A156018 -id:A156018 - cp's OEIS frontend

A156021 Numbers k such that k^1 + k^2 + k^3 + k^4 -+ 1 are twin primes.

1, 2, 12, 30, 44, 50, 63, 74, 110, 165, 177, 222, 239, 254, 327, 492, 519, 804, 942, 954, 1007, 1343, 1352, 1520, 1770, 2375, 2450, 2658, 2795, 2945, 2994, 3075, 3332, 3527, 3548, 3803, 3915, 3935, 4025, 4653, 4704, 4785, 4808, 4862, 5270, 5310, 5364, 5370

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 01 2009

			2 is a term since 2 + 2^2 + 2^3 + 2^4 - 1 = 29 and 2 + 2^2 + 2^3 + 2^4 + 1 = 31 are twin primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A125964, A156018.

[n: n in [1..6*10^3] | IsPrime(n^4+n^3+n^2+n-1) and IsPrime(n^4+n^3+n^2+n+1)]; // Vincenzo Librandi, Dec 26 2015

lst={};Do[p=(n^1+n^2+n^3+n^4);If[PrimeQ[p-1]&&PrimeQ[p+1],AppendTo[lst,n]],{n,8!}];lst

A156026 Lesser of twin primes of the form k^1 + k^2 + k^3 + k^4 - 1.

Original entry on oeis.org

3, 29, 22619, 837929, 3835259, 6377549, 16007039, 30397349, 147753209, 745720139, 987082979, 2439903209, 3276517919, 4178766089, 11468884079, 58714318139, 72695416559, 418374010739, 788251653689, 829180295189

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 01 2009

nonn

The corresponding values of k are 1, 2, 12, 30, 44, ... (A156021).

			29 is a term since 2 + 2^2 + 2^3 + 2^4 - 1 = 29 and 2 + 2^2 + 2^3 + 2^4 + 1 = 31 are twin primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001359, A125964, A156018, A156021.

lst={};Do[p=(n^1+n^2+n^3+n^4);If[PrimeQ[p1=p-1]&&PrimeQ[p2=p+1],AppendTo[lst,p1]],{n,8!}];lst
Select[Table[n+n^2+n^3+n^4-1,{n,1000}],AllTrue[{#,#+2},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 17 2019 *)

A156027 Greater of twin primes pairs of the form k^1 + k^2 + k^3 + k^4 - 1.

Original entry on oeis.org

5, 31, 22621, 837931, 3835261, 6377551, 16007041, 30397351, 147753211, 745720141, 987082981, 2439903211, 3276517921, 4178766091, 11468884081, 58714318141, 72695416561, 418374010741, 788251653691, 829180295191, 1029317536801, 3255573820801, 3343706188681

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 01 2009

nonn

The corresponding values of k: 1, 2, 12, 30, 44, 50, 63, 74, 110, 165, 177, 222, 239, 254, 327, 492, 519, 804, 942, 954,...

			2 + 2^2 + 2^3 + 2^4 - 1 = 29 and 2 + 2^2 + 2^3 + 2^4 + 1 = 31.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A125964, A156018, A156021, A156026.

[p+2:k in [1..1500] | IsPrime(p) and IsPrime(p+2) where p is k^1+k^2+k^3+k^4-1]; // Marius A. Burtea, Dec 21 2019

lst={};Do[p=(n^1+n^2+n^3+n^4);If[PrimeQ[p1=p-1]&&PrimeQ[p2=p+1],AppendTo[lst,p2]],{n,8!}];lst

More terms from Amiram Eldar, Dec 21 2019

A188269 Prime numbers of the form k^4 + k^3 + 4k^2 + 7k + 5 = k^4 + (k+1)^3 + (k+2)^2.

Original entry on oeis.org

5, 59, 348077, 10023053, 30414227, 55367063, 72452489, 85856933, 109346759, 182679473, 254112143, 305966369, 433051637, 727914497, 2029672529, 4178961167, 6528621257, 8346080159, 12783893813, 17220494579, 17993776223, 19618171127, 23673478589, 29448235247, 43333033853

Offset: 1

Rafael Parra Machio, Jun 09 2011

nonn

Bunyakovsky's conjecture implies that this sequence is infinite. - Charles R Greathouse IV, Jun 09 2011

All the terms in the sequence are congruent to 2 mod 3. - K. D. Bajpai, Apr 11 2014

			5 is prime and appears in the sequence because 0^4 + 1^3 + 2^2 = 5.
59 is prime and appears in the sequence because 2^4 + 3^3 + 4^2 = 59.
348077 = 24^4 + (24+1)^3 + (24+2)^2 = 24^4 + 25^3 + 26^2.
10023053 = 56^4 + (56+1)^3 + (56+2)^2 = 56^4 + 57^3 + 58^2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A088548, A088550, A156018.

select(isprime, [n^4+(n+1)^3+(n+2)^2$n=0..1000])[]; # K. D. Bajpai, Apr 11 2014

lst={};Do[If[PrimeQ[p=n^4+n^3+4*n^2+7*n+5], AppendTo[lst, p]],{n,200}];lst
Select[Table[n^4+n^3+4n^2+7n+5,{n,500}],PrimeQ] (* Harvey P. Dale, Jun 19 2011 *)

for(n=1,1e3,if(isprime(k=n^4+n^3+4*n^2+7*n+5),print1(k", "))) \\ Charles R Greathouse IV, Jun 09 2011

Duplicate Mathematica program deleted by Harvey P. Dale, Jun 19 2011

Missing term 5 inserted by Alois P. Heinz, Sep 21 2024

cp's OEIS Frontend

A156021 Numbers k such that k^1 + k^2 + k^3 + k^4 -+ 1 are twin primes.

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A156026 Lesser of twin primes of the form k^1 + k^2 + k^3 + k^4 - 1.

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A156027 Greater of twin primes pairs of the form k^1 + k^2 + k^3 + k^4 - 1.

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A188269 Prime numbers of the form k^4 + k^3 + 4*k^2 + 7*k + 5 = k^4 + (k+1)^3 + (k+2)^2.

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A188269 Prime numbers of the form k^4 + k^3 + 4k^2 + 7k + 5 = k^4 + (k+1)^3 + (k+2)^2.