A380604 Numbers k such that there is no number i such that A046144(i) = 2*k.

7, 13, 15, 17, 19, 21, 23, 25, 28, 29, 31, 33, 34, 35, 37, 38, 39, 43, 45, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 83, 85, 87, 91, 92, 93, 94, 97, 98, 99, 101, 103, 104, 105, 107, 109, 111, 112, 113, 114, 115, 117, 118

Offset: 1

Author

David James Sycamore, Jan 28 2025

Keywords

• nonn

Comments

2*a(n) are the even numbers which are not in A378508, namely numbers 2*m for which no number exists which has 2*m primitive roots. See A380594 for discussion of even numbers which are not in this sequence.

Examples

			 There is no x such that A046144(x) = 14, so 7 is a term in this sequence (see also A380594).
		

Links

• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
• David M. Bressoud, A Course in Computational Number Theory (web page), CNT.m, Computational Number Theory Mathematica package.

Crossrefs

Cf. A046144, A378508, A380594.

Programs

Mathematica

q[n_] := Count[Join @@ PhiInverse[PhiInverse[2*n]], ?(IntegerQ[PrimitiveRoot[#]] &)] == 0; Select[Range[120], q] (* _Amiram Eldar, Jan 28 2025, using David M. Bressoud's CNT.m *)

PARI

isA033948(n) = {my(f = factor(n)); lcm(znstar(f)[2]) == eulerphi(f); }
isok(k) = {my(v = invphi(2*k), w, c = 0); for(i = 1, #v, c += vecsum(apply(x -> isA033948(x), invphi(v[i])))); c == 0; } \\ Amiram Eldar, Jan 28 2025, using Max Alekseyev's invphi.gp