A098845 -id:A098845 - cp's OEIS frontend

A107639 Smallest prime a such that (ab)^2 + ab -1 is prime with b prime = 2^(2*n) - 2^n - 1, see A098845 for n.

5, 7, 31, 11, 5, 3, 809, 661, 139, 73, 521, 701, 131, 487, 89, 211, 733, 3011, 311, 8837, 2837, 8831, 1733, 401, 983, 17881, 1783, 19553, 13859

Offset: 1

Pierre CAMI, Jun 10 2005

			2^4-2^2-1=3=b(1) prime, (3*3)^2+3*3-1=89 prime so a(1)=3
2^8-2^4-1=239=b(2) prime, (7*239)^2+7*239-1=2800601 prime so a(2)=7

Cf. A098845.

A265481 Numbers k such that 9^k - 3^k - 1 is prime.

Original entry on oeis.org

1, 2, 3, 6, 7, 20, 35, 36, 140, 523, 1170, 1731, 1842, 3727, 3886, 9270, 11071, 13823, 14451, 27086, 27606, 31876, 78008

Offset: 1

Vincenzo Librandi, Dec 09 2015

For k = 1, 2, 3, 6, 7, 20, 35, the corresponding primes are 5, 71, 701, 530711, 4780781, 12157665455570144399, 2503155504993241551284026887086141.
a(n) is not of the form 4*k+5 (divisibility by 5) or 5*k+4 (divisibility by 11) or 9*k+4*(-1)^k (divisibility by 19).
Conjectures: a(n) is not of the form 7*k+4 or 8*k.
a(17) = 11071 mod 7 is 4, so the first half of the conjecture above is not true. - Robert Price, Sep 25 2019
a(23) = 78008 is divisible by 8, so the second half of the conjecture above is not true. - Robert Price, Sep 25 2019
a(24) > 2*10^5. - Robert Price, Apr 21 2020

			6 is in the sequence because 9^6 - 3^6 - 1 = 530711 is prime.

Cf. numbers n such that k^(2*n)-k^n-1 is prime: A098845 (k=2), this sequence (k=3), A265482 (k=4), A265483 (k=5), A265484 (k=6), A265485 (k=7), A265486 (k=8), A265487 (k=9), A265383 (k=10), A265683 (k=16).

[n: n in [0..500] | IsPrime(9^n-3^n-1)];

Select[Range[1500], PrimeQ[9^# - 3^# - 1] &]

is(n)=ispseudoprime(9^n - 3^n - 1) \\ Anders Hellström, Dec 09 2015

a(17)-a(23) from Robert Price, Sep 25 2019

A156589 a(n) = 4^n - 2^n - 1.

Original entry on oeis.org

-1, 1, 11, 55, 239, 991, 4031, 16255, 65279, 261631, 1047551, 4192255, 16773119, 67100671, 268419071, 1073709055, 4294901759, 17179738111, 68719214591, 274877382655, 1099510579199, 4398044413951, 17592181850111, 70368735789055

Offset: 0

M. F. Hasler, Feb 10 2009

Sequence A098845 lists indices of primes, i.e., a(n) prime <=> n=A098845(k) for some k.
Starting with n=2, binary numbers of the form (n-1)0(n) where n is the index and the number of 1's. It can also be formed by appending a 1 to the right of each term of A129868.
1/a(n) = Sum_{m>0} A000045(m)*2^(-n(m+1)) for n > 0. E.g., 1/a(4) = 0.0000 0001 0001 0010 0011 0101 1000 ... in binary. - Lee A. Newberg, Apr 12 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A000045, A098845, A129868.

[4^n-2^n-1: n in [0..30]]; // Vincenzo Librandi, Apr 13 2018

Table[4^n - 2^n - 1, {n, 0, 25}] (* Vincenzo Librandi, Apr 13 2018 *)

PARI
```
vector(99,n,4^n-2^n-1)
```

G.f.: ( 1-8*x+10*x^2 ) / ( (-1+x)*(2*x-1)*(4*x-1) ). - R. J. Mathar, Oct 21 2014

A191751 Least k such that (2^n-1)*2^n - k is a prime number.

Original entry on oeis.org

0, 1, 3, 1, 1, 5, 3, 11, 1, 1, 25, 29, 3, 13, 3, 7, 39, 1, 13, 23, 3, 5, 69, 11, 39, 13, 15, 31, 99, 83, 117, 31, 9, 11, 25, 67, 45, 1, 39, 47, 45, 71, 69, 77, 1, 131, 67, 101, 55, 1, 9, 41, 13, 43, 33, 233, 1, 113, 7, 29, 45, 55, 99, 41, 261, 5, 15, 343, 9

Offset: 1

Juri-Stepan Gerasimov, Jun 14 2011, Jun 15 2011

nonn

			a(1)=0 because (2^1-1)*2^1 - 0 =    2 is prime,
a(2)=1 because (2^2-1)*2^2 - 1 =   11 is prime,
a(3)=3 because (2^3-1)*2^3 - 3 =   53 is prime,
a(4)=1 because (2^4-1)*2^4 - 1 =  239 is prime,
a(5)=1 because (2^5-1)*2^5 - 1 =  991 is prime,
a(6)-5 because (2^6-1)*2^6 - 5 = 4027 is prime.

Jinyuan Wang, Table of n, a(n) for n = 1..1500 (terms 1..500 from Nathaniel Johnston)

Cf. A098845, A191001, A191620.

Cf. A020522 ((2^n-1)*2^n).

a := proc(n) local k: for k from 0 do if(isprime((2^n-1)*2^n-k))then return k: fi: od: end: seq(a(n), n=1..69); # Nathaniel Johnston, Jun 14 2011

lk[n_]:=Module[{c=2^n,k=0},While[!PrimeQ[c(c-1)-k],k++];k]; Array[lk,70] (* Harvey P. Dale, Jul 02 2018 *)

a(n) = my(x=(2^n-1)*2^n); x - precprime(x); \\ Michel Marcus, Feb 21 2019

A229927 Smallest k >= n such that (4^n-1)*2^k - 1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 8, 6, 7, 13, 11, 16, 103, 12, 16, 52, 26, 16, 26, 34, 38, 28, 23, 22, 26, 24, 50, 41, 30, 28, 32, 46, 31, 34, 202, 34, 35, 40, 47, 113, 46, 50, 44, 54, 58, 46, 51, 48, 130, 59, 64, 101, 60, 62, 94, 74, 88, 98, 71, 234, 67, 93, 83, 101, 308, 64, 92

Offset: 1

Pierre CAMI, Oct 03 2013

nonn

Conjecture: there is no "Riesel" number of the form 4^n-1; that is, a(n) exists for all n.

			(4^1-1)*2^1-1=5 prime so a(1)=1.
(4^2-1)*2^2-1=59 prime so a(2)=2.

Pierre CAMI, Table of n, a(n) for n = 1..2500

Cf. A098845.

sk[n_]:=Module[{k=n,c=4^n-1},While[!PrimeQ[c*2^k-1],k++];k]; Array[sk,70] (* Harvey P. Dale, Jul 30 2020 *)

A265383 Numbers k such that 10^k * (10^k - 1) - 1 is prime.

Original entry on oeis.org

1, 6, 9, 154, 253, 1114, 1390, 2618, 5611, 12871, 15286, 108609, 132574, 164369, 188484

Offset: 1

Serge Batalov, Dec 07 2015

The primes arising from this construction (e.g., 999998999999) are among the primes counted in A266148. In particular, it follows that A266148(a(n)) > 0. - David A. Corneth, May 19 2016

a(16) > 188484. - Ben Meekins, Sep 08 2018

			6 is in the sequence because 10^12 - 10^6 - 1 = 999998999999 is prime.

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).
M. Kamada, Near-repdigit-related prime numbers

Cf. similar sequences listed in A265481.
A098845: Similar sequence in base 2.
A183187: Numbers k such that 10^(2k+1)-10^k-1 is prime, palindromic.
A266148: Number of n-digit primes in which n-1 of the digits are 9's.

[n: n in [0..200] | IsPrime(10^n*(10^n-1)-1)]; // Vincenzo Librandi, Dec 08 2015

Select[Range[15000], PrimeQ[10^# (10^# - 1) - 1] &] (* Vincenzo Librandi, Dec 08 2015 *)

for(n=1,9999,if(ispseudoprime(10^n*(10^n-1)-1),print1(n", ")))

a(11) from Kazuyoshi Asao, Feb 11 2002
a(12) from Serge Batalov, Dec 25 2015
a(13) from Ben Meekins, Feb 16 2016
a(14) from Ben Meekins, Dec 17 2016
a(15) from Ben Meekins, Sep 08 2018

A265482 Numbers k such that 16^k - 4^k - 1 is prime.

Original entry on oeis.org

1, 2, 5, 9, 19, 25, 54, 104, 112, 120, 177, 317, 504, 540, 734, 780, 1649, 1923, 2715, 4308, 5917, 6494, 7305, 22653, 26888, 71448, 93834, 137027, 158472, 174648

Offset: 1

Vincenzo Librandi, Dec 10 2015

For k = 1, 2, 5, 9, 19, 25, the corresponding primes are 11, 239, 1047551, 68719214591, 75557863725639445512191, 1267650600228228275596796362751.
a(n) is not of the form 5*k+6 (divisibility by 11) or 9*k+8 (divisibility by 19) or 7*k+3*(-1)^k (divisibility by 29).
Conjecture: the odd terms are not of the form 8*k+7.
k is in the sequence iff 2*k is in A098845 (terms a(21)-a(30) are derived from that sequence). - Ray Chandler, Sep 25 2019

			5 is in the sequence because 16^5-4^5-1 = 1047551 is prime.

Cf. A098845, similar sequences listed in A265481.

[n: n in [0..500] | IsPrime(16^n-4^n-1)];

Select[Range[2500], PrimeQ[16^# - 4^# - 1] &]

is(n)=ispseudoprime(16^n-4^n-1) \\ Charles R Greathouse IV, Jun 13 2017

A265486 Numbers k such that 64^k - 8^k - 1 is prime.

Original entry on oeis.org

3, 6, 15, 19, 36, 75, 80, 118, 199, 336, 360, 520, 1282, 1810, 2872, 4870, 14467, 15102, 22499, 24675, 45615, 105648, 116432

Offset: 1

Vincenzo Librandi, Dec 12 2015

For k = 3, 6, 15, 19 the corresponding primes are 261631, 68719214591, 1237940039285345090527035391, 20769187434139310370006797241024511.

3*a(n) belongs to A098845 (the terms from a(18) to a(23) are derived from that sequence).

			3 is in the sequence because 64^3 - 8^3 - 1 = 261631 is prime.

Cf. A098845, similar sequences listed in A265481.

[n: n in [0..300] | IsPrime(64^n-8^n-1)];

Select[Range[1000], PrimeQ[64^# - 8^# - 1] &]

for(n=1, 1e3, if(ispseudoprime(64^n - 8^n - 1), print1(n, ", "))) \\ Altug Alkan, Dec 12 2015

A265683 Numbers n such that 16^n * (16^n - 1) - 1 is prime.

Original entry on oeis.org

1, 27, 52, 56, 60, 252, 270, 367, 390, 2154, 3247, 13444, 35724, 46917, 79236, 87324

Offset: 1

Shivam N. Patel, Dec 13 2015

Each of the corresponding numbers of the form 16^n * (16^n - 1) - 1 is almost a repdigit in base 16 (i.e., a repdigit with only one change).

n is in the sequence iff 4*n is in A098845. - Robert Israel, Dec 14 2015

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015)

Cf. A098845, similar sequences listed in A265481.

[n: n in [1..200] | IsPrime(256^n-16^n-1)]; // Vincenzo Librandi, Dec 14 2015

Select[Range[1000], PrimeQ[16^# (16^# - 1) - 1] &] (* Shivam Patel, Dec 13 2015 *)

is(n)=ispseudoprime(16^n * (16^n - 1) - 1) \\ Anders Hellström, Dec 13 2015

a(12)-a(16) computed from A098845 by Ray Chandler, Sep 25 2019

A255682 Numbers k such that 4^k - 2^k - 3 is prime.

Original entry on oeis.org

3, 7, 13, 15, 21, 73, 231, 301, 493, 895, 955, 4069, 5057, 5415, 15325, 19845, 21603, 24577, 37877, 40193, 44283

Offset: 1

Vincenzo Librandi, Mar 02 2015

a(22) > 150000. - Hans Havermann, Jul 18 2015

			3 is in this sequence because 4^3 - 2^3 - 3 = 53.

Cf. A098845.

[n: n in [1..800] | IsPrime(4^n-2^n-3)];

Select[Range[15000], PrimeQ[4^# - 2^# - 3] &]

for(n=1,500,if(ispseudoprime(4^n-2^n-3),print1(n,", "))) \\ Derek Orr, Mar 03 2015

a(15)-a(21) from Hans Havermann, Jul 18 2015

cp's OEIS Frontend

A107639 Smallest prime a such that (a*b)^2 + a*b -1 is prime with b prime = 2^(2*n) - 2^n - 1, see A098845 for n.

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A265481 Numbers k such that 9^k - 3^k - 1 is prime.

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Magma

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A156589 a(n) = 4^n - 2^n - 1.

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A191751 Least k such that (2^n-1)*2^n - k is a prime number.

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Maple

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A229927 Smallest k >= n such that (4^n-1)*2^k - 1 is prime.

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Mathematica

A265383 Numbers k such that 10^k * (10^k - 1) - 1 is prime.

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Magma

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Extensions

A265482 Numbers k such that 16^k - 4^k - 1 is prime.

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Magma

Mathematica

PARI

A265486 Numbers k such that 64^k - 8^k - 1 is prime.

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A107639 Smallest prime a such that (ab)^2 + ab -1 is prime with b prime = 2^(2*n) - 2^n - 1, see A098845 for n.