A141941 -id:A141941 - cp's OEIS frontend

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

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

A244769 Prime numbers ending in the prime number 43.

Original entry on oeis.org

43, 443, 643, 743, 1543, 2143, 2243, 2543, 2843, 3343, 3643, 3943, 4243, 4643, 4943, 5443, 5743, 5843, 6043, 6143, 6343, 7043, 7243, 7643, 8243, 8443, 8543, 9043, 9343, 9643, 9743, 10243, 10343, 11243, 11443, 11743, 12043, 12143, 12343, 12743, 13043, 14143

Offset: 1

Vincenzo Librandi, Jul 06 2014

Also primes of the form 100*n+43. Subsequence of A105854, A141941.

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

Cf. A105854, A141941.

Cf. similar sequences listed in A244763.

[n: n in PrimesUpTo(16000) | n mod 100 eq 43];

Select[Prime[Range[5, 6000]], Take[IntegerDigits[#], -2]=={4, 3} &]

select(x->(x % 100)==43, primes(2000)) \\ Michel Marcus, Jul 06 2014

A256177 Primes congruent to {8, 13, 18, 23} mod 25.

Original entry on oeis.org

13, 23, 43, 73, 83, 113, 163, 173, 193, 223, 233, 263, 283, 293, 313, 373, 383, 433, 443, 463, 523, 563, 593, 613, 643, 673, 683, 733, 743, 773, 823, 863, 883, 983, 1013, 1033, 1063, 1093, 1123, 1163, 1193, 1213, 1223, 1283, 1373, 1423, 1433, 1483, 1493

Offset: 1

Arkadiusz Wesolowski, Mar 18 2015

nonn

Union of A141933, A141937, A141941, and A141945.

These primes cannot be written as the sum of a triangular number and a square.

Cf. A141933, A141937, A141941, A141945.

[p: p in PrimesUpTo(1493) | p mod 25 in {8, 13, 18, 23}];

Select[Prime@Range[283], MemberQ[{8, 13, 18, 23}, Mod[#, 25]] &]

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

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A244769 Prime numbers ending in the prime number 43.

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A256177 Primes congruent to {8, 13, 18, 23} mod 25.

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Magma

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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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