A070689 -id:A070689 - cp's OEIS frontend

A176070 Numbers of the form k^3+k^2+k+1 that are the product of two distinct primes.

15, 85, 259, 1111, 4369, 47989, 65641, 291919, 2016379, 2214031, 3397651, 3820909, 5864581, 9305311, 13881841, 15687751, 16843009, 19756171, 22030681, 28746559, 62256349, 64160401, 74264821, 79692331, 101412319, 117889591, 172189309, 185518471, 191435329, 270004099, 328985791

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 07 2010

nonn

As k^3 + k^2 + k + 1 = (k + 1) * (k^2 + 1) and k <= 1 does not give a term, k + 1 and k^2 + 1 must be prime so k must be even. - David A. Corneth, May 30 2023

			15 is in the sequence as 15 = 3*5 = 2^3+2^2+2+1; 15 is a product of two distinct primes and of the form k^3 + k^2 + k + 1.

Cf. A002496, A006093, A006881, A053698, A070689, A174969, A176069, A237627 (semiprimes of that form).

f[n_]:=Last/@FactorInteger[n]=={1,1};Select[Array[ #^3+#^2+#+1&,7! ],f[ # ]&]

upto(n) = {my(res = List(), u = sqrtnint(n, 3) + 1); forprime(p = 3, u, c = (p-1)^2 + 1; if(isprime(c), listput(res, c*p))); res} \\ David A. Corneth, May 30 2023

a(n) = (A070689(n + 1) + 1) * (A070689(n + 1)^2 + 1). - David A. Corneth, May 30 2023

Name corrected by and more terms from David A. Corneth, May 30 2023

A279480 Numbers n such that n+1 and n^4+1 are primes.

Original entry on oeis.org

1, 2, 4, 6, 16, 28, 46, 82, 88, 106, 180, 198, 210, 228, 238, 276, 312, 352, 430, 442, 466, 498, 540, 556, 568, 600, 616, 690, 732, 738, 742, 760, 768, 772, 786, 810, 856, 928, 936, 952, 966, 996, 1038, 1150, 1152

Offset: 1

Ely Golden, Dec 12 2016

nonn

For any n > 1 in this sequence, (n+1)*(n^4+1) has the same nonzero digits as its prime factors in base n. - Ely Golden, Dec 12 2016

Ely Golden, Table of n, a(n) for n = 1..1000

Cf. A070689 (the similar sequence for n+1 and n^2+1)

Select[Range@ 2000, Times @@ Boole@ Map[PrimeQ, {# + 1, #^4 + 1}] == 1 &] (* Michael De Vlieger, Dec 13 2016 *)
Select[Range[2000],AllTrue[1+{#,#^4},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 13 2019 *)

list(lim)=my(v=List()); forprime(p=2,lim+1, if(isprime(1+(p-1)^4), listput(v, p-1))); Vec(v) \\ Charles R Greathouse IV, Dec 13 2016

c=1
index=1
while(index<=1000):
    if((is_prime(c+1))&(is_prime(c**4+1))):
        print(str(index)+" "+str(c))
        index+=1
    c+=1
print("complete")

a(n) >> n log^2 n. - Charles R Greathouse IV, Dec 13 2016

A285390 Numbers n such that n + 2, n^2 + 2^2, n^4 + 2^4, n^8 + 2^8 and n^16 + 2^16 are primes.

Original entry on oeis.org

1, 1938357, 12543605, 42052067, 42916485, 48724977, 89483837, 94041095, 94101375, 129136065, 138460937, 145198547, 153488937, 172741167, 182885915, 215999297, 217054557, 217734965, 249913685, 271516605, 277401597, 285433077, 295266797, 306256175

Offset: 1

Zak Seidov, Apr 18 2017

nonn

Are there any n such that n^32 + 2^32 is also prime?

a(87) = 2321204055 and a(156) = 5231769977 are the first 2 terms such that n^32 + 2^32 is prime. - Chai Wah Wu, Sep 03 2020

Chai Wah Wu, Table of n, a(n) for n = 1..6400

Subsequence of A192794. Cf. A070689.

A102368 Smallest k>0 such that n^k + 1 is not prime.

Original entry on oeis.org

3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2

Offset: 2

Reinhard Zumkeller, Feb 22 2005

nonn

a(odd) = 1.

Since n + 1 divides n^3 + 1, a(n) <= 3. - Robert Israel, Jun 15 2014

			n=10: 10^1+1=11=A000040(5), 10^2+1=101=A000040(26), but 10^3+1=1001=7*11*13, therefore a(10)=3.

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A070689: a(n) = 3.

A102368:= proc(n)
  if n::odd or not isprime(n+1) then 1
    elif isprime(n^2+1) then 3 else 2
  fi
end proc; # Robert Israel, Jun 15 2014

sk[n_]:=Module[{k=1},While[PrimeQ[n^k+1],k++];k]; Array[sk,110,2] (* Harvey P. Dale, Apr 09 2016 *)

A236692 Numbers k such that k+1, 2*k+1 and k^2+1 are primes.

Original entry on oeis.org

1, 2, 6, 36, 156, 210, 270, 306, 576, 690, 936, 966, 2136, 2310, 2550, 2706, 2850, 3390, 3966, 4026, 4176, 4260, 4566, 4590, 5226, 5430, 5850, 6120, 6216, 6360, 6420, 6546, 7410, 7536, 8940, 9126, 9240, 9276, 9900, 10530, 10836, 11286, 11586, 11886, 12390, 13680

Offset: 1

Alex Ratushnyak, Jan 30 2014

nonn

Intersection of A070689 and {b(n)=A005382(n)-1}.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A005382, A070689.

Select[Range[14000],AllTrue[{#+1,2#+1,#^2+1},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 27 2019 *)

isok(n) = isprime(n+1)&&isprime(2*n+1)&&isprime(n^2+1); \\ Colin Barker, Jan 30 2014

import sympy
from sympy import isprime
for n in range(100000):
    if isprime(n+1) and isprime(n*2+1) and isprime(n*n+1): print(n, end=', ')

A237191 Numbers k such that k+1, k+3, k^2+1, k^2+3, k^4+1, k^4+3 are six primes.

Original entry on oeis.org

2, 520360, 14320216, 30527896, 119668186, 120506050, 131448430, 142493926, 211158676, 254574706, 276368680, 306216940, 315122416, 421132180, 472731400, 506213890, 540271396, 616078786, 629310346, 646308250, 741176296, 752897860, 800587480, 851425030, 897745996

Offset: 1

Alex Ratushnyak, Feb 04 2014

nonn

Giovanni Resta, Table of n, a(n) for n = 1..1000

A subsequence of A067662, A070325, A070689, A080149.

from sympy import isprime
for n in range(0,1000000000,2):
    if isprime(n+1) and isprime(n*n+1) and isprime(n**4+1):
        if isprime(n+3) and isprime(n*n+3) and isprime(n**4+3):
            print(n, end=', ')

A188698 Numbers k such that 1+k, 1+k^2, 1+k^4 and 1+k^16 are all prime.

Original entry on oeis.org

1, 2, 690, 33190, 57106, 77140, 135606, 258990, 303430, 331140, 337536, 359230, 375646, 455526, 458326, 493396, 548226, 550540, 585106, 602056, 659250, 680830, 742306, 800406, 827680, 870240, 918340, 925390, 968320, 1203100, 1273890, 1455526, 1497576, 1605016

Offset: 1

Zak Seidov, Apr 09 2011

nonn

Subsequence of A070325, which itself is a subsequence of A070689, which itself is a subsequence of A006093.

			a(3) = 690 = A070689(32) = A070325(11) = A006093(125).

[n: n in [0..7000000]| IsPrime(1+n) and IsPrime(1+n^2) and IsPrime(1+n^4) and IsPrime(1+n^16)]; // Vincenzo Librandi, Apr 11 2011

a(1) = 1 prepended by Vincenzo Librandi, Apr 11 2011

A193266 Numbers k such that k + 3 and k^2 + 9 are both prime.

Original entry on oeis.org

2, 8, 10, 20, 38, 40, 58, 70, 98, 100, 110, 160, 170, 188, 190, 220, 308, 350, 380, 538, 568, 598, 628, 640, 680, 688, 698, 740, 748, 818, 850, 968, 1010, 1028, 1048, 1088, 1190, 1228, 1300, 1378, 1420, 1430, 1448, 1528, 1540, 1550, 1568, 1580, 1730, 1738

Offset: 1

Zak Seidov, Dec 19 2012

nonn

			2 is in the sequence since 2 + 3 = 5 and 2^2 + 9 = 13 are both prime;
8 is in the sequence since 8 + 3 = 11 and 8^2 + 9 = 73 are both prime.

Cf. A070689, A192794.

Select[Prime[Range[3,300]]-3,PrimeQ[#^2+9]&] (* Harvey P. Dale, Nov 11 2014 *)

{a=3;forstep(n=2,2000,2,if(isprime(n+a)&&isprime(n^2+a^2),print1(n",")))}

cp's OEIS Frontend

A176070 Numbers of the form k^3+k^2+k+1 that are the product of two distinct primes.

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A279480 Numbers n such that n+1 and n^4+1 are primes.

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A285390 Numbers n such that n + 2, n^2 + 2^2, n^4 + 2^4, n^8 + 2^8 and n^16 + 2^16 are primes.

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A102368 Smallest k>0 such that n^k + 1 is not prime.

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A236692 Numbers k such that k+1, 2*k+1 and k^2+1 are primes.

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A237191 Numbers k such that k+1, k+3, k^2+1, k^2+3, k^4+1, k^4+3 are six primes.

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Python

A188698 Numbers k such that 1+k, 1+k^2, 1+k^4 and 1+k^16 are all prime.

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Magma

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A193266 Numbers k such that k + 3 and k^2 + 9 are both prime.

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