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A373647
Positive integers that cannot be written as a sum of a practical number and a square.
1
14, 23, 35, 47, 59, 62, 71, 74, 86, 95, 98, 107, 110, 119, 131, 134, 138, 143, 155, 158, 167, 179, 182, 183, 191, 194, 195, 203, 206, 215, 218, 230, 239, 242, 251, 254, 263, 266, 275, 278, 282, 287, 299, 302, 311, 314, 318, 323, 327, 335, 338, 347, 350, 359
OFFSET
1,1
COMMENTS
Somu et al. (2023) proved that there are infinitely many such integers.
Somu and Tran (2024) proved a more general result, which states that infinitely many positive integers cannot be written as a sum of a practical number and an s-gonal number if s is congruent to 4 modulo 12.
LINKS
Sai Teja Somu, Ting Hon Stanford Li, and Andrzej Kukla, On some results on practical numbers, INTEGERS, 23, 2023.
Sai Teja Somu and Duc Van Khanh Tran, On sums of practical numbers and polygonal numbers, Journal of Integer Sequences, 27(5), 2024.
MATHEMATICA
Lim=360; sqlim=Sqrt[Lim];
PracticalQ[nn_] := Module[{f, p, e, prod=1, ok=True}, If[nn<1 || (nn>1 && OddQ[n]), False, If[nn==1, True, f=FactorInteger[nn]; {p, e} = Transpose[f]; Do[If[p[[i]] > 1+DivisorSigma[1, prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]]; prac= Select[Range[Lim], PracticalQ] ;
seq={}; Do[sq=i^2; sqi=prac+sq; AppendTo[seq, sqi], {i, 0, sqlim}] (* sums of squares and practical numbers *);
Complement[Range[Lim], Union[Flatten[seq]]] (* James C. McMahon, Jun 15 2024 *)
CROSSREFS
Sequence in context: A102876 A188166 A184220 * A026065 A316735 A010922
KEYWORD
nonn
AUTHOR
Duc Van Khanh Tran, Jun 12 2024
STATUS
approved