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A294307
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Positive integers m with m^k - 1 (k = 1,...,13) all practical.
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1
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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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.
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EXAMPLE
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a(1) = 169 since 169 is the first number m such that m - 1, m^2 - 1, ..., m^13 - 1 are all practical.
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MATHEMATICA
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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]
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CROSSREFS
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Cf. A005153, A294112, A294200, A294225.
Sequence in context: A112076 A305055 A069645 * A017534 A120904 A038596
Adjacent sequences: A294304 A294305 A294306 * A294308 A294309 A294310
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KEYWORD
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nonn
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AUTHOR
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Zhi-Wei Sun, Oct 27 2017
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STATUS
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approved
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