OFFSET
1,1
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.
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 *)
CROSSREFS
KEYWORD
nonn,hard
AUTHOR
Duc Van Khanh Tran, Jun 11 2024
STATUS
approved