Arnaud Vernier - cp's OEIS frontend

A283225 Primes prime(k) such that prime(k)^2 mod prime(k+2) is different from prime(k+2)^2 mod prime(k).

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 43, 47, 53, 59, 61, 73, 79, 83, 89, 109, 113, 137, 139, 199, 211, 241, 283, 293, 313, 317, 523, 1321, 1327

Offset: 1

Arnaud Vernier, Mar 03 2017

I conjecture that there are no other terms in this sequence.
A124129 is constructed in a similar way: by comparing the values of prime(k)^2 mod prime(k+1) and prime(k+1)^2 mod prime(k).
If it exists, then a(35) > 10^12. - Lucas A. Brown, Feb 11 2021

			a(10) = prime(10) = 29 is in the sequence because the remainder of the division of 29^2 = 841 by prime(12) = 37 is 27, which is different from the remainder of the division of 37^2 = 1369 by prime(10) = 29, which is 6.

Cf. A124129.

Select[Prime[Range[250]],PowerMod[#,2,NextPrime[#,2]] != PowerMod[ NextPrime[ #,2],2,#]&]  (* Harvey P. Dale, Nov 17 2020 *)

A178751 Numbers k such that in Z/kZ the equation x^y + 1 = 0 has only the trivial solutions with x == -1 (mod k).

Original entry on oeis.org

2, 3, 4, 6, 8, 12, 15, 16, 20, 24, 30, 32, 40, 48, 51, 60, 64, 68, 80, 96, 102, 120, 128, 136, 160, 192, 204, 240, 255, 256, 272, 320, 340, 384, 408, 480, 510, 512, 544, 640, 680, 768, 771, 816, 960, 1020, 1024, 1028, 1088, 1280, 1360, 1536, 1542, 1632, 1920, 2040

Offset: 1

Arnaud Vernier, Jun 09 2010, Jun 10 2010

nonn

It appears that odd terms 3, 15, 51, 255, 771, 3855, 13107, 65535, ... are given by A038192. - Michel Marcus, Aug 08 2013

This is the complement of A126949 in the numbers k > 1. (But it could be argued that the sequence should start with k = 1 as initial term.) It appears that for any a(j) in the sequence, 2*a(j) is also in the sequence. The primitive terms (not of the form a(j) = 2*a(m), m < j) are 2, 3, 15, 20, 51, 68, 255, 340, 771, 1028, .... (see A274003). - M. F. Hasler, Jun 06 2016

			In Z/3Z, the only solution to the equation x^y + 1 = 0 is x = 2 and y = 1. Whereas in Z/5Z, the equation has at least one nontrivial solution: 2^2 + 1 = 0.

Arnaud Vernier and Charles R Greathouse IV, Table of n, a(n) for n = 1..189 (first 78 terms from Vernier)

Cf. A038192, A126949, A274003.

is(n)=for(x=2,n-2,if(gcd(x,n)>1,next);my(t=Mod(x,n));while(abs(centerlift(t))>1,t*=x);if(t==-1,return(0)));n>1 \\ Charles R Greathouse IV, Aug 08 2013

A167415 Positive integers k such that there is no solution of the equation x^2 + y^2 + 3xy = 0 in Z/nZ except for the trivial one (0,0).

Original entry on oeis.org

2, 3, 6, 7, 13, 14, 17, 21, 23, 26, 34, 37, 39, 42, 43, 46, 47, 51, 53, 67, 69, 73, 74, 78, 83, 86, 91, 94, 97, 102, 103, 106, 107, 111, 113, 119, 127, 129, 134, 137, 138, 141, 146, 157, 159, 161, 163, 166, 167, 173, 182, 193, 194, 197, 201, 206, 214, 219

Offset: 1

Arnaud Vernier, Nov 03 2009

Prime numbers of this sequence are congruent to {2,3} modulo 5.

			The only solution of the equation x^2 + y^2 + 3*x*y = 0 in Z/2Z is (0,0).
4 is not in the sequence because 0^2 + 2^2 + 3*2*0 = 4 == 0 (mod 4). 5 is not in the sequence because 1^2 + 1^2 + 3*1*1 = 5 == 0 (mod 5). 10 is not in the sequence because 2^2 + 2^2 + 3*2*2 = 20 == 0 (mod 10). - _R. J. Mathar_, Jun 16 2019

Cf. A031363 (x^2 + y^2 + 3xy).

isA167415 := proc(n)
    local x,y ;
    for x from 0 to n-1 do
        for y from x to n-1 do
            if modp(x^2+y^2+3*x*y,n) = 0 and (x <> 0 or y <> 0) then
                return false;
            end if;
        end do:
    end do:
    true ;
end proc:
for n from 2 to 300 do
    if isA167415(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jun 16 2019

Name corrected by R. J. Mathar, Jun 16 2019 and Don Reble

A167181 Squarefree numbers such that all prime factors are == 3 mod 4.

Original entry on oeis.org

1, 3, 7, 11, 19, 21, 23, 31, 33, 43, 47, 57, 59, 67, 69, 71, 77, 79, 83, 93, 103, 107, 127, 129, 131, 133, 139, 141, 151, 161, 163, 167, 177, 179, 191, 199, 201, 209, 211, 213, 217, 223, 227, 231, 237, 239, 249, 251, 253, 263, 271, 283, 301, 307, 309, 311, 321, 329

Offset: 1

Arnaud Vernier, Oct 29 2009

Or, numbers that are not divisible by the sum of two squares (other than 1). - Clarified by Gabriel Conant, Apr 18 2016

If a term divides the sum of two squares, then it divides each of the two numbers individually. Moreover, only the numbers in this sequence have this property. See link for proof. - V Sai Prabhav, Jul 15 2025

Robert Israel, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Generalizations of Mertens's Formula and k-Free and s-Full Numbers with Prime Divisors in Arithmetic Progression, ResearchGate, 2024.
V Sai Prabhav, Proof for the comment

Cf. A002145, A004613, A004614, A005117, A231754.

Cf. A175647, A243379.

N:= 1000: # to get all terms <= N
S:= {1};
for p from 3 by 4 to N do
  if isprime(p) then
    S:= S union select(`<=`, map(t -> t*p, S),N)
  fi
od:
sort(convert(S,list)); # Robert Israel, Apr 18 2016

Select[Range@ 1000, #==1 || ({{3}, {1}} == Union /@ {Mod[ #[[1]], 4], #[[2]]} &@ Transpose@ FactorInteger@ #) &] (* Giovanni Resta, Apr 18 2016 *)

isok(n) = if (! issquarefree(n), return (0)); f = factor(n); for (i=1, #f~, if (f[i, 1] % 4 != 3, return (0))); 1 \\ Michel Marcus, Sep 04 2013

A005117 INTERSECT A004614. - R. J. Mathar, Nov 05 2009

The number of terms that do not exceed x is ~ c * x / sqrt(log(x)), where c = A243379/(2*sqrt(A175647)) = 0.4165140462... (Jakimczuk, 2024, Theorem 3.10, p. 26). - Amiram Eldar, Mar 08 2024

Edited by Zak Seidov, Oct 30 2009

Narrowed definition down to squarefree numbers - R. J. Mathar, Nov 05 2009

A093612 Primes of form 7^n-2.

Original entry on oeis.org

5, 47, 2399, 823541, 5764799, 13841287199, 4747561509941, 459986536544739960976799, 157775382034845806615042741, 97327453648743672783790144527749033795901408624680013074608083129650399

Offset: 1

Arnaud Vernier, May 23 2004

nonn

The exponents n are listed in A090669, cf. formula. [From M. F. Hasler, Nov 26 2009]

The next term (a(11)) has 83 digits. - Harvey P. Dale, Nov 14 2014

Cf. A014232.

Select[7^Range[90]-2,PrimeQ] (* Harvey P. Dale, Nov 14 2014 *)

a(n)=7^A090669(n)-2. [From M. F. Hasler, Nov 26 2009]

Terms beyond a(6) from M. F. Hasler, Nov 26 2009

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User: Arnaud Vernier

A283225 Primes prime(k) such that prime(k)^2 mod prime(k+2) is different from prime(k+2)^2 mod prime(k).

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A178751 Numbers k such that in Z/kZ the equation x^y + 1 = 0 has only the trivial solutions with x == -1 (mod k).

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A167415 Positive integers k such that there is no solution of the equation x^2 + y^2 + 3*x*y = 0 in Z/nZ except for the trivial one (0,0).

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A167181 Squarefree numbers such that all prime factors are == 3 mod 4.

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Author

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Programs

Maple

Mathematica

PARI

Formula

Extensions

A093612 Primes of form 7^n-2.

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Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

Extensions

A167415 Positive integers k such that there is no solution of the equation x^2 + y^2 + 3xy = 0 in Z/nZ except for the trivial one (0,0).