A294112 -id:A294112 - cp's OEIS frontend

A294200 Primes p such that 2^p - 2 is a practical number.

2, 3, 5, 7, 11, 13, 17, 19, 29, 31, 37, 41, 43, 53, 61, 67, 71, 73, 79, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 151, 157, 163, 181, 191, 193, 197, 199, 211, 223, 229, 233, 241, 251, 257, 271, 277, 281, 283, 307, 311, 313, 331, 337, 349, 353

Offset: 1

Zhi-Wei Sun, Oct 24 2017

nonn

Conjecture: The sequence has infinitely many terms. In other words, there are infinitely many practical numbers of the form 2^p - 2 with p prime.

By Fermat's little theorem, p divides 2^p - 2 for any prime p. Note that those 2^p - 1 with p prime are called Mersenne numbers.

			a(1) = 2 since 2 is prime and 2^2 - 2 = 2 is practical.
a(2) = 3 since 3 is prime and 2^3 - 2 = 6 is practical.

Zhi-Wei Sun, Table of n, a(n) for n = 1..77
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.

Cf. A000040, A000043, A000079, A005153, A294112.

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);
tab={};Do[If[pr[2^(Prime[k])-2],tab=Append[tab,Prime[k]]],{k,1,71}];Print[tab]

A294225 Practical numbers q with q + 2 and q^2 + 2 both practical.

Original entry on oeis.org

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

Zhi-Wei Sun, Oct 25 2017

nonn

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).

			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.

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.

Cf. A005153, A287681, A294112, A294200.

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]

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

Original entry on oeis.org

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

Zhi-Wei Sun, Oct 27 2017

nonn

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.

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

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.

Cf. A005153, A294112, A294200, A294225.

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]

cp's OEIS Frontend

A294200 Primes p such that 2^p - 2 is a practical number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A294225 Practical numbers q with q + 2 and q^2 + 2 both practical.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica