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Positive integers that cannot be written as a sum of a practical number and a hexagonal number.
1

%I #24 Jun 11 2024 03:59:26

%S 11,15,50,59,83,137,142,158,164,167,212,227,362,419,607,683,779,809,

%T 872,887,914,1097,1124,1187,1262,1412,1493,1514,1699,2189,2363,2462,

%U 2489,2522,2594,2963,3089,3527,3539,3572,3749,3764,4127,4232,4349,4457,4622,4694

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

%C 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.

%H Duc Van Khanh Tran, <a href="/A373443/b373443.txt">Table of n, a(n) for n = 1..101</a>

%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=4700;hexlim=Ceiling[Sqrt[Lim/2]];

%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[hex=i(2i-1);hexi=prac+hex;AppendTo[seq,hexi],{i,0,hexlim}] (* sums of hexagonal and practical numbers *);

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

%Y Cf. A000384, A005153.

%K nonn,hard

%O 1,1

%A _Duc Van Khanh Tran_, Jun 05 2024