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.

Showing 1-4 of 4 results.

A065903 Integers i > 1 for which there is no prime p such that i is a solution mod p of x^4 = 2.

Original entry on oeis.org

1689, 1741, 3306, 3894, 4362, 4587, 4999, 5754, 6025, 6371, 6668, 7012, 7982, 9054, 9158, 9695, 9742, 9832, 10056, 10664, 11005, 12027, 12385, 13676, 13895, 14026, 14059, 16104, 16239, 16903, 17050, 17153, 18079, 18202, 18642, 20349, 21060
Offset: 1

Views

Author

Klaus Brockhaus, Nov 28 2001

Keywords

Comments

Solutions mod p are represented by integers from 0 to p-1. The following equivalences holds for i > 1: There is a prime p such that i is a solution mod p of x^4 = 2 iff i^4 - 2 has a prime factor > i; i is a solution mod p of x^4 = 2 iff p is a prime factor of i^4 - 2 and p > i. i^4 - 2 has at most three prime factors > i. For i such that i^4 - 2 has one resp. two resp. three prime factors > i; cf. A065904 resp. A065905 resp. A065906.

Examples

			a(2) = 1741, since 1741 is (after 1689) the second integer i for which there are no primes p > i such that i is a solution mod p of x^4 = 2, or equivalently, 1741^4 - 2 = 9187452028559 = 7*7*79*887*1609*1663 has no prime factor > 1741. (cf. A065902).

Crossrefs

Cf. A040028, A065902, A065904, A065905, A065906.

Programs

  • PARI
    a065903(m) = local(c,n,f,a); c = 0; n = 2; while(c

Formula

a(n) = n-th integer i such that i^4 - 2 has no prime factor > i.

A065902 Smallest prime p such that n is a solution mod p of x^4 = 2, or 0 if no such prime exists.

Original entry on oeis.org

7, 79, 127, 7, 647, 2399, 23, 937, 4999, 14639, 1481, 28559, 19207, 23, 31, 47, 73, 18617, 79999, 194479, 117127, 5711, 165887, 73, 4663, 113, 233, 707279, 47, 40153, 524287, 191, 167, 257, 439, 267737, 45329, 2313439, 182857, 2825759, 1555847
Offset: 2

Views

Author

Klaus Brockhaus, Nov 28 2001

Keywords

Comments

Solutions mod p are represented by integers from 0 to p-1. The following equivalences holds for n > 1: There is a prime p such that n is a solution mod p of x^4 = 2 iff n^4 - 2 has a prime factor > n; n is a solution mod p of x^4 = 2 iff p is a prime factor of n^ 4 - 2 and p > n. n^4 - 2 has at most three prime factors > n, so these factors are the only primes p such that n is a solution mod p of x^4 = 2. The first zero is at n = 1689 (cf. A065903 ). For n such that n^4 - 2 has one resp. two resp. three prime factors > n; cf. A065904 resp. A065905 resp. A065906.

Examples

			a(16) = 31, since 16 is a solution mod 31 of x^4 = 2 and 16 is not a solution mod p of x^4 = 2 for primes p < 31. Although 16^4 = 2 (mod 7), prime 7 is excluded because 7 < 16 and 16 = 2 (mod 7).

Crossrefs

Cf. A040098, A065903, A065904, A065905, A065906.

Programs

  • PARI
    a065902(m) = local(n,f,a,j); for(n = 2,m,f = factor(n^4-2); a = matsize(f)[1]; j = 1; while(f[j,1]< = n&&jn,f[j,1],0),","))
a065902(45)

Formula

If n^4 - 2 has prime factors > n, then a(n) = smallest of these prime factors, else a(n) = 0.

A065904 Integers i > 1 for which there is one prime p such that i is a solution mod p of x^4 = 2.

Original entry on oeis.org

2, 3, 4, 6, 7, 9, 10, 11, 12, 13, 14, 19, 20, 21, 22, 23, 24, 26, 29, 31, 32, 37, 38, 39, 40, 41, 42, 43, 44, 49, 50, 52, 53, 54, 59, 60, 61, 62, 64, 65, 70, 72, 73, 74, 75, 77, 79, 80, 82, 83, 85, 87, 89, 93, 94, 95, 96, 97, 99, 100, 101, 103, 108, 109, 111, 116, 119, 121
Offset: 1

Views

Author

Klaus Brockhaus, Nov 28 2001

Keywords

Comments

Solutions mod p are represented by integers from 0 to p-1. The following equivalences holds for i > 1: There is a prime p such that i is a solution mod p of x^4 = 2 iff i^4 - 2 has a prime factor > i; i is a solution mod p of x^4 = 2 iff p is a prime factor of i^4 - 2 and p > i. i^4 - 2 has at most three prime factors > i. For i such that i^4 - 2 has no resp. two resp. three prime factors > i; cf. A065903 resp. A065905 resp. A065906.

Examples

			a(3) = 4, since 4 is (after 2 and 3) the third integer i for which there is one prime p > i (viz. 127) such that i is a solution mod p of x^4 = 2, or equivalently, 4^4 - 2 = 254 = 2*127 has one prime factor > 4 (cf. A065902).

Links

Crossrefs

Cf. A040028, A065902, A065903, A065905, A065906.

Programs

  • Maple
    filter:= n -> nops(select(`>`, numtheory:-factorset(n^4-2),n))=1:
select(filter, [$2..1000]); # Robert Israel, Jan 30 2017
  • Mathematica
    okQ[n_] := Length[Select[FactorInteger[n^4 - 2][[All, 1]], # > n&]] == 1;
Select[Range[2, 200], okQ] (* Jean-François Alcover, Mar 26 2019, after Robert Israel *)
  • PARI
    a065904(m) = local(c,n,f,a,s,j); c = 0; n = 2; while(cn,s = concat(s,f[j,1]))); if(matsize(s)[2] == 1,print1(n,","); c++); n++)
a065904(70)

Formula

a(n) = n-th integer i such that i^4 - 2 has one prime factor > i.

A065906 Integers i > 1 for which there are three primes p such that i is a solution mod p of x^4 = 2.

Original entry on oeis.org

15, 48, 55, 197, 206, 221, 235, 283, 297, 408, 444, 472, 489, 577, 578, 623, 641, 677, 701, 703, 763, 854, 930, 1049, 1081, 1134, 1140, 1159, 1160, 1201, 1253, 1303, 1311, 1328, 1374, 1385, 1415, 1458, 1459, 1495, 1501, 1517, 1557, 1585, 1714, 1723, 1726
Offset: 1

Views

Author

Klaus Brockhaus, Nov 28 2001

Keywords

Comments

Solutions mod p are represented by integers from 0 to p-1. The following equivalences holds for i > 1: There is a prime p such that i is a solution mod p of x^4 = 2 iff i^4 - 2 has a prime factor > i; i is a solution mod p of x^4 = 2 iff p is a prime factor of i^4 - 2 and p > i. i^4 - 2 has at most three prime factors > i. For i such that i^4 - 2 has no resp. one resp. two prime factors > i cf. A065903 resp. A065904 resp. A065905.

Examples

			a(3) = 55, since 55 is (after 15 and 48) the third integer i for which there are three primes p > i (viz. 73, 103 and 1217) such that i is a solution mod p of x^4 = 2, or equivalently, 55^4 - 2 = 9150623 = 73*103*1217 has three prime factors > 4. (cf. A065902).

Crossrefs

Cf. A040028, A065902, A065903, A065904, A065905.

Programs

  • PARI
    a065906(m) = local(c,n,f,a,s,j); c = 0; n = 2; while(cn,s = concat(s,f[j,1]))); if(matsize(s)[2] == 3,print1(n,","); c++); n++)
a065906(50)

Formula

a(n) = n-th integer i such that i^4 - 2 has three prime factors > i.
Showing 1-4 of 4 results.

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