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 1-8 of 8 results.

A377177 Primes p such that -7/2 is a primitive root modulo p.

Original entry on oeis.org

11, 17, 29, 31, 37, 41, 43, 47, 73, 89, 103, 107, 109, 149, 167, 179, 197, 257, 277, 311, 313, 317, 347, 353, 367, 373, 383, 389, 409, 433, 479, 491, 499, 503, 521, 541, 557, 571, 577, 593, 601, 607, 647, 653, 659, 683, 701, 719, 727, 761, 769, 821, 839, 857, 883, 887, 907, 929, 937, 947, 983
Offset: 1

Views

Author

Jianing Song, Oct 18 2024

Keywords

Comments

If p is a term in this sequence, then -7/2 is not a square modulo p (i.e., p is in A191061).
Conjecture: this sequence has relative density equal to Artin's constant (A005596) with respect to the set of primes.

Links

Crossrefs

Primes p such that +a/2 is a primitive root modulo p: A320384 (a=3), A377174 (a=5), A377176 (a=7), A377178 (a=9).
Primes p such that -a/2 is a primitive root modulo p: A377172 (a=3), A377175 (a=5), this sequence (a=7), A377179 (a=9).
Cf. A191061, A005596.

Programs

A377172 Primes p such that -3/2 is a primitive root modulo p.

Original entry on oeis.org

17, 23, 37, 41, 43, 47, 67, 89, 109, 113, 137, 139, 157, 163, 167, 191, 229, 233, 239, 257, 263, 277, 283, 311, 349, 353, 359, 379, 383, 397, 421, 449, 479, 503, 521, 523, 541, 547, 569, 571, 593, 599, 613, 619, 641, 647, 661, 719, 733, 739, 743, 757, 761, 787, 809, 811, 839, 853, 857, 859, 863, 877, 887, 911, 929, 953, 977, 983
Offset: 1

Views

Author

Jianing Song, Oct 18 2024

Keywords

Comments

If p is a term in this sequence, then -3/2 is not a square modulo p (i.e., p is in A191059).
Conjecture: this sequence has relative density equal to Artin's constant (A005596) with respect to the set of primes.

Links

Crossrefs

Primes p such that +a/2 is a primitive root modulo p: A320384 (a=3), A377174 (a=5), A377176 (a=7), A377178 (a=9).
Primes p such that -a/2 is a primitive root modulo p: this sequence (a=3), A377175 (a=5), A377177 (a=7), A377179 (a=9).
Cf. A191059, A005596.

Programs

  • PARI
    forprime(p=5, 10^3, if(znorder(Mod(-3/2, p))==p-1, print1(p, ", ")));

A377174 Primes p such that 5/2 is a primitive root modulo p.

Original entry on oeis.org

11, 17, 23, 47, 59, 73, 101, 103, 109, 113, 137, 139, 149, 167, 179, 211, 223, 229, 233, 257, 263, 269, 313, 337, 349, 353, 367, 379, 383, 389, 419, 421, 433, 461, 487, 499, 503, 509, 593, 607, 617, 647, 659, 661, 673, 727, 743, 811, 821, 823, 829, 857, 859, 863, 887, 941, 953, 967, 971, 977, 983
Offset: 1

Views

Author

Jianing Song, Oct 18 2024

Keywords

Comments

If p is a term in this sequence, then 5/2 is not a square modulo p (i.e., p is in A038880).
Conjecture: this sequence has relative density equal to Artin's constant (A005596) with respect to the set of primes.

Links

Crossrefs

Primes p such that +a/2 is a primitive root modulo p: A320384 (a=3), this sequence (a=5), A377176 (a=7), A377178 (a=9).
Primes p such that -a/2 is a primitive root modulo p: A377172 (a=3), A377175 (a=5), A377177 (a=7), A377179 (a=9).
Cf. A038880, A005596.

Programs

  • PARI
    forprime(p=7, 10^3, if(znorder(Mod(5/2, p))==p-1, print1(p, ", ")));

A377175 Primes p such that -5/2 is a primitive root modulo p.

Original entry on oeis.org

3, 17, 31, 43, 67, 71, 73, 79, 83, 101, 107, 109, 113, 137, 149, 163, 191, 199, 227, 229, 233, 239, 257, 269, 271, 283, 307, 311, 313, 337, 347, 349, 353, 359, 389, 421, 431, 433, 439, 443, 461, 467, 479, 509, 547, 563, 587, 593, 599, 617, 631, 661, 673, 683, 719, 821, 827, 829, 839, 857, 907, 911, 919, 941, 947, 953, 977
Offset: 1

Views

Author

Jianing Song, Oct 18 2024

Keywords

Comments

If p is a term in this sequence, then -5/2 is not a square modulo p (i.e., p is in A296925).
Conjecture: this sequence has relative density equal to Artin's constant (A005596) with respect to the set of primes.

Links

Crossrefs

Primes p such that +a/2 is a primitive root modulo p: A320384 (a=3), A377174 (a=5), A377176 (a=7), A377178 (a=9).
Primes p such that -a/2 is a primitive root modulo p: A377172 (a=3), this sequence (a=5), A377177 (a=7), A377179 (a=9).
Cf. A296925, A005596.

Programs

  • PARI
    print1(3, ", "); forprime(p=7, 10^3, if(znorder(Mod(-5/2, p))==p-1, print1(p, ", ")));

A377176 Primes p such that 7/2 is a primitive root modulo p.

Original entry on oeis.org

3, 17, 19, 23, 29, 37, 41, 59, 73, 79, 83, 89, 109, 127, 139, 149, 191, 197, 227, 239, 251, 257, 263, 277, 283, 307, 313, 317, 353, 359, 373, 389, 409, 419, 431, 433, 467, 487, 521, 523, 541, 557, 563, 577, 587, 593, 599, 601, 619, 643, 653, 691, 701, 761, 769, 821, 857, 863, 919, 929, 937, 967, 991
Offset: 1

Views

Author

Jianing Song, Oct 18 2024

Keywords

Comments

If p is a term in this sequence, then 7/2 is not a square modulo p (i.e., p is in A038886).
Conjecture: this sequence has relative density equal to Artin's constant (A005596) with respect to the set of primes.

Links

Crossrefs

Primes p such that +a/2 is a primitive root modulo p: A320384 (a=3), A377174 (a=5), this sequence (a=7), A377178 (a=9).
Primes p such that -a/2 is a primitive root modulo p: A377172 (a=3), A377175 (a=5), A377177 (a=7), A377179 (a=9).
Cf. A038886, A005596.

Programs

  • PARI
    print1(3, ", "); forprime(p=11, 10^3, if(znorder(Mod(-5/2, p))==p-1, print1(p, ", ")));

A377178 Primes p such that 9/2 is a primitive root modulo p.

Original entry on oeis.org

5, 13, 19, 29, 43, 53, 59, 61, 83, 101, 107, 109, 149, 157, 173, 179, 197, 227, 229, 251, 269, 277, 283, 293, 317, 331, 347, 373, 389, 419, 443, 461, 467, 491, 509, 523, 547, 557, 563, 587, 613, 619, 653, 661, 677, 683, 691, 701, 709, 733, 739, 757, 773, 787, 797, 821, 829, 853, 883, 907, 947, 971
Offset: 1

Views

Author

Jianing Song, Oct 18 2024

Keywords

Comments

If p is a term in this sequence, then 9/2 is not a square modulo p (i.e., p is in A003629).
Conjecture: this sequence has relative density equal to Artin's constant (A005596) with respect to the set of primes.

Links

Crossrefs

Primes p such that +a/2 is a primitive root modulo p: A320384 (a=3), A377174 (a=5), A377176 (a=7), this sequence (a=9).
Primes p such that -a/2 is a primitive root modulo p: A377172 (a=3), A377175 (a=5), A377177 (a=7), A377179 (a=9).
Cf. A003629, A005596.

Programs

  • PARI
    forprime(p=5, 10^3, if(znorder(Mod(9/2, p))==p-1, print1(p, ", ")));

A377179 Primes p such that -9/2 is a primitive root modulo p.

Original entry on oeis.org

5, 13, 23, 29, 31, 47, 53, 61, 71, 79, 101, 109, 149, 151, 157, 167, 173, 191, 197, 199, 223, 229, 239, 263, 269, 277, 293, 311, 317, 359, 367, 373, 383, 389, 461, 463, 479, 487, 503, 509, 557, 599, 613, 647, 653, 661, 677, 701, 709, 719, 733, 743, 757, 773, 797, 821, 823, 829, 839, 853, 863, 887, 911, 967, 983, 991
Offset: 1

Views

Author

Jianing Song, Oct 18 2024

Keywords

Comments

If p is a term in this sequence, then -9/2 is not a square modulo p (i.e., p is in A003628).
Conjecture: this sequence has relative density equal to Artin's constant (A005596) with respect to the set of primes.

Links

Crossrefs

Primes p such that +a/2 is a primitive root modulo p: A320384 (a=3), A377174 (a=5), A377176 (a=7), A377178 (a=9).
Primes p such that -a/2 is a primitive root modulo p: A377172 (a=3), A377175 (a=5), A377177 (a=7), this sequence (a=9).
Cf. A003628, A005596.

Programs

  • PARI
    forprime(p=5, 10^3, if(znorder(Mod(-9/2, p))==p-1, print1(p, ", ")));

A320383 Multiplicative order of 3/2 modulo n-th prime.

Original entry on oeis.org

2, 6, 10, 4, 16, 3, 11, 7, 30, 36, 40, 21, 23, 13, 58, 12, 33, 7, 36, 26, 82, 88, 8, 25, 102, 106, 108, 112, 126, 130, 136, 69, 74, 150, 156, 81, 83, 86, 178, 36, 95, 96, 49, 66, 5, 222, 226, 228, 232, 119, 30, 250, 256, 131, 67, 270, 276, 40, 141, 73, 51, 155, 156, 79, 11, 168, 346, 348, 352, 179, 366, 124
Offset: 3

Views

Author

Jianing Song, Oct 12 2018

Keywords

Comments

Let p = prime(n). a(n) is the smallest positive k such that p divides 3^k - 2^k. Obviously, a(n) divides p - 1. If a(n) = p - 1, then p is listed in A320384.
If p == 1, 5, 19, 23 (mod 24), then 3/2 is a quadratic residue modulo p, so a(n) divides (p - 1)/2.
By Zsigmondy's theorem, for each k >=2 there is a prime that divides 3^k-2^k but not 3^j-2^j for j < k. Therefore each integer >= 2 appears in the sequence at least once. - Robert Israel, Apr 20 2021

Examples

			Let ord(n,p) be the multiplicative order of n modulo p.
3/2 == 4 (mod 5), so a(3) = ord(4,5) = 2.
3/2 == 5 (mod 7), so a(4) = ord(5,7) = 6.
3/2 == 7 (mod 11), so a(5) = ord(7,11) = 10.
3/2 == 8 (mod 13), so a(6) = ord(8,13) = 4.

Links

Crossrefs

Cf. A001047, A211242, A320384.

Programs

  • Maple
    f:= proc(n) local p; p:= ithprime(n); numtheory:-order(3/2 mod p,p) end proc:
map(f, [$3..100]); # Robert Israel, Apr 20 2021
  • Mathematica
    a[n_] := With[{p = Prime[n]}, Do[If[Divisible[3^k - 2^k, p], Return[k]], {k, Rest@Divisors[p-1]}]];
Table[a[n], {n, 3, 100}] (* Jean-François Alcover, Feb 10 2023 *)
  • PARI
    forprime(p=5,10^3,print1(znorder(Mod(3/2,p)),", ")) \\ Joerg Arndt, Oct 13 2018
Showing 1-8 of 8 results.

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