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

12, 15, 20, 21, 24, 28, 30, 33, 35, 36, 39, 40, 42, 44, 45, 48, 51, 52, 55, 56, 57, 60, 63, 65, 66, 68, 69, 70, 72, 75, 76, 77, 78, 80, 84, 85, 87, 88, 90, 91, 92, 93, 95, 96, 99, 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

Offset: 1

Author

N. J. A. Sloane, Aug 09 2021

Keywords

• nonn

Comments

Conjectures: (i) For all k in this sequence, A047994(k) >= A344005(k).

(ii) Equals composite numbers with {18, 2*p (p prime), p^i (p prime, i >= 2)} deleted.

The second conjecture asserts that this is equal to A265128 with {0, 1, 18} deleted.

I believe I have a proof of both conjectures, although I have not yet written out all the details.

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

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

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..8020

Crossrefs

Cf. A047994, A265128, A344005, A345992, A354928 (complement).

Positions of nonzeros in A346607. Positions of zeros in A354924.

Setwise difference A265128 \ ({0,1} U A138929). (conjectured).

Intersection of A024619 and A230078 (conjectured).

Programs

PARI

A047994(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^f[2, i])-1); };
A344005(n) = for(m=1, oo, if((m*(m+1))%n==0, return(m))); \\ From A344005
A354924(n) = (A047994(n)==A344005(n));
isA346608(n) = !A354924(n); \\ Antti Karttunen, Jun 13 2022

PARI

isA346608conjectured(n) = ((n>1) && !isprimepower(n) && ((n%2) || !isprimepower(n/2)));