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A333232 Terms of A051488 that do not belong to A083207. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 22 06:54 EDT 2022. Contains 353933 sequences. (Running on oeis4.)