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.

A327655 Intersection of A327653 and A327654.

Original entry on oeis.org

119, 649, 1189, 4187, 12871, 14041, 16109, 23479, 24769, 28421, 31631, 34997, 38503, 41441, 48577, 50545, 56279, 58081, 59081, 61447, 75077, 91187, 95761, 96139, 116821, 127937, 146329, 148943, 150281, 157693, 170039, 180517, 188501, 207761, 208349, 244649, 281017, 311579, 316409
Offset: 1

Views

Author

Jianing Song, Sep 20 2019

Keywords

Comments

Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n) = m*x(n-1) + x(n-2) for k >= 2. For primes p, we have (a) p divides x(p-((m^2+4)/p)); (b) x(p) == ((m^2+4)/p) (mod p), where (D/p) is the Kronecker symbol. This sequence gives composite numbers k such that gcd(k, m^2+4) = 1 and that conditions similar to (a) and (b) hold for k simultaneously, where m = 2.
If k is not required to be coprime to m^2 + 4 (= 13), then there are 322 such k <= 10^5 and 1381 such k <= 10^6, while there are only 24 terms <= 10^5 and 72 terms <= 10^6 in this sequence.

Examples

			119 divides A006190(120) as well as A006190(119) + 1, so 119 is a term.

Crossrefs

m m=1 m=2 m=3
k | x(k-Kronecker(m^2+4,k))* A081264 U A141137 A327651 A327653
k | x(k)-Kronecker(m^2+4,k) A049062 A099011 A327654
both A212424 A327652 this seq
* k is composite and coprime to m^2 + 4.
Cf. A006190, A011583 ({Kronecker(13,n)}).

Programs

  • PARI
    seqmod(n, m)=((Mod([3, 1; 1, 0], m))^n)[1, 2]
isA327655(n)=!isprime(n) && seqmod(n, n)==kronecker(13,n) && !seqmod(n-kronecker(13,n), n) && gcd(n,13)==1 && n>1

A327653 Composite numbers k coprime to 13 such that k divides A006190(k-Kronecker(13,k)).

Original entry on oeis.org

10, 119, 649, 1189, 1763, 3599, 4187, 5559, 6681, 12095, 12403, 12685, 12871, 12970, 14041, 14279, 15051, 16109, 19043, 22847, 23479, 24769, 26795, 28421, 30743, 30889, 31631, 31647, 33919, 34997, 37949, 38503, 39203, 41441, 46079, 48577, 49141, 50523, 50545, 53301, 56279, 58081, 58589
Offset: 1

Views

Author

Jianing Song, Sep 20 2019

Keywords

Comments

Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n) = m*x(n-1) + x(n-2) for k >= 2. For primes p, we have (a) p divides x(p-((m^2+4)/p)); (b) x(p) == ((m^2+4)/p) (mod p), where (D/p) is the Kronecker symbol. This sequence gives composite numbers k such that gcd(k, m^2+4) = 1 and that a condition similar to (a) holds for k, where m = 3.
If k is not required to be coprime to m^2 + 4 (= 13), then there are 360 such k <= 10^5 and 1506 such k <= 10^6, while there are only 62 terms <= 10^5 and 197 terms <= 10^6 in this sequence.
Also composite numbers k coprime to 13 such that A322907(k) divides k - Kronecker(13,k).

Examples

			A006190(9) = 12970 which is divisible by 10, so 10 is a term.

Links

Crossrefs

m m=1 m=2 m=3
k | x(k-Kronecker(m^2+4,k))* A081264 U A141137 A327651 this seq
k | x(k)-Kronecker(m^2+4,k) A049062 A099011 A327654
both A212424 A327652 A327655
* k is composite and coprime to m^2 + 4.
Cf. A006190, A322907, A011583 ({Kronecker(13,n)}).

Programs

  • PARI
    seqmod(n, m)=((Mod([3, 1; 1, 0], m))^n)[1, 2]
isA327653(n)=!isprime(n) && !seqmod(n-Kronecker(13,n), n) && gcd(n,13)==1 && n>1

A327651 Composite numbers k coprime to 8 such that k divides Pell(k - Kronecker(8,k)), Pell = A000129.

Original entry on oeis.org

35, 169, 385, 779, 899, 961, 1121, 1189, 2419, 2555, 2915, 3107, 3827, 6083, 6265, 6441, 6601, 6895, 6965, 7801, 8119, 8339, 9179, 9809, 9881, 10403, 10763, 10835, 10945, 13067, 14027, 14111, 15179, 15841, 18241, 18721, 19097, 20833, 20909, 22499, 23219, 24727, 26795, 27869, 27971
Offset: 1

Views

Author

Jianing Song, Sep 20 2019

Keywords

Comments

Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n) = m*x(n-1) + x(n-2) for k >= 2. For primes p, we have (a) p divides x(p-((m^2+4)/p)); (b) x(p) == ((m^2+4)/p) (mod p), where (D/p) is the Kronecker symbol. This sequence gives composite numbers k such that gcd(k, m^2+4) = 1 and that a condition similar to (a) holds for k, where m = 2.
If k is not required to be coprime to m^2 + 4 (= 8), then there are 1232 such k <= 10^5 and 4973 such k <= 10^6, while there are only 83 terms <= 10^5 and 245 terms <= 10^6 in this sequence.
Also composite numbers k coprime to 8 such that A214028(k) divides k - Kronecker(8,k).

Examples

			Pell(36) = 21300003689580 is divisible by 35, so 35 is a term.

Links

Crossrefs

m m=1 m=2 m=3
k | x(k-Kronecker(m^2+4,k))* A081264 U A141137 this seq A327653
k | x(k)-Kronecker(m^2+4,k) A049062 A099011 A327654
both A212424 A327652 A327655
* k is composite and coprime to m^2 + 4.
Cf. A000129, A214028, A091337 ({Kronecker(8,n)}).

Programs

  • PARI
    pellmod(n, m)=((Mod([2, 1; 1, 0], m))^n)[1, 2]
isA327651(n)=!isprime(n) && !pellmod(n-kronecker(8,n), n) && gcd(n,8)==1 && n>1

A327652 Intersection of A099011 and A327651.

Original entry on oeis.org

169, 385, 961, 1121, 3827, 6265, 6441, 6601, 7801, 8119, 10945, 13067, 15841, 18241, 19097, 20833, 24727, 27971, 29953, 31417, 34561, 35459, 37345, 38081, 39059, 42127, 45961, 47321, 49105, 52633, 53041, 55969, 56953, 58241, 62481, 74305, 79361, 81361, 84587, 86033, 86241, 101311, 107801
Offset: 1

Views

Author

Jianing Song, Sep 20 2019

Keywords

Comments

Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n) = m*x(n-1) + x(n-2) for k >= 2. For primes p, we have (a) p divides x(p-((m^2+4)/p)); (b) x(p) == ((m^2+4)/p) (mod p), where (D/p) is the Kronecker symbol. This sequence gives composite numbers k such that gcd(k, m^2+4) = 1 and that conditions similar to (a) and (b) hold for k simultaneously, where m = 2.
If k is not required to be coprime to m^2 + 4 (= 8), then there are 1190 such k <= 10^5 and 4847 such k <= 10^6, while there are only 41 terms <= 10^5 and 119 terms <= 10^6 in this sequence.

Examples

			169 divides Pell(168) as well as Pell(169) - 1, so 169 is a term.

Links

Crossrefs

m m=1 m=2 m=3
k | x(k-Kronecker(m^2+4,k))* A081264 U A141137 A327651 A327653
k | x(k)-Kronecker(m^2+4,k) A049062 A099011 A327654
both A212424 this seq A327655
* k is composite and coprime to m^2 + 4.
Cf. A000129, A091337 ({Kronecker(8,n)}).

Programs

  • PARI
    pellmod(n, m)=((Mod([2, 1; 1, 0], m))^n)[1, 2]
isA327652(n)=!isprime(n) && pellmod(n, n)==kronecker(8,n) && !pellmod(n-kronecker(8,n), n) && gcd(n,8)==1 && n>1
Showing 1-4 of 4 results.

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