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.

Previous Showing 31-39 of 39 results.

A240767 Numbers n such that n^k + (n-1)^k + ... + 3^k + 2^k is prime for some natural number k.

Original entry on oeis.org

2, 3, 4, 7, 8, 11, 12, 16
Offset: 1

Views

Author

Derek Orr, Apr 12 2014

Keywords

Comments

a(9) > 19. See A240766 for more information.
a(n) is also the n-values such that A240766(n) is nonzero.
It is known that a(n) must be == 3 mod 4 or 0 mod 4 (except a(1) = 2) due to the parity of the sum. If an n-value is congruent to 1 mod 4 or 2 mod 4, the sum will always be even and thus, not prime.
It is known that 31, 36, 40, 43, 47, 56, 67, 83, and 171 are members of this sequence.
If n-1 is not squarefree, then n is not a member of this sequence.

Examples

			2^k is prime for at least one k (and only one k in this instance; k = 1). Thus, 2 is a member of this sequence.
3^k+2^k is prime for at least one k (see A082101). Thus, 3 is a member of this sequence.

Crossrefs

Cf. A081507, A082101, A240766.

Programs

  • PARI
    a(n)=for(k=1,4000,if(ispseudoprime(sum(i=2,n,i^k)),return(k)))
n=1; while(n<200,if(a(n),print(a(n)));n+=1)

A386618 Primes of the form 2^k + 13^k.

Original entry on oeis.org

2, 173, 815730977
Offset: 1

Views

Author

Vincenzo Librandi, Aug 17 2025

Keywords

Comments

If 13^k + 2^k is prime then k is either 0 or a power of 2. The corresponding values of k for a(1)-a(4) are 0, 2, 8 and 512. The fourth value is too long to enter.

Links

Crossrefs

Cf. A082101, A094475, A094476, A094477, A176946, A176947.

Programs

  • Magma
    [a: n in [0..200] | IsPrime(a) where a is 13^n+2^n ];
  • Mathematica
    Select[Table[2^n+13^n,{n,0,600}],PrimeQ]

A094316 Primes p for which 2^j+p^j is also prime for j in {0,2,8,512}.

Original entry on oeis.org

13, 4133, 1831343, 2320583, 3828673, 9173893, 23658377, 24037537, 42489677, 56253203, 78222863, 96325093, 99846337, 110453773, 110468653, 117748427, 122173187, 130937467, 138072163, 146981537, 174978913, 184050553, 186927817
Offset: 1

Views

Author

Labos Elemer, Jun 02 2004

Keywords

Examples

			Smallest such prime is 13 and the relevant four primes are
2, 173, 815730977 and a 571-digit prime.

Crossrefs

Cf. A082101, A094473-A094499.

Programs

  • Mathematica
    {ta=Table[0, {100}], u=1}; {exponents, {a, b, c, d}={0, 2, 8, 512}} Do[s0=Prime[j]^a+2^a;s1=Prime[j]^b+2^b;s2=Prime[j]^c+2^c;s3=Prime[j]^d+2^d; If[PrimeQ[s0]&&PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s3], Print[{j, Prime[j]}];ta[[u]]=Prime[j];u=u+1], {j, 1, 1000000}] ta

Extensions

a(6)-a(23) from Donovan Johnson, Oct 12 2008

A094478 Primes of form 2^j + 59^j.

Original entry on oeis.org

2, 61, 12117377, 464798130469793589516643498190087912509935907401081390977
Offset: 1

Views

Author

Labos Elemer, Jun 01 2004

Keywords

Comments

The number j must be zero or a power of 2. Checked j being powers of two through 2^21. Thus a(5) > 10^2900000. Primes of this magnitude are rare (about 1 in 6.7 million), so chance of finding one is remote with today's computer algorithms and speeds. - Robert Price, Apr 28 2013

Examples

			j=0: p=1+1=2;
j=1: p=2+59=61;
j=4: p=16+12117361=12117377;
j=32: p=2^32+59^32=464798130469793589516643498190087912509935907401081390977;
the j exponents are powers of 2.

Crossrefs

Cf. A082101, A094473-A094480.

A094483 Primes of form 2^j + 179^j.

Original entry on oeis.org

2, 181, 1026625697, 1110832290554380967776058484990830657
Offset: 1

Views

Author

Labos Elemer, Jun 01 2004

Keywords

Comments

No additional terms through j=1000. - Harvey P. Dale, Apr 24 2013
The number j must be zero or a power of 2. Checked j being powers of two through 2^19. Thus a(5) > 10^2300000. Primes of this magnitude are rare (about 1 in 5.4 million), so chance of finding one is remote with today's computer algorithms and speeds. - Robert Price, May 05 2013

Crossrefs

Cf. A082101, A094473-A094482.

Programs

  • Mathematica
    Select[Table[2^j+179^j, {j,0,30}], PrimeQ] (* Harvey P. Dale, Apr 24 2013 *)

A094484 Primes of form 2^j + 461^j.

Original entry on oeis.org

2, 463, 45165175457, 4161163747708008324368372925882377717624897
Offset: 1

Views

Author

Labos Elemer, Jun 01 2004

Keywords

Comments

The number j must be zero or a power of 2. Checked j being powers of two through 2^19. Thus a(5) > 10^2700000. Primes of this magnitude are rare (about 1 in 6.4 million), so chance of finding one is remote with today's computer algorithms and speeds. - Robert Price, Apr 29 2013

Examples

			The relevant exponents are powers of 2: 0, 1, 4, 16; a(4) = 65536+461^16 = 4161163747708008324368372925882377717624897.

Crossrefs

Cf. A082101, A094473-A094482.

A094493 Primes p such that 2^j+p^j are primes for j=0,1,2,16.

Original entry on oeis.org

43577, 84317, 93887, 108377, 124247, 346667, 379997, 431867, 461297, 579197, 681257, 819317, 863867, 889037, 1143047, 1146797, 1271027, 1306817, 1518707, 1775867, 1926647, 1948517, 2119937, 2177447, 2348807, 2491607, 2604557
Offset: 1

Views

Author

Labos Elemer, Jun 01 2004

Keywords

Comments

Primes of 2^j+p^j form are a generalization of Fermat-primes. 1^j is replaced by p^j. This is strongly supported by the observation that corresponding j-exponents are apparently powers of 2 like for the 5 known Fermat primes. See A094473-A094491.

Examples

			For j=0: 1+1=2 is prime; other conditions are:
because of p^1+2=prime; 3rd and 4th conditions are as
follows: prime=p^2+4 and prime=65536+p^16.

Crossrefs

Cf. A082101, A094473-A094491.

Programs

  • Mathematica
    {ta=Table[0, {100}], u=1}; Do[s0=2;s1=2+Prime[j]^1;s2=4+Prime[j]^2;s16=65536+Prime[j]^16 If[PrimeQ[s0]&&PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s16], Print[{j, Prime[j]}];ta[[u]]=Prime[j];u=u+1], {j, 1, 1000000}]
Select[Prime[Range[2*10^5]],AllTrue[Table[2^k+#^k,{k,{0,1,2,16}}],PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 05 2021 *)

A161558 Primes of form 2^k+3^k-4.

Original entry on oeis.org

31, 271, 2311, 94151567431, 847322163871, 450284043329950831, 239299329793567483011391, 13915193059764305948125655305497609574930528737031, 123329495011708990974900261530856061081804307325717309329809036625289391
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jun 13 2009

Keywords

Comments

Generated by k = 3, 5, 7, 23, 25, 37, 49, 103, 149, 211,....

Crossrefs

Cf. A082101

Programs

  • Mathematica
    lst={};Do[If[PrimeQ[p=2^n+3^n-4],AppendTo[lst,p]],{n,6!}];lst
Select[Table[2^k+3^k-4,{k,200}],PrimeQ] (* Harvey P. Dale, Apr 05 2019 *)

Extensions

Comment added by R. J. Mathar, Oct 04 2009
One more term (a(9)) from Harvey P. Dale, Apr 05 2019

A173640 Primes of form n+2^n+3^n.

Original entry on oeis.org

2, 101, 60083, 11610630703530923996233764322611619865107483053157900065365853867349888133476404509
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Nov 23 2010

Keywords

Comments

For a(5), n > 10000. - Daniel Starodubtsev, Aug 04 2019

Crossrefs

Cf. A082101, A082182, A128447, A128448, A128449, A128450, A128451, A100840, A100841

Programs

  • Mathematica
    Select[Table[n+2^n+3^n,{n,0,6!}],PrimeQ[#]&]
Previous Showing 31-39 of 39 results.

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