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

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.

Sequence in context: A012440 A058172 A122808 * A225841 A203509 A009562

Adjacent sequences: A294222 A294223 A294224 * A294226 A294227 A294228

Keyword

nonn

Author

Zhi-Wei Sun, Oct 25 2017

Status

approved