in A354924.
%Y A346608 Setwise difference A265128 \ ({0,1} U A138929). (conjectured).
%Y A346608 Intersection of A024619 and A230078 (conjectured).
%K A346608 nonn
%O A346608 1,1
%A A346608 _N. J. A. Sloane_, Aug 09 2021