A373647 Positive integers that cannot be written as a sum of a practical number and a square.

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

Author

Duc Van Khanh Tran, Jun 12 2024

Keywords

• nonn

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

• Duc Van Khanh Tran, Table of n, a(n) for n = 1..10000
• 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.

Crossrefs

Cf. A000290, A005153.

Programs

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 *)