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.

A140444 Primes congruent to 1 (mod 14).

Original entry on oeis.org

29, 43, 71, 113, 127, 197, 211, 239, 281, 337, 379, 421, 449, 463, 491, 547, 617, 631, 659, 673, 701, 743, 757, 827, 883, 911, 953, 967, 1009, 1051, 1093, 1163, 1289, 1303, 1373, 1429, 1471, 1499, 1583, 1597, 1667, 1709, 1723, 1877, 1933, 2003, 2017, 2087
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Jun 26 2008

Keywords

Comments

From Federico Provvedi, May 24 2018: (Start)
Also primes congruent to 1 (mod 7).
For every prime p > 2, primes congruent to 1 (mod p) are also congruent to 1 (mod 2*p).
Conjecture: The monic polynomial P(x) = (x+1)^7/x - 1/x = ((x+1)^7-1)/x is irreducible but factorizable over Galois field (mod a(n)) with exactly 6 distinct irreducible factors of degree 1. Examples:
P(x) == (5 + x) (6 + x) (7 + x) (10 + x) (14 + x) (23 + x) (mod 29)
P(x) == (3 + x) (9 + x) (23 + x) (28 + x) (33 + x) (40 + x) (mod 43)
P(x) == (24 + x) (27 + x) (35 + x) (40 + x) (42 + x) (52 + x) (mod 71)
P(x) == (5 + x) (8 + x) (65 + x) (84 + x) (86 + x) (98 + x) (mod 113)
... (End).
Primes in A131877. - Eric Chen, Jun 14 2018

Links

Crossrefs

Cf. A045437, A045458, A045471, A045473.
A090613 gives prime index.
Cf. A090614.
Cf. A131877.
Primes congruent to 1 (mod k): A000040 (k=1), A065091 (k=2), A002476 (k=3 and 6), A002144 (k=4), A030430 (k=5 and 10), this sequence (k=7 and 14), A007519 (k=8), A061237 (k=9 and 18), A141849 (k=11 and 22), A068228 (k=12), A268753 (k=13 and 26), A132230 (k=15 and 30), A094407 (k=16), A129484 (k=17 and 34), A141868 (k=19 and 38), A141881 (k=20), A124826 (k=21 and 42), A212374 (k=23 and 46), A107008 (k=24), A141927 (k=25 and 50), A141948 (k=27 and 54), A093359 (k=28), A141977 (k=29 and 58), A142005 (k=31 and 62), A133870 (k=32).

Programs

  • GAP
    Filtered(Filtered([1..2300],n->n mod 14=1),IsPrime); # Muniru A Asiru, Jun 27 2018
  • Magma
    [p: p in PrimesUpTo(3000)|p mod 14 in {1}]; // Vincenzo Librandi, Dec 18 2010
  • Maple
    select(isprime,select(n->modp(n,14)=1,[$1..2300])); # Muniru A Asiru, Jun 27 2018
  • Mathematica
    Select[Prime[Range[500]], Mod[#, 14] == 1 &]  (* Harvey P. Dale, Mar 21 2011 *)
  • PARI
    is(n)=isprime(n) && n%14==1 \\ Charles R Greathouse IV, Jul 02 2016

Formula

a(n) ~ 6n log n. - Charles R Greathouse IV, Jul 02 2016

Extensions

Simpler definition from N. J. A. Sloane, Jul 11 2008

A172469 Primes congruent to +/-1 or +/-7 modulo 25.

Original entry on oeis.org

7, 43, 101, 107, 149, 151, 157, 193, 199, 251, 257, 293, 307, 349, 401, 443, 449, 457, 499, 557, 593, 599, 601, 607, 643, 701, 743, 751, 757, 857, 907, 1049, 1051, 1093, 1151, 1193, 1201, 1249, 1301, 1307, 1399, 1451, 1493, 1499, 1543, 1549, 1601, 1607
Offset: 1

Views

Author

Katherine E. Stange, Feb 03 2010

Keywords

Comments

Equivalently, primes p such that the smallest extension of F_p containing the 5th roots of unity also contains the 25th roots of unity.
In this respect, the sequence is the n=5 instance of a family of sequences. For n=3, see A129805, and for n=2, see A002144.
Equivalently, the primes p for which, if p^t = 1 mod 5, then p^t = 1 mod 25.

Programs

  • Python
    from itertools import count, islice
from sympy import isprime
def A172469_gen(): # generator of terms
    yield from (7, 43)
    for n in count(50,50):
        for m in (1,7,43,49):
            if isprime(n+m):
                yield n+m
A172469_list = list(islice(A172469_gen(),48)) # Chai Wah Wu, Apr 28 2025

Formula

A141927 U A141932 U A141946 U A141941. [From R. J. Mathar, Feb 05 2010]

Extensions

More terms from R. J. Mathar, Feb 05 2010

A381253 Prime numbers whose constant congruence speed of tetration is greater than 1.

Original entry on oeis.org

5, 7, 43, 101, 107, 149, 151, 157, 193, 199, 251, 257, 293, 307, 349, 401, 443, 449, 457, 499, 557, 593, 599, 601, 607, 643, 701, 743, 751, 757, 857, 907, 1049, 1051, 1093, 1151, 1193, 1201, 1249, 1301, 1307, 1399, 1451, 1493, 1499, 1543, 1549, 1601, 1607, 1657
Offset: 1

Views

Author

Gabriele Di Pietro and Marco Ripà, Apr 17 2025

Keywords

Comments

The only positive integers with a constant congruence speed greater than 1 (see A373387) are necessarily congruent to 1, 7, 43, or 49 modulo 50.
As a result, 36% of positive integers have a constant congruence speed of at least 2, while 20% of primes have a constant congruence speed greater than 1. In the interval (1, 10^4), there are 1229 prime numbers, 247 of whom have a constant congruence speed of at least 2.
Moreover, as a consequence of Dirichlet's theorem on arithmetic progressions, Theorem 3 of "The congruence speed formula" (see Links) proves that, for any given positive integer k, there are infinitely many primes characterized by a constant congruence speed of (exactly) k.

Examples

			a(1) = 5 since 5 is the smallest prime number with a constant congruence speed of at least 2.

References

  • Marco Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011. ISBN 978-88-6178-789-6

Links

Crossrefs

Also 5 together with A172469.
Union of {5}, A141927, A141932, A141941, A141946.
Cf. A317905, A321131, A373387, A382862.

Programs

  • Python
    from sympy import isprime
valid_mod_50 = {1, 7, 43, 49}
result = [5]
n = 6
while len(result) < 1000:
    if isprime(n) and n % 50 in valid_mod_50:
        result.append(n)
    n += 1
print(result)

Formula

a(1) = 5. For n >= 2, a(n) = A172469(n-1).
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.