A294307 Positive integers m with m^k - 1 (k = 1,...,13) all practical.

169, 625, 729, 1089, 1681, 3969, 4225, 5929, 6241, 6561, 6889, 8647, 9409, 11449, 14641, 15625, 16129, 18769, 20449, 22201, 24649, 27561, 28561, 30625, 32761, 33331, 33489, 33661, 34969, 35209, 35721, 38071, 38809, 39601, 41209, 42025, 43681, 43969, 44521, 47089, 47961, 50625, 51529, 55225, 58081

Offset: 1

Author

Zhi-Wei Sun, Oct 27 2017

Keywords

• nonn

Comments

Conjecture: For any positive integer n, there are infinitely many positive integers m with m^k - 1 (k = 1,...,n) all practical.

This is true for n = 2. In fact, by a result of Melfi, there are infinitely many positive integers m such that m - 1 and m + 1 are both practical and hence (m-1)*(m+1) = m^2 - 1 is also practical.

Examples

			a(1) = 169 since 169 is the first number m such that m - 1, m^2 - 1, ..., m^13 - 1 are all practical.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..200
• 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.

Crossrefs

Cf. A005153, A294112, A294200, A294225.

Programs

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-1]&&pr[n^2-1]&&pr[n^3-1]&&pr[n^4-1]&&pr[n^5-1]&&pr[n^6-1]&&pr[n^7-1]&&pr[n^8-1]&&pr[n^9-1]&&pr[n^(10)-1]&&pr[n^(11)-1]&&pr[n^(12)-1]&&pr[n^(13)-1]
tab={};Do[If[pq[k],tab=Append[tab,k]],{k,1,59000}];Print[tab]