A141932 -id:A141932 - cp's OEIS frontend

A141928 Primes congruent to 2 mod 25.

2, 127, 227, 277, 577, 677, 727, 827, 877, 977, 1277, 1327, 1427, 1627, 1777, 1877, 2027, 2377, 2477, 2677, 2777, 2927, 3527, 3677, 3727, 3877, 4027, 4127, 4177, 4327, 4877, 5077, 5227, 5477, 5527, 5827, 5927, 6277, 6427, 6577, 6827, 6977, 7027, 7127, 7177

Offset: 1

N. J. A. Sloane, Jul 11 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A141930, A141931, A141932.

[p: p in PrimesUpTo(8000) | p mod 25 eq 2 ]; // Vincenzo Librandi, Aug 16 2012

Select[Range[2, 20000, 25], PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2011 *)

is(n)=isprime(n) && n%25==2 \\ Charles R Greathouse IV, Jul 03 2016

{2} UNION A142466. - R. J. Mathar, Jul 20 2008

a(n) ~ 20n log n. - Charles R Greathouse IV, Jul 03 2016

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

Katherine E. Stange, Feb 03 2010

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.

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

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

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

Gabriele Di Pietro and Marco Ripà, Apr 17 2025

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.

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

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

Marco Ripà, The congruence speed formula, Notes on Number Theory and Discrete Mathematics, 2021, 27(4), 43—61.
Marco Ripà, Twelve Python Programs to Help Readers Test Peculiar Properties of Integer Tetration, ResearchGate, 2024. See pp. 22-23, 27.
Marco Ripà and Luca Onnis, Number of stable digits of any integer tetration, Notes on Number Theory and Discrete Mathematics, 2022, 28(3), 441—457.
Wikipedia, Tetration.

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

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)

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

A383672 Squarefree numbers k such that k^2+1 is not squarefree.

Original entry on oeis.org

7, 38, 41, 43, 57, 70, 82, 93, 107, 118, 143, 157, 182, 193, 218, 239, 251, 257, 282, 293, 307, 318, 327, 357, 382, 393, 407, 418, 437, 443, 457, 482, 493, 515, 518, 543, 557, 577, 582, 593, 606, 607, 618, 643, 682, 707, 718, 743, 746, 757, 782, 793, 807, 818, 829, 843, 857, 893

Offset: 1

Alexandre Herrera, May 04 2025

nonn

			38 = 2*19 is squarefree but 38*38 + 1 = 1445 = 5*17*17 is not squarefree.

Intersection of A005117 and A049532.
Includes A141932 and A141941.
Cf. A080666, A224718.

filter:= proc(n) numtheory:-issqrfree(n) and not numtheory:-issqrfree(n^2+1) end proc:
select(filter, [$1..1000]); # Robert Israel, May 04 2025

Select[Range[900],SquareFreeQ[#] && !SquareFreeQ[#^2+1] &] (* Stefano Spezia, May 04 2025 *)

isok(k) = issquarefree(k) && !issquarefree(k^2+1); \\ Michel Marcus, May 04 2025

from sympy import factorint
def is_squarefree(n):
    return all(exponent == 1 for exponent in factorint(n).values())
print([a for a in range(1,900) if is_squarefree(a) and not(is_squarefree(a*a + 1))])

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A141928 Primes congruent to 2 mod 25.

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

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

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A383672 Squarefree numbers k such that k^2+1 is not squarefree.

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