A065903 -id:A065903 - cp's OEIS frontend

A083955 Numbers n > 1 such that n^5 - 2 has no prime factor > n.

3557, 12038, 14810, 15424, 28456, 30742, 31540, 37665, 45602, 46883, 47879, 48152, 52196, 52617, 55265, 57902, 68306, 69032, 74925, 76262, 79562, 79984, 84569, 90442, 104867, 104956, 107213, 112570, 114614, 119477, 127634, 131072, 132466

Offset: 1

Klaus Brockhaus, May 09 2003

nonn

Also integers n > 1 for which there is no prime p > n such that x = n is a solution mod p of x^5 = 2, since the following equivalences hold for n > 1: There is a prime p > n such that n is a solution mod p of x^5 = 2 iff n^5 - 2 has a prime factor > n; n is a solution mod p of x^5 = 2 iff p is a prime factor of n^5 - 2 and p > n.

			12038 is a term since 12038^5 - 2 = 252796871460867395166 = 2*3*3*3*263*571*641*911*5849*9127 has no prime factor > 12038.

Cf. A040159, A040160, A065903.

t = {}; Do[If[Max[First/@FactorInteger[n^5-2]]Jayanta Basu, May 20 2013 *)
Select[Range[2,133000],Max[FactorInteger[#^5-2][[;;,1]]]<=#&] (* Harvey P. Dale, Mar 02 2023 *)

{for(n=2,133000,f=factor(n^5-2); if(f[matsize(f)[1],1]<=n,print1(n,",")))}

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

Klaus Brockhaus, Nov 28 2001

nonn

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.

			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).

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

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)

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

Klaus Brockhaus, Nov 28 2001

nonn

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.

			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).

Robert Israel, Table of n, a(n) for n = 1..10000

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

filter:= n -> nops(select(`>`, numtheory:-factorset(n^4-2),n))=1:
select(filter, [$2..1000]); # Robert Israel, Jan 30 2017

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 *)

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)

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

A065905 Integers i > 1 for which there are two primes p such that i is a solution mod p of x^4 = 2.

Original entry on oeis.org

5, 8, 16, 17, 18, 25, 27, 28, 30, 33, 34, 35, 36, 45, 46, 47, 51, 56, 57, 58, 63, 66, 67, 68, 69, 71, 76, 78, 81, 84, 86, 88, 90, 91, 92, 98, 102, 104, 105, 106, 107, 110, 112, 113, 114, 115, 117, 118, 120, 122, 123, 125, 126, 127, 131, 132, 133, 134, 135, 136, 137

Offset: 1

Klaus Brockhaus, Nov 28 2001

nonn

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. three prime factors > i cf. A065903 resp. A065904 resp. A065906.

			a(3) = 16, since 16 is (after 5 and 8) the third integer i for which there are two primes p > i (viz. 31 and 151) such that i is a solution mod p of x^4 = 2, or equivalently, 16^4 - 2 = 65534 = 2*7*31*151 has two prime factors > 4. (cf. A065902).

Cf. A040028, A065902, A065903, A065904, A065906.

a065905(m) = local(c,n,f,a,s,j); c = 0; n = 2; while(cn,s = concat(s,f[j,1]))); if(matsize(s)[2] == 2,print1(n,","); c++); n++)
a065905(65)

a(n) = n-th integer i such that i^4 - 2 has two prime factors > 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

Klaus Brockhaus, Nov 28 2001

nonn

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.

			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).

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

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)

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

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A083955 Numbers n > 1 such that n^5 - 2 has no prime factor > n.

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

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

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A065905 Integers i > 1 for which there are two primes p such that i is a solution mod p of x^4 = 2.

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

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