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

%I A373647 #11 Jun 16 2024 02:09:48
%S A373647 14,23,35,47,59,62,71,74,86,95,98,107,110,119,131,134,138,143,155,158,
%T A373647 167,179,182,183,191,194,195,203,206,215,218,230,239,242,251,254,263,
%U A373647 266,275,278,282,287,299,302,311,314,318,323,327,335,338,347,350,359
%N A373647 Positive integers that cannot be written as a sum of a practical number and a square.
%C A373647 Somu et al. (2023) proved that there are infinitely many such integers.
%C A373647 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.
%H A373647 Duc Van Khanh Tran, <a href="/A373647/b373647.txt">Table of n, a(n) for n = 1..10000</a>
%H A373647 Sai Teja Somu, Ting Hon Stanford Li, and Andrzej Kukla, <a href="https://math.colgate.edu/~integers/x68/x68.pdf">On some results on practical numbers</a>, INTEGERS, 23, 2023.
%H A373647 Sai Teja Somu and Duc Van Khanh Tran, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Somu/somu5.html">On sums of practical numbers and polygonal numbers</a>, Journal of Integer Sequences, 27(5), 2024.
%t A373647 Lim=360;sqlim=Sqrt[Lim];
%t A373647 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] ;
%t A373647 seq={};Do[sq=i^2;sqi=prac+sq;AppendTo[seq,sqi],{i,0,sqlim}] (* sums of squares and practical numbers *);
%t A373647 Complement[Range[Lim],Union[Flatten[seq]]] (* _James C. McMahon_, Jun 15 2024 *)
%Y A373647 Cf. A000290, A005153.
%K A373647 nonn
%O A373647 1,1
%A A373647 _Duc Van Khanh Tran_, Jun 12 2024