A265481 -id:A265481 - cp's OEIS frontend

A098845 Numbers k such that 4^k - 2^k - 1 is prime.

2, 4, 5, 9, 10, 18, 38, 45, 50, 57, 108, 161, 208, 224, 225, 240, 354, 597, 634, 1008, 1080, 1468, 1525, 1560, 3298, 3329, 3846, 4129, 5430, 8616, 11834, 12988, 14610, 43401, 45306, 53776, 54449, 67497, 74025, 122449, 136845, 142896, 164541, 171157, 187668, 274054, 316944, 349296

Offset: 1

Pierre CAMI, Oct 10 2004; extended several times: Jun 01 2005, Jun 19 2006, May 03 2007

nonn

All primes certified using PFGW from primeform group. - Pierre CAMI, Mar 07 2005
No terms 2, 3, 7, 12, 13 or 15 (mod 20) except 2. - Robert Israel, Dec 08 2015, updated by Fabrice Lavier, Jan 10 2019
Using such "Goldilocks" primes (a term coined by Mike Hamburg) as modulus facilitates use of Karatsuba multiplication in elliptic-curve cryptography. - Francois R. Grieu, Mar 25 2021

Chris Caldwell, The largest known primes
Mike Hamburg, Ed448-Goldilocks, a new elliptic curve, Cryptology ePrint Archive, Report 2015/625.

Cf. similar sequences listed in A265481.

[n: n in [0..1000] | IsPrime(2^n*(2^n-1)-1)]; // Vincenzo Librandi, Dec 08 2015

select(t -> isprime(4^t-2^t-1), [$1..1000]); # Robert Israel, Dec 08 2015

Select[Range[15000], PrimeQ[4^# - 2^# - 1] &] (* Vincenzo Librandi, Dec 08 2015 *)

for(n=1, 1e3, if(ispseudoprime(4^n-2^n-1), print1(n, ", "))) \\ Altug Alkan, Dec 08 2015

from sympy import isprime
for n in range(1,1000):
    if isprime(4**n-2**n-1):
        print(n, end=', ') # Stefano Spezia, Jan 11 2019

Extended to a(44) = 349296 (2^698592 - 2^349296 - 1 is a 210298-digit certified prime) by Pierre CAMI, Jan 11 2009
Definition simplified by Pierre CAMI, May 10 2012
a(30) corrected by Robert Israel, Dec 14 2015
4 missing terms between a(41) = 136845 and what is now a(46) = 274054 added by Fabrice Lavier, Jan 10 2019

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

A265483 Numbers k such that 25^k - 5^k - 1 is prime.

Original entry on oeis.org

1, 2, 4, 15, 16, 24, 57, 206, 284, 1290, 1722, 1862, 1866, 3271, 5306, 5474, 15401, 18729, 34757, 42842, 63930, 89967

Offset: 1

Vincenzo Librandi, Dec 11 2015

For k = 1, 2, 4, 15, 16, the corresponding primes are  19, 599, 389999, 931322574584960937499, 23283064365234374999999.
a(n) is not of the form 5*m + 3 (divisibility by 11) or 9*m + 10 (divisibility by 19), 7*m + (-1)^m + 7 (divisibility by 29) or 29*m + 27 (divisibility by 59).
a(23) > 10^5. - Robert Price, Dec 12 2019

			4 is in the sequence because 25^4-5^4-1 = 389999 is prime.

Cf. similar sequences listed in A265481.

[n: n in [0..300] | IsPrime(25^n-5^n-1)];

Select[Range[6000], PrimeQ[25^# - 5^# - 1] &]

is(n)=ispseudoprime(25^n - 5^n - 1) \\ Anders Hellström, Dec 11 2015

a(17)-a(22) from Robert Price, Dec 12 2019

A265484 Numbers k such that 36^k - 6^k - 1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 24, 63, 81, 92, 165, 232, 236, 591, 669, 1343, 2973, 3630, 5600, 7019, 17169, 31945, 52172

Offset: 1

Vincenzo Librandi, Dec 12 2015

For k = 1, 2, 3, 4, 24 the corresponding primes are 29, 1259, 46439, 1678319, 22452257707354557235348829785471057919.

a(22) > 10^5. - Robert Price, Jan 06 2020

			4 is in the sequence because 36^4 - 6^4 - 1 = 1678319 is prime.

Cf. similar sequences listed in A265481.

[n: n in [0..300] | IsPrime(36^n-6^n-1)];

Select[Range[1000], PrimeQ[36^# - 6^# - 1] &]

for(n=1, 1e3, if(ispseudoprime(36^n - 6^n - 1), print1(n, ", "))) \\ Altug Alkan, Dec 12 2015

a(14)-a(18) from Ray Chandler, Sep 25 2019

a(18) corrected and a(19)-a(21) added by Robert Price, Jan 06 2020

A265485 Numbers k such that 49^k - 7^k - 1 is prime.

Original entry on oeis.org

1, 2, 12, 14, 110, 2574, 10964, 24257, 44364

Offset: 1

Vincenzo Librandi, Dec 12 2015

For k = 1, 2, 12, 14 the corresponding primes are 41, 2351, 191581231366725127199, 459986536544061737903951.

a(10) > 10^5. - Robert Price, Dec 25 2019

			2 is in the sequence because 49^2 - 7^2 - 1 = 2351 is prime.

Cf. similar sequences listed in A265481.

[n: n in [0..1000] | IsPrime(49^n - 7^n - 1)];

Select[Range[2000], PrimeQ[49^# - 7^# - 1] &]

for(n=1, 1e3, if(ispseudoprime(49^n - 7^n - 1), print1(n, ", "))) \\ Altug Alkan, Dec 12 2015

a(6) from Ray Chandler, Sep 25 2019

a(7)-a(9) from Robert Price, Dec 25 2019

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

A265487 Numbers k such that 81^k - 9^k - 1 is prime.

Original entry on oeis.org

1, 3, 10, 18, 70, 585, 921, 1943, 4635, 13543, 13803, 15938, 39004

Offset: 1

Vincenzo Librandi, Dec 12 2015

2*a(n) is in A265481.
For k = 1, 3, 10, 18 the corresponding primes are 71, 530711, 12157665455570144399, 22528399544939174261745512577773519.
a(n) is not of the form 5*k+2 (divisibility by 11), 9*k+2 (divisibility by 19), 7*k+2*(-1)^k+7 (divisibility by 29), 15*k+2 (divisibility by 31) or 29*k+8 (divisibility by 59).
a(14) > 10^5. - Robert Price, Apr 21 2020

			3 is in the sequence because 81^3 - 9^3 - 1 = 530711 is prime.

Cf. similar sequences listed in A265481.

[n: n in [0..500] | IsPrime(81^n-9^n-1)];

Select[Range[1000], PrimeQ[81^# - 9^# - 1] &]

for(n=1, 1e3, if(ispseudoprime(81^n - 9^n - 1), print1(n, ", "))) \\ Altug Alkan, Dec 12 2015

a(9) from Altug Alkan, Dec 12 2015

a(10)-a(13) computed from A265481 by Ray Chandler, Sep 25 2019

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

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A098845 Numbers k such that 4^k - 2^k - 1 is prime.

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

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

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A265483 Numbers k such that 25^k - 5^k - 1 is prime.

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A265484 Numbers k such that 36^k - 6^k - 1 is prime.

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A265485 Numbers k such that 49^k - 7^k - 1 is prime.

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

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