OFFSET
1,1
COMMENTS
Somu et al. (2023) proved that there are infinitely many such integers.
Somu and Tran (2024) proved a more general result, which states that infinitely many positive integers cannot be written as a sum of a practical number and an s-gonal number if s is congruent to 4 modulo 12.
LINKS
Duc Van Khanh Tran, Table of n, a(n) for n = 1..10000
Sai Teja Somu, Ting Hon Stanford Li, and Andrzej Kukla, On some results on practical numbers, INTEGERS, 23, 2023.
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=360; sqlim=Sqrt[Lim];
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[sq=i^2; sqi=prac+sq; AppendTo[seq, sqi], {i, 0, sqlim}] (* sums of squares and practical numbers *);
Complement[Range[Lim], Union[Flatten[seq]]] (* James C. McMahon, Jun 15 2024 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Duc Van Khanh Tran, Jun 12 2024
STATUS
approved