A294225 - OEIS

A294225

Practical numbers q with q + 2 and q^2 + 2 both practical.

2, 4, 520, 2560, 3100, 4648, 6448, 6784, 7252, 11128, 12400, 15496, 19264, 26128, 26752, 26860, 28768, 31648, 32368, 36160, 37408, 41728, 45400, 48760, 53248, 53584, 54832, 57148, 58828, 63544, 66820, 68440, 68500, 73948, 74176, 80512, 81508, 84208, 93184, 94300, 106780, 112288, 113968, 118528, 131068

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OFFSET

1,1

COMMENTS

Conjecture: The sequence has infinitely many terms.

In 1996 G. Melfi proved that there are infinitely many positive integers q with q and q + 2 both practical.

As any practical number greater than 2 is a multiple of 4 or 6, when q > 2, q + 2 and q^2 + 2 are all practical, we must have q^2 + 2 == 0 (mod 6), hence q is not divisible by 3 and thus 4 | q and 6 | (q + 2), therefore q == 4 (mod 12).

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..2500

G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210.

Zhi-Wei Sun, Conjectures on representations involving primes, in: M. B. Nathanson (ed.), Combinatorial and Additive Number Theory II: CANT, New York, NY, USA, 2015 and 2016, Springer Proc. in Math. & Stat., Vol. 220, Springer, New York, 2017.

EXAMPLE

a(1) = 2 since 2, 2 + 2 = 4 and 2^2 + 2 = 6 are all practical.

a(2) = 4 since 4, 4 + 2 = 6 and 4^2 + 2 = 18 are all practical.

MATHEMATICA

f[n_]:=f[n]=FactorInteger[n];

Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2]);

Con[n_]:=Con[n]=Sum[If[Part[Part[f[n], s+1], 1]<=DivisorSigma[1, Product[Pow[n, i], {i, 1, s}]]+1, 0, 1], {s, 1, Length[f[n]]-1}];

pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0);

pq[n_]:=pq[n]=pr[n]&&pr[n+2]&&pr[n^2+2];

tab={}; Do[If[pq[k], tab=Append[tab, k]], {k, 1, 132000}]; Print[tab]

CROSSREFS

Cf. A005153, A287681, A294112, A294200.