cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A210506 Numbers k such that (11^k - 2^k)/9 is prime.

Original entry on oeis.org

2, 5, 11, 13, 331, 599, 18839, 23747, 24371, 29339, 32141, 67421
Offset: 1

Views

Author

Robert Price, Jan 25 2013

Keywords

Comments

All terms are prime.
a(13) > 10^5.

Crossrefs

Cf. A004063, A028491, A057468, A059801, A121877, A128024, A128025, A128026, A128027, A128028, A128029, A128030, A128031, A128032.

Programs

  • Mathematica
    Select[ Prime[ Range[1, 100000] ], PrimeQ[ (11^# - 2^#)/9 ]& ]
  • PARI
    is(n)=isprime((11^n-2^n)/9) \\ Charles R Greathouse IV, Feb 17 2017

A058013 Smallest prime p such that (n+1)^p - n^p is prime.

Original entry on oeis.org

2, 2, 2, 3, 2, 2, 7, 2, 2, 3, 2, 17, 3, 2, 2, 5, 3, 2, 5, 2, 2, 229, 2, 3, 3, 2, 3, 3, 2, 2, 5, 3, 2, 3, 2, 2, 3, 3, 2, 7, 2, 3, 37, 2, 3, 5, 58543, 2, 3, 2, 2, 3, 2, 2, 3, 2, 5, 3, 4663, 54517, 17, 3, 2, 5, 2, 3, 3, 2, 2, 47, 61, 19, 23, 2, 2, 19, 7, 2, 7, 3, 2, 331, 2, 179, 5, 2, 5, 3, 2, 2
Offset: 1

Views

Author

Robert G. Wilson v, Nov 13 2000

Keywords

Comments

The terms a(47) and a(60) [were] unknown. The sequence continues at a(48): 2, 3, 2, 2, 3, 2, 2, 3, 2, 5, 3, 4663, a(60)=?, continued at a(61): 17, 3, 2, 5, 2, 3, 3, 2, 2, 47, 61, 19, 23, 2, 2, 19, 7, 2, 7, 3, 2, 331, 2, 179, 5, 2, 5, 3, 2, 2. - Hugo Pfoertner, Aug 27 2004
In September and November 2005, Jean-Louis Charton found a(60)=54517 and a(47)=58543, respectively. Earlier, Mike Oakes found a(106)=7639 and a(124)=5839. All these large values of a(n) yield probable primes. - T. D. Noe, Dec 05 2005, Sep 18 2008
a(106) = 6529 and a(124) = 5167 are true.
a(137) is probably 196873 from prime of this form discovered by Jean-Louis Charton in December 2009 and reported to Henri Lifchitz's PRP Top. - Robert Price, Feb 17 2012
a(138) through a(150) is 2,>32401,2,2,3,8839,5,7,2,3,5,271,13. - Robert Price, Feb 17 2012
a(276)=88301, a(139)>240000 and a(256)>100000. - Jean-Louis Charton, Jun 27 2012
Three more terms found, a(325)=81943, a(392)=64747, a(412)=56963 and also a(139)>260000, a(295)>100000, a(370)>100000, a(373)>100000. 29 unknown terms < 1000 remain. - Jean-Louis Charton, Aug 15 2012
Three more terms a(577)=55117, a(588)=60089 and a(756)=96487. - Jean-Louis Charton, Dec 13 2012
Three more (PRP) terms a(845)=83761, a(897)=48311, a(918)=54919. - Jean-Louis Charton, Dec 31 2013.
Some of the results were computed using the PrimeFormGW (PFGW) primality-testing program. - Hugo Pfoertner, Nov 14 2019

Links

Crossrefs

Cf. A065913, A103794, A115596, A247244.
Cf. A000043, A057468, A059801, A059802, A062572-A062666, A215538.

Programs

  • Mathematica
    lmt = 10000; f[n_] := Block[{p = 2}, While[p < lmt && !PrimeQ[(n+1)^p - n^p], p = NextPrime@ p]; If[p > lmt, 0, p]]; Do[ Print[{n, f[n] // Timing}], {n, 1000}] (* Robert G. Wilson v, Dec 01 2014 *)
spp[n_]:=Module[{p=2},While[!PrimeQ[(n+1)^p-n^p],p=NextPrime[p]];p]; Array[spp,90] (* Harvey P. Dale, Jul 01 2025 *)
  • PARI
    a(n)=forprime(p=2,default(primelimit),if(ispseudoprime((n+1)^p-n^p),return(p))) \\ Charles R Greathouse IV, Feb 20 2012

Formula

a((p-1)/2) = 2 for odd primes p. - Alexander Adamchuk, Dec 01 2006

Extensions

More terms from T. D. Noe, Dec 05 2005
Typo in first Mathematica program corrected by Ray Chandler, Feb 22 2017

A128067 Numbers k such that (3^k + 7^k)/10 is prime.

Original entry on oeis.org

3, 13, 31, 313, 3709, 7933, 14797, 30689, 38333
Offset: 1

Views

Author

Alexander Adamchuk, Feb 14 2007

Keywords

Comments

All terms are primes.
a(10) > 10^5. - Robert Price, Oct 03 2012

Crossrefs

Cf. A007658 = numbers n such that (3^n + 1)/4 is prime. Cf. A057469 = numbers n such that (3^n + 2^n)/5 is prime. Cf. A122853 = numbers n such that (3^n + 5^n)/8 is prime. Cf. A128066, A128068, A128069, A128070, A128071, A128072, A128073, A128074, A128075. Cf. A059801 = numbers n such that 4^n - 3^n is prime. Cf. A121877 = numbers n such that (5^n - 3^n)/2 is a prime. Cf. A128024, A128025, A128026, A128027, A128028, A128029, A128030, A128031, A128032.

Programs

  • Mathematica
    k=7; Do[ p=Prime[n]; f=(3^p+k^p)/(k+3); If[ PrimeQ[f], Print[p]], {n,1,100} ]
  • PARI
    is(n)=isprime((3^n+7^n)/10) \\ Charles R Greathouse IV, Feb 17 2017

Extensions

More terms from Ryan Propper, Apr 02 2007
a(7)-a(9) from Robert Price, Oct 03 2012

A128069 Numbers k such that (3^k + 10^k)/13 is prime.

Original entry on oeis.org

3, 19, 31, 101, 139, 167, 1097, 43151, 60703, 90499
Offset: 1

Views

Author

Alexander Adamchuk, Feb 14 2007

Keywords

Comments

All terms are primes.
Next term is greater than 6700. - Stefan Steinerberger, May 11 2007
a(11) > 10^5. - Robert Price, Jan 15 2013

Crossrefs

Cf. A007658 = numbers n such that (3^n + 1)/4 is prime. Cf. A057469 = numbers n such that (3^n + 2^n)/5 is prime. Cf. A122853 = numbers n such that (3^n + 5^n)/8 is prime. Cf. A128066, A128067, A128068, A128070, A128071, A128072, A128073, A128074, A128075. Cf. A059801 = numbers n such that 4^n - 3^n is prime. Cf. A121877 = numbers n such that (5^n - 3^n)/2 is a prime. Cf. A128024, A128025, A128026, A128027, A128028, A128029, A128030, A128031, A128032.

Programs

  • Mathematica
    k=10; Do[ p=Prime[n]; f=(3^p+k^p)/(k+3); If[ PrimeQ[f], Print[p]], {n,1,100} ]
  • PARI
    is(n)=isprime((3^n+10^n)/13) \\ Charles R Greathouse IV, Feb 17 2017

Extensions

a(7) from Alexander Adamchuk, Feb 14 2007
a(8)-a(10) from Robert Price, Jan 15 2013

A128070 Numbers k such that (3^k + 11^k)/14 is prime.

Original entry on oeis.org

3, 103, 271, 523, 23087, 69833
Offset: 1

Views

Author

Alexander Adamchuk, Feb 14 2007

Keywords

Comments

All terms are primes.
a(7) > 10^5. - Robert Price, Mar 04 2013

Crossrefs

Cf. A007658 (numbers k such that (3^k + 1)/4 is prime).
Cf. A057469 (numbers k such that (3^k + 2^k)/5 is prime).
Cf. A122853 (numbers k such that (3^k + 5^k)/8 is prime).
Cf. A128066, A128067, A128068, A128069, A128071, A128072, A128073, A128074, A128075.
Cf. A059801 (numbers k such that 4^k - 3^k is prime).
Cf. A121877 (numbers k such that (5^k - 3^k)/2 is prime).
Cf. A128024, A128025, A128026, A128027, A128028, A128029, A128030, A128031, A128032.

Programs

  • Mathematica
    k=11; Do[ p=Prime[n]; f=(3^p+k^p)/(k+3); If[ PrimeQ[f], Print[p]], {n,1,100} ]
  • PARI
    is(n)=isprime((3^n+11^n)/14) \\ Charles R Greathouse IV, Feb 17 2017

Extensions

a(5)-a(6) from Robert Price, Mar 04 2013

A128068 Numbers k such that (3^k + 8^k)/11 is prime.

Original entry on oeis.org

5, 163, 191, 229, 271, 733, 21059, 25237
Offset: 1

Views

Author

Alexander Adamchuk, Feb 14 2007

Keywords

Comments

All terms are primes.
a(9) > 10^5. - Robert Price, Mar 06 2013

Crossrefs

Cf. A007658 = numbers n such that (3^n + 1)/4 is prime. Cf. A057469 = numbers n such that (3^n + 2^n)/5 is prime. Cf. A122853 = numbers n such that (3^n + 5^n)/8 is prime. Cf. A128066, A128067, A128069, A128070, A128071, A128072, A128073, A128074, A128075. Cf. A059801 = numbers n such that 4^n - 3^n is prime. Cf. A121877 = numbers n such that (5^n - 3^n)/2 is a prime. Cf. A128024, A128025, A128026, A128027, A128028, A128029, A128030, A128031, A128032.

Programs

  • Mathematica
    k=8; Do[ p=Prime[n]; f=(3^p+k^p)/(k+3); If[ PrimeQ[f], Print[p]], {n,1,100} ]
  • PARI
    is(n)=isprime((3^n+8^n)/11) \\ Charles R Greathouse IV, Feb 17 2017

Extensions

a(6) from Alexander Adamchuk, Feb 14 2007
a(7)-a(8) from Robert Price, Mar 06 2013

A062577 Numbers k such that 11^k - 10^k is prime.

Original entry on oeis.org

3, 5, 19, 311, 317, 1129, 4253, 7699, 18199, 35153, 206081
Offset: 1

Views

Author

Mike Oakes, May 18 2001, May 19 2001

Keywords

Comments

Terms greater than 1000 may correspond to unproven strong pseudoprimes.

Links

Crossrefs

Cf. A000043, A057468, A059801, A059802, A062572-A062666.

Programs

Extensions

Two more terms 18199 and 35153 from Jean-Louis Charton, Sep 02 2009
New term 206081 found by Jean-Louis Charton in October 2011
Edited by M. F. Hasler, Sep 16 2013

A087490 Primes p such that 4^p - 3^p is composite.

Original entry on oeis.org

5, 11, 13, 19, 23, 29, 31, 37, 41, 43, 47, 53, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 293, 307
Offset: 1

Views

Author

Cino Hilliard, Oct 26 2003

Keywords

Comments

Primes not in A059801. - Robert Israel, Nov 03 2024

Links

Crossrefs

Cf. A005061, A059801.
Primes p such that k^p - (k-1)^p is composite: A087489 (k=3), this sequence (k=4), A087685 (k=5), A087749 (k=6), A087759 (k=7), A087763 (k=8), A087894 (k=9), A087895 (k=10).

Programs

  • Maple
    filter:= p -> isprime(p) and not isprime(4^p-3^p):
select(filter, [seq(i,i=3..1000,2)]); # Robert Israel, Nov 03 2024
  • Mathematica
    Select[Prime[Range[70]],CompositeQ[4^#-3^#]&] (* Harvey P. Dale, Mar 14 2025 *)
  • PARI
    apmb(a,b,n) = { forprime(x=2,n, y=a^x-b^x; if(!ispseudoprime(y), print1(x","); ) ) }

Extensions

Offset corrected by Mohammed Yaseen, Jul 18 2022

A062576 Numbers k such that 10^k - 9^k is prime.

Original entry on oeis.org

2, 3, 7, 11, 19, 29, 401, 709, 2531, 15787, 66949, 282493
Offset: 1

Views

Author

Mike Oakes, May 18 2001, May 19 2001

Keywords

Comments

Terms > 1000 are often only strong pseudoprimes.
All terms are prime. - Alexander Adamchuk, Apr 27 2008

Examples

			10^2 - 9^2 = 100 - 81 = 19, which is prime, hence 2 is in the sequence.
10^3 - 9^3 = 1000 - 729 = 271, which is prime, hence 3 is in the sequence.
10^4 - 9^4 = 10000 - 6561 = 3439 = 19 * 181, which is not prime, hence 4 is not in the sequence.

Links

Crossrefs

Cf. A000043, A057468, A059801, A059802, A059803 (9^n-8^n is prime), A062572-A062666.
Cf. A016189 = 10^n - 9^n, and A199819 (primes of this form).

Programs

Extensions

Three more terms 15787, 66949 and 282493 found by Jean-Louis Charton in 2004 and 2007

A125713 Smallest odd prime p such that (n+1)^p - n^p is prime.

Original entry on oeis.org

3, 3, 3, 3, 5, 3, 7, 7, 3, 3, 3, 17, 3, 3, 43, 5, 3, 1607, 5, 19, 127, 229, 3, 3, 3, 13, 3, 3, 149, 3, 5, 3, 23, 3, 5, 83, 3, 3, 37, 7, 3, 3, 37, 5, 3, 5, 58543, 3, 3, 7, 29, 3, 479, 5, 3, 19, 5, 3, 4663, 54517, 17, 3, 3, 5, 7, 3, 3, 17, 11, 47, 61, 19, 23, 3, 5, 19, 7, 5, 7, 3, 3
Offset: 1

Views

Author

Alexander Adamchuk, Dec 01 2006, Feb 15 2007

Keywords

Comments

Corresponding smallest primes of the form (n+1)^p - n^p, where p = a(n) is an odd prime, are listed in A121091(n+1) = {7, 19, 37, 61, 4651, 127, 1273609, 2685817, 271, 331, 397, 6431804812640900941, 547, 631, ...}. a(n) = A058013(n) for n = {4, 7, 10, 12, 13, 16, 17, 19, 22, 24, 25, 27, 28, 31, ...} = A047845(n) = (n-1)/2, where n runs through odd nonprimes (A014076), for n>1. a(97) = 7. a(99)..a(112) = {5, 43, 5, 13, 7, 5, 3, 6529, 59, 3, 5, 5, 113, 5}. a(114) = 139. a(117)..a(129) = {7, 13, 3, 5, 5, 7, 3, 5167, 3, 41, 59, 3, 3}. a(131) = 101. a(n) is currently unknown for n = {113, 115, 116, 130, 132, ...}.
a(96) = 1307, a(98) = 709.
a(137) is probably 196873 from a prime of this form discovered by Jean-Louis Charton in December 2009 and reported to Henri Lifchitz's PRP Top. - Robert Price, Feb 17 2012
a(138) through a(150) are 113, >32401, 3, 7, 3, 8839, 5, 7, 13, 3, 5, 271, 13. - Robert Price, Feb 17 2012
a(137) = 196873 confirmed by Fischer link; a(139) > 260000. - Ray Chandler, Feb 26 2017

Links

Crossrefs

Cf. A058013 (smallest prime p such that (n+1)^p - n^p is prime).
Cf. A065913 (smallest prime of form (n+1)^k - n^k).
Cf. A121091 (smallest nexus prime of the form n^p - (n-1)^p, where p is odd prime).
Cf. A047845, A014076.
Cf. A062585 (numbers n such that k^n - (k-1)^n is prime, where k is 19).
Cf. A000043, A057468, A059801, A059802, A062572-A062666.
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