OFFSET
1,1
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.
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.
EXAMPLE
a(1) = 169 since 169 is the first number m such that m - 1, m^2 - 1, ..., m^13 - 1 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-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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Oct 27 2017
STATUS
approved