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 1-3 of 3 results.

A385188 Primes p == 3 (mod 4) such that the multiplicative order of 2+-i modulo p in Gaussian integers (A385165) is not divisible by 2 or 3.

Original entry on oeis.org

599, 691, 1291, 1451, 2411, 3851, 4919, 5051, 5479, 5531, 5879, 6599, 7079, 7691, 8011, 8039, 11491, 13291, 14011, 15091, 15971, 16651, 17359, 18731, 19211, 19531, 20731, 22651, 23971, 24611, 25639, 25679, 26251, 32051, 32359, 32531, 32771, 32971, 35879, 37039, 37571, 38011, 38371
Offset: 1

Views

Author

Jianing Song, Jun 20 2025

Keywords

Comments

Primes p == 3 (mod 4) are precisely the rational primes in the ring of Gaussian integers.
5 is a quadratic residue of integers modulo p for p being a term of this sequence. (See A385165).

Examples

			5479 is a term since (2+-i)^125081 == 1 (mod 5479), and 125081 is divisible by neither 2 nor 3.

Links

Crossrefs

Cf. A385165, A385179, A385219 (the actual multiplicative orders).
this sequence < A385169 < A385180 < A385167 < intersection of A122869 and A385168, where Ax < Ay means that Ax is a subsequence of Ay.
Also a subsequence of A385191.

Programs

  • PARI
    ord(p) = my(d = divisors((p+1)*znorder(Mod(5, p)))); for(i=1, #d, if(Mod([2, -1; 1, 2], p)^d[i] == 1, return(d[i]))) \\ for a prime p == 3 (mod 4), returns ord(2+-i, p)
isA385188(p) = isprime(p) && p%4==3 && ord(p)%2 && ord(p)%3

A385217 Odd multiplicative orders of 2+-i modulo primes p == 3 (mod 4).

Original entry on oeis.org

13695, 40755, 7475, 19895, 43995, 117855, 138075, 13185, 69445, 87725, 308505, 220665, 567645, 80735, 1103355, 1321125, 1386945, 507795, 1594005, 130995, 205975, 2051325, 2092035, 2216565, 2703975, 1368315, 2750685, 504095, 3039345, 212605, 3342405, 125081, 1274665, 3991725, 152205, 4279275
Offset: 1

Views

Author

Jianing Song, Jun 22 2025

Keywords

Comments

Primes p == 3 (mod 4) are precisely the rational primes in the ring of Gaussian integers.
Odd elements in A385165.
By definition, a(n) is the multiplicative order of 2+-i modulo A385169(n).

Examples

			a(49) = 635253 since it is the multiplicative order of 5 modulo A385169(49) = 8731, and it is odd.
a(50) = 57447 since it is the multiplicative order of 5 modulo A385169(50) = 8839, and it is odd.
a(51) = 2029953 since it is the multiplicative order of 5 modulo A385169(51) = 9011, and it is odd.

Links

Crossrefs

Cf. A385165, A385169 (corresponding primes), A385218, A385219.

Programs

  • PARI
    ord(p) = my(d = divisors((p+1)*znorder(Mod(5, p)))); for(i=1, #d, if(Mod([2, -1; 1, 2], p)^d[i] == 1, return(d[i]))) \\ for a prime p == 3 (mod 4), returns ord(2+-i, p)
forprime(p=3, 1e4, if(p%4==3 && ord(p)%2, print1(ord(p), ", ")))

A385218 Multiplicative orders of 2+-i modulo p == 3 (mod 4) that are congruent to 2 modulo 4.

Original entry on oeis.org

30, 4290, 3710, 3150, 20090, 164430, 21114, 22490, 59514, 43494, 244650, 65110, 819930, 932190, 1011030, 1266750, 1405410, 533830, 1864590, 135470, 2266530, 79002, 946970, 3863190, 1039890, 4952850, 170178, 566202, 6277530, 1324930, 3091690, 9397290, 214314, 5054610, 3467950, 3511090
Offset: 1

Views

Author

Jianing Song, Jun 22 2025

Keywords

Comments

Primes p == 3 (mod 4) are precisely the rational primes in the ring of Gaussian integers.
Elements in A385165 that are congruent to 2 modulo 4.
By definition, a(n) is the multiplicative order of 2+-i modulo A385179(n).

Examples

			a(7) = 21114 since it is the multiplicative order of 5 modulo A385179(7) = 919, and it is congruent to 2 modulo 4.

Links

Crossrefs

Cf. A385165, A385179 (corresponding primes), A385217, A385219.

Programs

  • PARI
    ord(p) = my(d = divisors((p+1)*znorder(Mod(5, p)))); for(i=1, #d, if(Mod([2, -1; 1, 2], p)^d[i] == 1, return(d[i]))) \\ for a prime p == 3 (mod 4), returns ord(2+-i, p)
forprime(p=3, 1e4, if(p%4==3 && ord(p)%4==2, print1(ord(p), ", ")))
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.