A333232 Terms of A051488 that do not belong to A083207.

5865, 7395, 10005, 15045, 28815, 37995, 45645, 50235, 99705, 134895, 170085, 275655, 310845, 347565, 391935, 436305, 470235, 486795, 521985, 530265, 590295, 613785, 627555, 635205, 658155, 662745, 707115, 791265, 797385, 830415, 835635, 873885, 887655, 979455, 994755

Offset

1,1

Links

Giovanni Resta, Table of n, a(n) for n = 1..10000

K. P. S. Bhaskara Rao and Yuejian Peng, On Zumkeller Numbers, Journal of Number Theory, Volume 133, Issue 4, April 2013, pp. 1135-1155.

Mathematica

(* First 200000 terms of A051488 *)
a051488=Select[Range[200000], EulerPhi[#]<EulerPhi[#-EulerPhi[#]]&];
(* Fast test to sift out the "easiest" Zumkeller numbers, see Proposition 17, Rao/Peng link *)
d[n_]:=Divisors[n]; fQ[n_]:=EvenQ[DivisorSigma[1, n]];
gQ[n_]:=Union[Table[d[n][[i+1]]<=2*d[n][[i]], {i, 1, Length[d[n]]-1}]]=={True}; znQ[n_]:=fQ[n]&&gQ[n]; t1=Select[a051488, !znQ[#]&];
(* Comprehensive test to sift out the remaining Zumkeller numbers, code by T. D. Noe at A083207 *)
zQ[n_]:=Module[{d=Divisors[n], t, ds, x}, ds=Plus@@d; If[Mod[ds, 2]>0, False, t=CoefficientList[Product[1+x^i, {i, d}], x]; t[[1+ds/2]]>0]]; t2=Select[t1, !zQ[#]&]

Crossrefs

Cf. A000010, A051488, A083207.

Sequence in context: A104927 A183843 A212478 * A053340 A291597 A233945

Adjacent sequences: A333229 A333230 A333231 * A333233 A333234 A333235

Keyword

nonn

Author

Ivan N. Ianakiev, Mar 12 2020

Extensions

Terms a(12) and beyond from Giovanni Resta, Mar 12 2020

Status

approved