A033203 -id:A033203 - cp's OEIS frontend

A240920 Prime numbers that occur as divisors of numbers of the form m^2 + 5.

2, 3, 5, 7, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 103, 107, 109, 127, 149, 163, 167, 181, 223, 227, 229, 241, 263, 269, 281, 283, 307, 347, 349, 367, 383, 389, 401, 409, 421, 443, 449, 461, 463, 467, 487, 503, 509, 521, 523, 541, 547, 563

Offset: 1

J. Lowell, Aug 02 2014

Conjecture: a prime number is in this sequence if and only if its next-to-last digit is even.

The law of quadratic reciprocity shows an odd prime is in the sequence if and only if it is 1, 3, 5, 7 or 9 (mod 20). This proves the above conjecture, so the sequence is the union of {2, 5} and A139513. - Jens Kruse Andersen, Aug 09 2014

			23 is in the sequence because it divides 8^2+5=69 with m=8.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000
David Lowry-Duda, Unexpected Conjectures about -5 Modulo Primes, College Mathematics Journal, Vol. 46, No. 1 (Jan 2015), pp.56-57.

Cf. A002313 (k=1 or k=4), A033203 (k=2), A045331 (k=3), A139513.

isA240920 := proc(p)
    local n;
    if isprime(p) then
        for n from 0 to p do
            if modp(n^2+5,p) = 0 then
                return true;
            end if;
        end do:
        false;
    else
        false;
    end if;
end proc:
for i from 1 to 600 do
    p := ithprime(i) ;
    if isA240920(p) then
        printf("%d,",p);
    end if;
end do:

select(p->issquare(Mod(-5,p)), primes(100)) \\ Charles R Greathouse IV, Nov 29 2016

a(n) ~ 2n log n. - Charles R Greathouse IV, Nov 29 2016

A243595 Primes p such that 3 + 2*p^2 is also prime.

Original entry on oeis.org

2, 5, 7, 23, 37, 43, 47, 83, 103, 107, 113, 127, 197, 373, 433, 463, 467, 523, 547, 587, 593, 617, 733, 743, 797, 863, 877, 887, 953, 1097, 1163, 1213, 1297, 1427, 1493, 1567, 1583, 1657, 1667, 1693, 1783, 1877, 1987, 2053, 2063, 2087, 2207, 2357, 2557, 2753

Offset: 1

Zak Seidov, Jun 07 2014

Corresponding primes 3 + 2*p^2 are congruent to 5 modulo 6.

			2 is in the sequence because 3+2*2^2 = 11 is prime; also, for the comment, 11 = 6+5.
5 is in the sequence because 3+2*5^2 = 53 is prime, also 53 = 6*8+5.
7 is in the sequence because 3+2*7^2 = 101 is prime, also 101 = 6*16+5.

Bruno Berselli, Table of n, a(n) for n = 1..1000

Cf. A002476, A033203, A153507, A161724.

[p: p in PrimesUpTo(4000) | IsPrime(3+2*p^2)]; // Bruno Berselli, Jun 07 2014

Select[Prime[Range[500]], PrimeQ[3 + 2 #^2] &] (* Bruno Berselli, Jun 07 2014 *)

s=[]; forprime(p=2, 4000, if(isprime(3+2*p^2), s=concat(s, p))); s \\ Colin Barker, Jun 07 2014

[p for p in primes(4000) if is_prime(3+2*p^2)] # Bruno Berselli, Jun 07 2014

A163184 Primes of the form 8k + 1 dividing 2^j + 1 for some odd j.

Original entry on oeis.org

281, 617, 1033, 1049, 1097, 1193, 1481, 1553, 1753, 1777, 2281, 2393, 2473, 2657, 2833, 2857, 3049, 3529, 3673, 3833, 4049, 4153, 4217, 4273, 4457, 4937, 5113, 5297, 5881, 6121, 6449, 6481, 6521, 6529, 6569, 6761, 6793, 6841, 7121, 7129, 7481, 7577, 7817, 8081, 8233, 8537, 9001, 9137, 9209, 9241

Offset: 1

Christopher J. Smyth, Jul 22 2009

Each term p has the form 2^r*j + 1, where r >= 3, j is odd, and ord_p(-2) divides j.

			281 is in the sequence as 281 = 2^3*35 + 1 and 281 | 2^35 + 1.

Set difference of A163183 and A007520.

Subsequence of A033203, A051071, A051073, A051077, A051085, A051101.

with(numtheory):A:=NULL:p:=2: for c to 500 do p:=nextprime(p);if order(-2,p) mod 2=1 and p mod 8 = 1 then A:=A,p;;fi;od:A;

More terms from Max Alekseyev, Sep 29 2016

A214588 Primes p such that p mod 16 < 8.

Original entry on oeis.org

2, 3, 5, 7, 17, 19, 23, 37, 53, 67, 71, 83, 97, 101, 103, 113, 131, 149, 151, 163, 167, 179, 181, 193, 197, 199, 211, 227, 229, 241, 257, 263, 277, 293, 307, 311, 337, 353, 359, 373, 389, 401, 419, 421, 433, 439, 449, 467, 487, 499, 503, 547, 563, 577, 593, 599

Offset: 1

Brad Clardy, Jul 22 2012

nonn

Original definition: Primes p such that p XOR 8 = p + 8.
This is an example of a class of primes p such that p XOR n = p + n.
A002144 is the case where n=2, there are no cases where n=3, in A033203 n=4, 2 is the only p for n=5, in A007519 n=6, there are no cases where n=7. This sequence occurs when n=8.
In general if n is an odd number in A004767 there are no primes, if n is an odd number in A016813, then 2 is the only prime, and if n is an even number (A005843) there is a set of primes that satisfies the relationship p XOR n = p + n.

			103 is in the sequence because 103 mod 16 is 7 which is less than 8. - _Indranil Ghosh_, Jan 18 2017

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Cf. A002144, A033203, A007519, A004767, A016813, A005843.

XOR := func;
for n in [2 .. 1000] do
   if IsPrime(n)  then  pn:=n;
      if (XOR(pn,8) eq pn+8) then pn; end if;
   end if;
end for;

Select[Prime[Range[200]],Mod[#,16]<8&] (* Harvey P. Dale, Jan 11 2018 *)

is_A214588(p)={ !bittest(p,3) & isprime(p) } \\ M. F. Hasler, Jul 24 2012

forprime(p=1,699, bittest(p,3) || print1(p",")) \\ M. F. Hasler, Jul 24 2012

from sympy import isprime
i=1
while i<=600:
    if  isprime(i)==True and (i%16)<8:
        print(i, end=", ")
    i+=1 # Indranil Ghosh, Jan 18 2017

A360154 Primes of the form m^2 + 2k^2 such that m^2 + 2(k+1)^2 is also prime.

Original entry on oeis.org

11, 41, 83, 107, 113, 227, 347, 443, 521, 563, 593, 641, 827, 929, 953, 1091, 1187, 1193, 1259, 1409, 1427, 1553, 1601, 1697, 1811, 1979, 2003, 2297, 2339, 2393, 2699, 2801, 2819, 3011, 3089, 3209, 3251, 3449, 3467, 3929, 3947

Offset: 1

Ludovic Schwob, Jan 28 2023

nonn

Primes of the form m^2 + 2*k^2 are norms of prime elements of Z[i*sqrt(2)]. Prime couples of the form (m^2 + 2*k^2, m^2 + 2*(k+1)^2) correspond to primes in Z[i*sqrt(2)] differing from i*sqrt(2).

A prime cannot be simultaneously the lesser of one such couple and the greater of another.

			The first 3 prime couples of the form (m^2 + 2*k^2, m^2 + 2*(k+1)^2) are (11,17) = (3^2 + 2*1^2, 3^2 + 2*2^2), (41,59) = (3^2 + 2*4^2, 3^2 + 2*5^2) and (83,89) = (9^2 + 2*1^2, 9^2 + 2*2^2).

See A360155 for greater values.
Cf. A000040 (prime numbers).
Cf. A033203 (primes of form m^2 + 2*k^2).

If (m^2 + 2*k^2, m^2 + 2*(k+1)^2) is a prime couple, then m is congruent to 3 modulo 6 and k is congruent to 1 modulo 3.

A360155 Primes of the form m^2 + 2(k+1)^2 such that m^2 + 2k^2 is also prime.

Original entry on oeis.org

17, 59, 89, 131, 137, 233, 401, 449, 587, 617, 659, 683, 857, 971, 1019, 1097, 1217, 1283, 1361, 1481, 1499, 1571, 1667, 1787, 1889, 2081, 2129, 2411, 2441, 2531, 2729, 2843, 2969, 3137, 3203, 3257, 3371, 3491, 3617, 4019, 4073

Offset: 1

Ludovic Schwob, Jan 28 2023

nonn

Primes of the form m^2 + 2*k^2 are the norms of prime elements of Z[i*sqrt(2)]. Pairs of primes of the form (m^2 + 2*k^2, m^2 + 2*(k+1)^2) correspond to primes in Z[i*sqrt(2)] differing by i*sqrt(2).

A prime cannot simultaneously be the lesser of such a pair and the greater of another.

			The first 3 such prime pairs are
  (11,17) = (3^2 + 2*1^2, 3^2 + 2*2^2) with m=3 and k=1,
  (41,59) = (3^2 + 2*4^2, 3^2 + 2*5^2) with m=3 and k=4,
  (83,89) = (9^2 + 2*1^2, 9^2 + 2*2^2) with m=9 and k=1.

See A360154 for lesser primes.
Cf. A000040 (prime numbers).
Cf. A033203 (primes of the form m^2 + 2*k^2).

If m^2 + 2*k^2 and m^2 + 2*(k+1)^2 are primes, then m == 3 (mod 6) and k == 1 (mod 3).

A363410 a(n)= 1/sqrt(2) * the imaginary part of Product_{k = 1..n} (1 + k*sqrt(-2)).

Original entry on oeis.org

0, 1, 3, -6, -90, 45, 5607, 8316, -616572, -2517075, 106354215, 779869134, -26562900078, -299503403199, 9075456298755, 144911485323000, -4066415773786872, -87372799002303111, 2313066895842715947, 64609858869087786210, -1627745411473223627970

Offset: 0

Peter Bala, Jun 01 2023

Compare with A105751(n) = the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-1).
Moll (2012) studied the prime divisors of the terms of A105750 - the real part of Product_{k = 0..n} 1 + k*sqrt(-1) - and divided the primes into three classes. Numerical calculation suggests that a similar division holds in this case.
Type 1: primes p that do not divide any element of the sequence {a(n)}.
We conjecture that in this case, unlike in A105750, the set of type 1 primes is empty; that is, every prime p divides some term of this sequence.
Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. An example is given below.
We conjecture that the set of type 2 primes consists of primes p == 1 or 3 (mod 8), equivalently, rational primes that split in the field extension Q(sqrt(-2)) of Q. See A033200.
Moll's conjecture 5.5 extends to this sequence: for primes p of type 2, the p-adic valuation v_p(a(n)) ~ n/(p - 1) as n -> oo.
Type 3: primes p such that the sequence of p-adic valuations {v_p(a(n)) : n >= 0} exhibits an oscillatory behavior (this phrase is not precisely defined). An example is given below.
We conjecture that the set of type 3 primes consists of primes p == 5 or 7 (mod 8), equivalently, primes that remain inert in the field extension Q(sqrt(-2)) of Q, together with the prime p = 2, which ramifies in Q(sqrt(-2)). See A033203.

			Type 2 prime p = 3: the sequence of 3-adic valuations [v_3(n) : n = 1..100] =  [0, 1, 1, 2, 2, 2, 3, 4, 4, 4, 5, 5, 6, 6, 6, 9, 9, 9, 10, 10, 10, 11, 11, 11, 13, 13, 13, 14, 14, 14, 15, 15, 15, 17, 17, 17, 18, 18, 18, 19, 19, 19, 22, 22, 22, 23, 23, 23, 24, 24, 24, 26, 26, 26, 27, 27, 27, 28, 28, 28, 30, 30, 30, 31, 31, 31, 32, 32, 32, 36, 36, 36, 37, 37, 37, 38, 38, 38, 41, 40, 40, 42, 41, 41, 42, 42, 42, 44, 44, 44, 45, 45, 45, 46, 46, 46, 49, 49, 49, 50].
Note that v_3(a(100)) = 50 = 100/(3 - 1), in agrement with the asymptotic growth for type 2 primes conjectured above.
Type 3 prime p = 5: the sequence of 5-adic valuations [v_5(n) : n = 1..100] = [0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 3, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2], showing the oscillatory behavior for type 3 primes conjectured above.

Victor H. Moll, An arithmetic conjecture on a sequence of arctangent sums, 2012, see f_n.

Cf. A105750, A105751, A363409 - A363416.

a := proc(n) option remember; if n = 0 then 0 elif n = 1 then 1 else ( (2*n - 1)*a(n-1) - n*(2*n^2 - 4*n + 3)*a(n-2) )/(n - 1) end if; end:
seq(a(n), n = 0..20);

(n - 1)*a(n) = (2*n - 1)*a(n-1) - n*(2*n^2 - 4*n + 3)*a(n-2) with a(0) = 0 and a(1) = 1.

a(n) = Sum_{k = 0..floor((n+1)/2)} (-2)^k*Stirling1(n+1,n+1-2*k).

A201544 Odd numbers of the form x^2 + 2*y^2 with positive integers x and y.

Original entry on oeis.org

3, 9, 11, 17, 19, 27, 33, 41, 43, 51, 57, 59, 67, 73, 75, 81, 83, 89, 97, 99, 107, 113, 121, 123, 129, 131, 137, 139, 147, 153, 163, 171, 177, 179, 187, 193, 201, 209, 211, 219, 225, 227, 233, 241, 243, 249, 251, 257, 267, 275, 281, 283, 289, 291, 297, 307

Offset: 1

Zak Seidov, Dec 02 2011

nonn

All terms == {1,3} mod 8. Terms that are not multiple of some previous term are prime numbers (see A033203, except for the first term 2 there).

For the numbers with positive proper representations see A225771 without member 1, the subsequence without 75 = 3*5^2, 147 = 3*7^2, 225 = (3*5)^2, 275 = 5^2*11, ... - Wolfdieter Lang, Jan 14 2025

Intersection of A005408 and A154777.

Cf. A033200 (primes), A033203, A225771.

cp's OEIS Frontend

A240920 Prime numbers that occur as divisors of numbers of the form m^2 + 5.

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A243595 Primes p such that 3 + 2*p^2 is also prime.

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A163184 Primes of the form 8k + 1 dividing 2^j + 1 for some odd j.

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A214588 Primes p such that p mod 16 < 8.

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A360154 Primes of the form m^2 + 2*k^2 such that m^2 + 2*(k+1)^2 is also prime.

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A360155 Primes of the form m^2 + 2*(k+1)^2 such that m^2 + 2*k^2 is also prime.

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A363410 a(n)= 1/sqrt(2) * the imaginary part of Product_{k = 1..n} (1 + k*sqrt(-2)).

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A201544 Odd numbers of the form x^2 + 2*y^2 with positive integers x and y.

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A360154 Primes of the form m^2 + 2k^2 such that m^2 + 2(k+1)^2 is also prime.

A360155 Primes of the form m^2 + 2(k+1)^2 such that m^2 + 2k^2 is also prime.