A373421 Positive integers that cannot be written as a sum of a practical number and a pentagonal number.

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

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..13.

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=10^4; penlim=Ceiling[Sqrt[2Lim/3]];
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[pen=i(3i-1)/2; peni=prac+pen; AppendTo[seq, peni], {i, 0, penlim}] (* sums of pentagonal and practical numbers *);
Complement[Range[Lim], Union[Flatten[seq]]] (* James C. McMahon, Jun 10 2024 *)

Crossrefs

Cf. A000326, A005153.

Sequence in context: A115679 A337610 A109373 * A267329 A120138 A373613

Adjacent sequences: A373417 A373418 A373419 * A373422 A373423 A373424

Keyword

nonn,more,hard

Author

Duc Van Khanh Tran, Jun 04 2024

Status

approved