A007519 -id:A007519 - cp's OEIS frontend

A139510 Primes of the form x^2 + 30x*y + y^2 for x and y nonnegative.

137, 193, 401, 617, 641, 953, 1009, 1129, 1289, 1297, 1801, 1913, 2129, 2137, 2377, 2473, 2657, 2713, 2801, 3049, 3257, 3313, 3329, 3593, 3889, 4001, 4057, 4153, 4201, 4337, 4649, 4657, 4729, 4817, 4937, 4993, 5009, 5153, 5209, 5441, 5657, 5849, 5881

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139489, A007645, A068228, A007519, A033212, A033212, A107152, A107008, A033215, A107145, A139490, A139491.

a = {}; w = 30; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a] (*Artur Jasinski*)

A139511 Primes of the form x^2 + 31x*y + y^2 for x and y nonnegative.

Original entry on oeis.org

67, 103, 181, 199, 223, 313, 397, 463, 487, 499, 631, 643, 661, 691, 709, 883, 991, 1021, 1039, 1093, 1153, 1213, 1321, 1483, 1543, 1567, 1741, 1747, 1753, 1831, 1879, 2017, 2029, 2083, 2113, 2137, 2179, 2203, 2269, 2311, 2377, 2539, 2557, 2677, 2731

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139489, A007645, A068228, A007519, A033212, A033212, A107152, A107008, A033215, A107145, A139490, A139491.

a = {}; w = 31; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a] (*Artur Jasinski*)

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

A233906 Primes of the form (3^k mod k^3) + 1, in order of increasing k.

Original entry on oeis.org

2, 1499, 1783, 9719, 9311, 67883, 134947, 203317, 189433, 560171, 438533, 943849, 640973, 578827, 2172383, 28687, 1505657, 7595033, 2822971, 1242379, 22899523, 9232219, 5730031, 12336083, 3487607, 35451433, 12174803, 10234079, 84459019, 68736683, 44671169, 85507057

Offset: 1

K. D. Bajpai, Dec 17 2013

nonn

			1499 is in the sequence because (3^13 mod 13^3) + 1 = 1499 which is prime.
9719 is in the sequence because (3^29 mod 29^3) + 1 = 9719 which is prime.

K. D. Bajpai, Table of n, a(n) for n = 1..1000

Cf. A000040 (prime numbers).

Cf. A007519 (primes congruent to 1 mod 8).

KD := proc() local a; a:=3^n mod n^3 + 1; if isprime(a) then RETURN (a); fi; end: seq(KD(), n=1..1000);

A281902 Numbers n which have the form n = k^2 - k + (i^2-i)/2 + 2ki, k>=1, i>=0.

Original entry on oeis.org

0, 2, 5, 6, 9, 11, 12, 14, 17, 19, 20, 24, 27, 29, 30, 32, 35, 36, 39, 41, 42, 44, 46, 50, 51, 53, 54, 55, 56, 57, 62, 65, 66, 69, 71, 72, 74, 75, 77, 80, 82, 84, 87, 89, 90, 95, 96, 100, 101, 104, 107, 109, 110, 111, 116, 117, 119, 120, 122, 126, 127, 128

Offset: 1

Vladimir Shevelev, Feb 01 2017

nonn

If prime p has form 8*n+1, then n is a member.

Peter J. C. Moses, Table of n, a(n) for n = 1..5000

Cf. A007519.

8k^2+8n+1 is perfect square.

More terms from Peter J. C. Moses, Feb 01 2017

A301619 Primes congruent to 65 (mod 192).

Original entry on oeis.org

257, 449, 641, 1217, 1409, 1601, 2753, 3137, 3329, 4289, 4481, 4673, 5441, 6977, 7937, 8513, 9281, 9473, 9857, 10433, 11393, 11777, 11969, 12161, 13121, 13313, 13697, 14081, 14657, 15233, 15809, 16001, 16193, 17729, 17921, 19073, 19457, 19841, 21377, 21569

Offset: 1

Felix Fröhlich, Mar 24 2018

In other words, primes of the form 192*k+65 for k > 0.

R. Scott and R. Styer, On p^x - q^y = c and related three term exponential Diophantine equations with prime bases, Journal of Number Theory, Vol. 105, No. 2 (2004), 212-234. See p. 218.

Subsequence of A002144 (primes of form 4*k+65) and A007519 (primes of form 8*k+65).

Cf. primes congruent to 65 (mod k): A142068 (k=66), A142137 (k=74), A142221 (k=82), A142271 (k=86), A142369 (k=94), A142427 (k=98), A142485 (k=102), A142542 (k=106), A142670 (k=114), A142733 (k=118), A142802 (k=122), A142890 (k=126), A105129 (k=128).

[p: p in PrimesUpTo(25000) | p mod 192 in {65}]; // Vincenzo Librandi, Jan 04 2020

Select[Prime[Range[2500]], MemberQ[{65}, Mod[#, 192]] &] (* Vincenzo Librandi, Jan 04 2020 *)

forprime(p=1, 5e4, if(Mod(p, 192)==65, print1(p, ", ")))

A353012 Numbers N such that gcd(N - d, N*d) >= d^2, where d = A000005(N) is the number of divisors of N.

Original entry on oeis.org

1, 2, 136, 156, 328, 444, 584, 600, 712, 732, 776, 876, 904, 1096, 1164, 1176, 1308, 1544, 1864, 1884, 1928, 2056, 2172, 2248, 2316, 2504, 2601, 2696, 2748, 2824, 2892, 2904, 3208, 3240, 3249, 3272, 3324, 3464, 3592, 3656, 3756, 4044, 4056, 4168, 4188, 4476, 4552, 4616

Offset: 1

M. F. Hasler, Apr 15 2022

nonn

As d^2 | N-d we have N = k*d^2 + d for some k >= 0 and d > 1. So gcd(k*d^2 + d - d, (N*d^2 + d)*d) = gcd(k*d^2, k*d^3 + d^2) = gcd(k*d^2, d^2) = d^2. So for any N such that d^2 | gcd(N - d, N*d) we have gcd(N - d, N*d) = d^2. - David A. Corneth, Apr 20 2022

Since gcd(N - d, N*d) is never larger than d^2 (if N = n*g, d = f*g with gcd(n,f) = 1, then gcd(N - d, N*d) = g*gcd(n-f,n*f*g) = g*gcd(n-f, f*f*g) <= g*g, since by assumption, no factor of f divides n), so one can also replace "=" by ">=" in the definition.

			N = 1 is in the sequence because d(N) = 1, gcd(1 - 1, 1*1) = 1 = d^2.
N = 2 is in the sequence because d(N) = 2, gcd(2 - 2, 2*2) = 4 = d^2.
N = 136 = 8*17 is in the sequence because d(N) = 4*2 = 8, gcd(8*17 - 8, 8*17*8) = gcd(8*16, 8*8*17) = 8*8 = d^2. Similarly for N = 8*p with any prime p = 8*k + 1.
N = 156 = 2^2*3*13 is in the sequence because d(n) = 3*2*2 = 12, gcd(12*13 - 12, 12*13*12) = gcd(12*12, 12*12*13) = 12*12 = d^2. Similarly for any N = 12*p with prime p = 12*k + 1.
More generally, when N = m*p^k with p^k == 1 (mod m) and m = (k+1)*d(m), then d(N) = d(m)*(k+1) = m and gcd(n - d, n*d) = gcd(m*p^k - m, m*p^k*m) = m*gcd(p^k - 1, p^k*m) = m^2. This holds for m = 8 and 12 with k = 1, for m = 9, 18 and 24 with k = 2, etc: see sequence A033950 for the m-values.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000005 (number of divisors), A352483 (numerator of (n-d)/(n*d)), A352482 (denominator), A049820 (n - d), A146566 (n*d is divisible by n-d), A033950 (refactorable or tau numbers: d(n) | n, supersequence of this).

Select[Range[4650], GCD[#1 - #2, #1 #2] == #2^2 & @@ {#, DivisorSigma[0, #]} &] (* Michael De Vlieger, Apr 21 2022 *)

select( {is(n, d=numdiv(n))=gcd(n-d,d^2)==d^2}, [1..10^4])

For all m in A033950, the sequence contains all numbers m*p^k with k = m/d(m) - 1, and p^k == 1 (mod m), in particular 8*A007519 and 12*A068228 (k = 1, m = 8 and 12), 9*A129805^2, 18*A129805^2 and 24*A215848^2 (k = 2, m = 9, 18 and 24, A^2 = {x^2, x in A}), etc.

A376349 Number of isomorphism classes k of groups G of order p*2^n when G contains a unique Sylow p subgroup and the maximal 2^m dividing p-1 is such that 2^m >= 2^n.

Original entry on oeis.org

1, 2, 5, 15, 54, 247, 1684, 21820, 1118964

Offset: 0

Miles Englezou, Sep 19 2024

A Sylow p subgroup is a subgroup of order p^r that necessarily exists when r is a maximal power of p. It is not necessarily unique, but when it is unique it is normal in G.
The condition that G of order p*2^n contains a unique Sylow p subgroup places an upper bound on the number of isomorphism classes of G; it is equivalent to stating that the minimal 2^r such that 2^r == 1 (mod p) be such that 2^r > 2^n. The condition that the maximal 2^m dividing p-1, i.e. for p == 1 (mod 2^m), is such that 2^m >= 2^n ensures a lower bound which is equal to the upper bound. See the Miles Englezou link for a proof.
If we relax the two conditions and just consider an arbitrary odd prime p and the number of isomorphism classes for |G| = p*2^n, it is likely that the set of such numbers is unique to p. Since every odd prime has a minimal 2^r such that 2^r == 1 (mod p) (a consequence of Fermat's little theorem), when 2^r = 2^n for |G| = p*2^n, the number of isomorphism classes will differ from a(n) due to the existence of groups where the Sylow p subgroup is not unique.

			a(2) = 5 since D_(p*2^2), C_(p*2^2), C_(p*2^1) x C_2, and two semidirect products C_p : C_4 are all the groups of order p*2^2 for p satisfying the two conditions.
Table showing minimal 2^r and maximal 2^m (as defined in the Comments) for some primes:
---------------------------------------------------------------------------
p |      Minimal 2^r == 1 (mod p)       |   Maximal 2^m, p == 1 (mod 2^m)  |
---------------------------------------------------------------------------
2 |             2^0  = 1                |              2^0 = 1             |
3 |             2^2  = 4                |              2^1 = 2             |
5 |             2^4  = 16               |              2^2 = 4             |
7 |             2^3  = 8                |              2^1 = 2             |
11|             2^10 = 1024             |              2^1 = 2             |
13|             2^12 = 4096             |              2^2 = 4             |
17|             2^8  = 256              |              2^4 = 16            |
19|             2^18 = 262144           |              2^1 = 2             |
23|             2^11 = 2048             |              2^1 = 2             |
29|             2^28 = 268435456        |              2^2 = 4             |
31|             2^5  = 32               |              2^1 = 2             |
37|             2^36 = 68719476736      |              2^2 = 4             |
---------------------------------------------------------------------------
Table of primes satisfying 2^r > 2^n, and 2^m >= 2^n:
-------------------------------------------------------------------------------
   2^n   |                          primes                           |   a(n)  |
-------------------------------------------------------------------------------
2^0 = 1  |  all primes                                    = A000040  | 1       |
2^1 = 2  |  all primes > 2                                = A065091  | 2       |
2^2 = 4  |  5, 13, 17, 29, 37, 41, 53, ...                = A002144  | 5       |
2^3 = 8  |  17, 41, 73, 89, 97, 113, 137, ...             = A007519  | 15      |
2^4 = 16 |  17, 97, 113, 193, 241, 257, 337 ...           = A094407  | 54      |
2^5 = 32 |  97, 193, 257, 353, 449, 577, 641, ...         = A133870  | 247     |
2^6 = 64 |  193, 257, 449, 577, 641, 769, 1153, ...       = A142925  | 1684    |
2^7 = 128|  257, 641, 769, 1153, 1409, 2689, 3329, ...    = A208177  | 21820   |
2^8 = 256|  257, 769, 3329, 7937, 9473, 14081, 14593 ...  = A105131  | 1118964 |
-------------------------------------------------------------------------------

Miles Englezou, Proof

Cf. A000001, A000040, A065091, A002144, A007519, A094407, A133870, A142925, A208177, A105131, A139669, A014664, A023506, A058384.

S:=[];
for i in [0..8] do
    n:=7681*2^i; # 7681 is an appropriate prime for reproducing up to a(8)
    S:=Concatenation(S,[NrSmallGroups(n)]);
od;
Print(S);

a(n) = A000001(p*2^(n)) for every p satisfying the two conditions mentioned in Comments.

A385449 Irregular triangle, read by rows: row n gives the pair of proper positive fundamental solutions (x, y) of the form x^2 - 2*y^2 representing -A057126(n).

Original entry on oeis.org

1, 1, 4, 3, 1, 2, 5, 4, 2, 3, 6, 5, 1, 3, 9, 7, 3, 4, 7, 6, 1, 4, 13, 10, 4, 5, 8, 7, 3, 5, 11, 9, 2, 5, 14, 11, 5, 6, 9, 8, 1, 5, 17, 13, 6, 7, 10, 9, 1, 6, 21, 16, 5, 7, 13, 11, 7, 8, 11, 10, 4, 7, 16, 13, 3, 7, 19, 15, 2, 7, 22, 17, 1, 7, 25, 19, 8, 9, 12, 11, 5, 8, 17, 14, 7, 9, 15, 13, 3, 8, 23, 18, 9, 10, 13, 12

Offset: 1

Wolfdieter Lang, Jul 11 2025

The number of (x, y) pairs in row n is 1 for n = 1 and 2, and 2^P, with P = P1 + P7, where P1 and P7 are the number of prime factors 1 modulo 8 and 7 modulo 8, respectively, of A057126(n), for n >= 3.
See A057126 for comments concerning its representation by x^2 - 2*y^2.
The numbers A057126 are given by 2^e_2 * Product_{i=1..P1} p_{1,i}^e_{1,i} * Product_{j=1..P7} p_{7,j}^e_{7,j}, with the odd primes p_{1,i} and p_{7,j} congruent to 1 and 7 modulo 8, respectively. See A007519 and A007522 for these odd primes. Together with 2 these primes are given in A038873, and without 2 in A001132. The exponents are e_2 = 0 or 1, and e_{1,i} and e_{7,j} are nonnegative. The a(1) = 1 is obtained if all exponents vanish. For the proof see Lemma 18 of the linked W. Lang paper, pp. 22 - 23.
The general solutions are obtained from each fundamental solution by application of integer powers of the matrix Auto' = Mat([3,4], [2,3]). See the linked paper eq (28), p. 14, and eq. (40), p. 17 for D = 2, and k = A057126(n). For the explicit form of the powers of Auto' in terms of Chebyshev polynomials S(n, 6) = A001109(n+1) see there eq. (38), and Lemma 10, eq. (43), p. 17.
The conversion to the pair of proper solutions (X, Y) of X^2 - 2*Y^2 = A057126(n) is given by (X, Y) = (2*y - x, x - y). This may result in solutions with negative Y values. They are then transformed to the fundamental positive proper solutions via the mentioned matrix Auto'. See the right part of the example below. For this conversion see also the Nov 09 2009 comment in A035251 by Franklin T. Adams-Watters.

			n, A057126(n) /k  1  2   3  4 ...   2^P | (X, Y) = (2*y - x, x - y)
-------------------------------------------------------------------
1,  1           | 1  1               1  |  1   0 (3   2)
2,  2           | 4  3               1  |  2   1
3,  7           | 1  2,  5  4        2  |  3  -1 (5   3),  3  1
4, 14 = 2*7     | 2  3,  6  5        2  |  4  -1 (8   5),  4  1
5, 17           | 1  3,  9  7        2  |  5  -2 (7   4),  5  2
6, 23           | 3  4,  7  6        2  |  5  -1 (11  7),  5, 1
7, 31           | 1  4, 13 10        2  |  7  -3 (9   5),  7  3
8, 34 = 2*17    | 4  5,  8  7        2  |  6  -1 (14  9),  6  1
9, 41           | 3  5, 11  9        2  |  7  -2 (13  8),  7  2
10, 46 = 2*23   | 2  5, 14 11        2  |  8  -3 (12  7),  8  3
11, 47          | 5  6,  9  8        2  |  7  -1 (17 11),  7  1
12, 49 = 7^2    | 1  5, 17 13        2  |  9  -4 (11  6),  9  4
13, 62 = 2*31   | 6  7, 10  9        2  |  8  -1 (20 13),  8  1
14, 71          | 1  6, 21 16        2  | 11  -5 (13  7), 11  5
15, 73          | 5  7, 13 11        2  |  9  -2 (19 12),  9  2
16, 79          | 7  8, 11 10        2  |  9  -1 (23 15),  9  1
17, 82 = 2*41   | 4  7, 16 13        2  | 10  -3 (18 11), 10  3
18, 89          | 3  7, 19 15        2  | 11  -4 (17 10), 11  4
19, 94 = 2*47   | 2  7, 22 17        2  | 12  -5 (16  9), 12  5
20, 97          | 1  7, 25 19        2  | 13  -6 (15  8), 13  6
21, 98 = 2*7^2  | 8  9, 12 11        2  | 10  -1 (26 17), 10  1
...
The corresponding fundamental positive proper solutions of X^2 - 2*Y^2 = +119 are: [13 -5 (19 11), 13, 5] and [11 -1 (29 19), 11 1].

Wolfdieter Lang, On Positive Integer Descartes-Steiner Curvature Quintuplets, arXiv:2503.08631 [nucl-th], 2025.

Cf. A001109, A007519, A007522, A035251, A057126, A038873.

A035198 From a Dirichlet series.

Original entry on oeis.org

1, 9, 17, 25, 41, 73, 81, 89, 97, 113, 121, 137, 153, 169, 193, 225, 233, 241, 257, 281, 289, 313, 337, 353, 361, 369, 401, 409, 425, 433, 449, 457, 521, 569, 577, 593, 601, 617, 625, 641, 657, 673, 697, 729, 761, 769, 801, 809, 841, 857, 873, 881, 929, 937

Offset: 0

N. J. A. Sloane

Contribution from R. J. Mathar, Jul 16 2010: (Start)
The Dirichlet function is (z_1(s))^2*z_3(2*s)*z_5(2*s) = 1+ 2/9^s+4/17^s+2/25^s+4/41^s+..,
where z_1(s) = prod_{p in A007519} Zeta(s,p) = 1+2/17^s+2/41^s+2/73^s+ ...(see A004625),
z_3(s) = prod_{p in A007520} Zeta(s,p) = 1+2/3^s+2/9^s+2/11^s+2/19^s+2/27^s+4/33^s+..,
z_5(s) = prod_{p in A007521} Zeta(s,p) = 1+2/5^s+2/13^s+...+4/65^s+2/101^s+..., Zeta(s,p)=(1+p^(-s))/(1-p^(-s)). (End)

P. A. B. Pleasants, M. Baake, J. Roth, Planar coincidences for N-fold symmetry J. Math. Phys. 37 (1996) 1029.

More terms from R. J. Mathar, Jul 16 2010

More terms from Sean A. Irvine, Sep 29 2020

cp's OEIS Frontend

A139510 Primes of the form x^2 + 30x*y + y^2 for x and y nonnegative.

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A139511 Primes of the form x^2 + 31x*y + y^2 for x and y nonnegative.

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A214588 Primes p such that p mod 16 < 8.

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A233906 Primes of the form (3^k mod k^3) + 1, in order of increasing k.

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A281902 Numbers n which have the form n = k^2 - k + (i^2-i)/2 + 2*k*i, k>=1, i>=0.

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A301619 Primes congruent to 65 (mod 192).

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A353012 Numbers N such that gcd(N - d, N*d) >= d^2, where d = A000005(N) is the number of divisors of N.

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A376349 Number of isomorphism classes k of groups G of order p*2^n when G contains a unique Sylow p subgroup and the maximal 2^m dividing p-1 is such that 2^m >= 2^n.

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A281902 Numbers n which have the form n = k^2 - k + (i^2-i)/2 + 2ki, k>=1, i>=0.