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A373633
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Positive integers that cannot be written as a sum of a practical number and a 15-gonal number.
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1
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10, 11, 14, 15, 22, 26, 34, 38, 52, 53, 59, 68, 76, 77, 92, 107, 116, 117, 125, 131, 134, 149, 152, 158, 164, 173, 179, 184, 185, 187, 188, 206, 212, 227, 230, 236, 245, 248, 251, 254, 259, 268, 269, 283, 293, 299, 317, 326, 332, 347, 356, 371, 389, 398, 403
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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MATHEMATICA
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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 *);
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CROSSREFS
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KEYWORD
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nonn,hard
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AUTHOR
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STATUS
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approved
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