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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)
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OFFSET
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1,1
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LINKS
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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.
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MATHEMATICA
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(* 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[#]&]
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CROSSREFS
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Cf. A000010, A051488, A083207.
Sequence in context: A104927 A183843 A212478 * A053340 A291597 A233945
Adjacent sequences: A333229 A333230 A333231 * A333233 A333234 A333235
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KEYWORD
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nonn
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AUTHOR
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Ivan N. Ianakiev, Mar 12 2020
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EXTENSIONS
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Terms a(12) and beyond from Giovanni Resta, Mar 12 2020
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STATUS
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approved
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