A152093 -id:A152093 - cp's OEIS frontend

A153091 a(n) = least positive k such that k5^n(5^n-1)+j is prime, with j = -1 or 1 or both.

1, 1, 3, 1, 2, 5, 5, 1, 2, 2, 18, 12, 12, 7, 1, 1, 4, 1, 9, 2, 36, 10, 70, 1, 3, 16, 6, 3, 2, 9, 74, 4, 6, 19, 20, 8, 14, 2, 2, 62, 3, 29, 47, 11, 47, 16, 58, 1, 49, 18, 51, 3, 12, 5, 18, 23, 1, 19, 54, 7, 35, 12, 7, 1, 12, 3, 5, 121, 70, 89, 12, 61, 33, 36, 9, 17, 135, 35, 21, 23, 20, 86, 18

Offset: 1

Pierre CAMI, Dec 18 2008

nonn

			For n = 1, 1*5^1*(5^1-1)-1 = 19 is prime, so a(1) = 1.
For n = 2, 1*5^2*(5^2-1)-1 = 599 is prime, as well as 1*5^2*(5^2-1)+1 = 601, so a(2) = 1.
For n = 3, k = 3 is the least k satisfying the required condition: 3*5^3*(5^3-1)-1 = 46499 is prime, so a(3) = 3.

Pierre CAMI, Table of n, a(n) for n = 1..1000.

Cf. A152414, A152090, A152092, A152093, A152094, A152095.

A153091[n_] := Module[{k = 0}, While[NoneTrue[++k*# + {-1, 1}, PrimeQ]] & [5^n*(5^n - 1)]; k];
Array[A153091, 100] (* Paolo Xausa, Jun 30 2025 *)

Limit_{n->oo} ( (Sum_{i=1..n} a(i)) / (n*(n+1)/2) ) = 13*log(5)/40.
Limit_{n->oo} ( (Sum_{i=1..n} a(2*i)) / (n*(n+1)) ) = log(5)/4.
Limit_{n->oo} ( (Sum_{i=1..n} a(2*i+1)) / (n*(n+2)) ) = 2*log(5)/5.

a(5) corrected by Paolo Xausa, Jun 30 2025

A153090 Least k(n) such that k(n)3^n(3^n-1)+j is prime with j= -1 or 1 or both.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 14, 5, 5, 5, 1, 7, 6, 5, 7, 12, 1, 5, 1, 6, 29, 23, 20, 8, 6, 6, 9, 2, 10, 18, 19, 13, 57, 1, 1, 9, 10, 8, 5, 8, 8, 26, 5, 5, 6, 39, 41, 6, 9, 50, 6, 32, 6, 4, 8, 2, 79, 28, 23, 33, 78, 31, 35, 19, 32, 46, 7, 6, 116, 39, 7, 20, 6, 35, 8

Offset: 1

Pierre CAMI, Dec 18 2008

nonn

Sum_{n=1..k} a(n) / Sum_{n=1..k} n tends to 2*log(3)/7.

			1*3^1*(3^1-1)-1=5 prime as 7 so k(1)=1 1*3^2*(3^2-1)-1=71 prime as 73 so k(2)=1

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A152414, A152091, A152092, A152093, A152094, A152095.

lk[n_]:=Module[{c=3^n (3^n-1),k=1},While[NoneTrue[k*c+{1,-1},PrimeQ],k++];k]; Array[lk,90] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 23 2020 *)

Corrected by Harvey P. Dale, Dec 23 2020

A153092 Least k(n) such that k(n)6^n(6^n-1)+j is prime with j= -1 or 1 or both.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 3, 2, 20, 4, 5, 2, 9, 2, 27, 7, 12, 3, 3, 2, 3, 6, 6, 1, 18, 8, 2, 17, 2, 14, 55, 1, 18, 4, 59, 18, 30, 26, 32, 3, 14, 59, 42, 35, 40, 22, 7, 17, 26, 6, 28, 3, 15, 11, 6, 32, 30, 18, 14, 4, 85, 3, 1, 65, 13, 64, 7, 18, 40, 8, 68, 5, 5, 6, 107, 7, 88, 25, 6, 3, 1, 21, 8, 12, 9

Offset: 1

Pierre CAMI, Dec 18 2008

nonn

When n increases sum k(n) for i=1 to n / sum n for i=1 to n tends to 2*log(6)/9.

			1*6^1*(6^1-1)-1=29 prime as 31 so k(1)=1.

Pierre CAMI, Table of n, a(n) for n = 1..1000.

Cf. A152414, A152090, A152091, A152093, A152094, A152095.

lkn[n_]:=Module[{c=6^n (6^n-1),k=1},While[NoneTrue[k*c+{1,-1},PrimeQ],k++];k]; Array[lkn,90] (* Harvey P. Dale, Feb 29 2024 *)

A153094 Least k(n) such that k(n)m^n(m^n-1)+j is prime with j= -1 or 1 or both and 1

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 2, 1, 1, 1, 4, 1, 1, 1, 3, 5, 2, 1, 1, 3, 2, 2, 2, 9, 6, 1, 6, 4, 1, 1, 4, 1, 2, 2, 3, 6, 6, 5, 1, 5, 5, 1, 2, 1, 10, 3, 5, 1, 3, 2, 1, 4, 5, 2, 2, 3, 1, 1, 2, 3, 5, 5, 6, 1, 12, 3, 5, 6, 1, 2, 9, 3, 4, 1, 1, 5, 2, 3, 4, 7, 2, 2, 16, 8, 5, 1, 5, 5, 6, 3, 2, 11, 2, 2

Offset: 1

Pierre CAMI, Dec 18 2008

nonn

when n increases sum k(n) for i=1 to n / sum n for i=1 to n tends to 0.05, 0

			1*2^1*(2^1-1)+1=3 prime so k(1)=1

Pierre CAMI, Table of n, a(n) for n = 1..1000

A152414, A152090, A152091, A152092, A152093, A152095

A153095 Least m(n) such that k(n)m(n)^n(m(n)^n-1)+j is prime with j= -1 or 1 or both and least possible k(n) with 1

Original entry on oeis.org

2, 2, 3, 2, 2, 3, 3, 5, 2, 2, 2, 3, 8, 7, 5, 4, 2, 2, 4, 3, 6, 2, 2, 5, 4, 7, 6, 4, 3, 2, 2, 2, 2, 6, 3, 3, 7, 2, 2, 2, 5, 2, 2, 3, 2, 2, 2, 5, 7, 2, 9, 5, 4, 4, 2, 2, 2, 8, 7, 4, 8, 4, 6, 5, 4, 5, 5, 4, 3, 9, 2, 2, 6, 3, 8, 8, 4, 2, 4, 8, 6, 2, 2, 4, 4, 5, 6, 4, 4, 8, 4, 6, 4, 7, 3, 4, 6, 7, 5, 2

Offset: 1

Pierre CAMI, Dec 18 2008

nonn

			1*2^1*(2^1-1)+1=3 prime so m(1)=2 1*2^2*(2^2-1)-1=11 as 13 so m(2)=2 1*2^3*(2^3-1)-1=55 composite 1*2^3*(2^3-1)+1=57 composite 1*3^3*(3^3-1)-1=71 prime as 73 so m(3)=3

Pierre CAMI, Table of n, a(n) for n = 1..1000

A152414, A152090, A152091, A152092, A152093, A152094

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cp's OEIS Frontend

A153091 a(n) = least positive k such that k*5^n*(5^n-1)+j is prime, with j = -1 or 1 or both.

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A153090 Least k(n) such that k(n)*3^n*(3^n-1)+j is prime with j= -1 or 1 or both.

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A153092 Least k(n) such that k(n)*6^n*(6^n-1)+j is prime with j= -1 or 1 or both.

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A153094 Least k(n) such that k(n)*m^n*(m^n-1)+j is prime with j= -1 or 1 or both and 1

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A153095 Least m(n) such that k(n)*m(n)^n*(m(n)^n-1)+j is prime with j= -1 or 1 or both and least possible k(n) with 1

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A153091 a(n) = least positive k such that k5^n(5^n-1)+j is prime, with j = -1 or 1 or both.

A153090 Least k(n) such that k(n)3^n(3^n-1)+j is prime with j= -1 or 1 or both.

A153092 Least k(n) such that k(n)6^n(6^n-1)+j is prime with j= -1 or 1 or both.

A153094 Least k(n) such that k(n)m^n(m^n-1)+j is prime with j= -1 or 1 or both and 1

A153095 Least m(n) such that k(n)m(n)^n(m(n)^n-1)+j is prime with j= -1 or 1 or both and least possible k(n) with 1