A373443 Positive integers that cannot be written as a sum of a practical number and a hexagonal number.

11, 15, 50, 59, 83, 137, 142, 158, 164, 167, 212, 227, 362, 419, 607, 683, 779, 809, 872, 887, 914, 1097, 1124, 1187, 1262, 1412, 1493, 1514, 1699, 2189, 2363, 2462, 2489, 2522, 2594, 2963, 3089, 3527, 3539, 3572, 3749, 3764, 4127, 4232, 4349, 4457, 4622, 4694

Offset

1,1

Comments

Somu and Tran (2024) conjectured that there are finitely many such integers. It was also conjectured that 1332329 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..101

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=4700; hexlim=Ceiling[Sqrt[Lim/2]];
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[hex=i(2i-1); hexi=prac+hex; AppendTo[seq, hexi], {i, 0, hexlim}] (* sums of hexagonal and practical numbers *);
Complement[Range[Lim], Union[Flatten[seq]]] (* James C. McMahon, Jun 10 2024 *)

Crossrefs

Cf. A000384, A005153.

Sequence in context: A357626 A241679 A219389 * A217085 A267123 A115779

Adjacent sequences: A373440 A373441 A373442 * A373444 A373445 A373446

Keyword

nonn,hard

Author

Duc Van Khanh Tran, Jun 05 2024

Status

approved