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A294225 Practical numbers q with q + 2 and q^2 + 2 both practical. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 27 16:48 EDT 2022. Contains 354110 sequences. (Running on oeis4.)