A373421 - OEIS

%I #27 Jun 10 2024 14:59:47

%S 10,15,22,27,103,114,186,244,494,619,1154,1854,2671

%N Positive integers that cannot be written as a sum of a practical number and a pentagonal number.

%C Somu and Tran (2024) conjectured that there are finitely many such integers. It was also conjectured that 2671 is the largest such integer. This conjecture was checked up to 10^8.

%H 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 Lim=10^4;penlim=Ceiling[Sqrt[2Lim/3]];

%t 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 seq={};Do[pen=i(3i-1)/2;peni=prac+pen;AppendTo[seq,peni],{i,0,penlim}] (* sums of pentagonal and practical numbers *);

%t Complement[Range[Lim],Union[Flatten[seq]]] (* _James C. McMahon_, Jun 10 2024 *)

%Y Cf. A000326, A005153.

%K nonn,more,hard

%O 1,1

%A _Duc Van Khanh Tran_, Jun 04 2024