A373633 Positive integers that cannot be written as a sum of a practical number and a 15-gonal number.

10, 11, 14, 15, 22, 26, 34, 38, 52, 53, 59, 68, 76, 77, 92, 107, 116, 117, 125, 131, 134, 149, 152, 158, 164, 173, 179, 184, 185, 187, 188, 206, 212, 227, 230, 236, 245, 248, 251, 254, 259, 268, 269, 283, 293, 299, 317, 326, 332, 347, 356, 371, 389, 398, 403

Offset: 1

Author

Duc Van Khanh Tran, Jun 11 2024

Keywords

• nonn
• hard

Comments

Somu and Tran (2024) conjectured that there are finitely many such integers. It was also conjectured that 1486748 is the largest such integer. This conjecture was checked up to 10^8.

Links

• Duc Van Khanh Tran, Table of n, a(n) for n = 1..767
• 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. A005153, A051867.

Programs

Mathematica

Lim=403;Lim15=Ceiling[Sqrt[2Lim/13]];
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[p15=i(13i-11)/2;p15i=prac+p15;AppendTo[seq,p15i],{i,0,Lim15}] (* sums of 15gonal and practical numbers *);
Complement[Range[Lim],Union[Flatten[seq]]] (* James C. McMahon, Jun 12 2024 *)