A141158 -id:A141158 - cp's OEIS frontend

A030430 Primes of the form 10*n+1.

11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 211, 241, 251, 271, 281, 311, 331, 401, 421, 431, 461, 491, 521, 541, 571, 601, 631, 641, 661, 691, 701, 751, 761, 811, 821, 881, 911, 941, 971, 991, 1021, 1031, 1051, 1061, 1091, 1151, 1171, 1181, 1201, 1231, 1291

Offset: 1

Warut Roonguthai

Also primes of form 5*n+1 or equivalently 5*n+6.
Primes p such that the arithmetic mean of divisors of p^4 is an integer: A000203(p^4)/A000005(p^4) = C. - Ctibor O. Zizka, Sep 15 2008
Being a subset of A141158, this is also a subset of the primes of form x^2-5*y^2. - Tito Piezas III, Dec 28 2008
5 is quadratic residue of primes of this form. - Vincenzo Librandi, Jun 25 2014
Primes p such that 5 divides sigma(p^4), cf. A274397. - M. F. Hasler, Jul 10 2016

Michael B. Porter, Table of n, a(n) for n = 1..100000
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 519.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.

Cf. A024912, A045453, A049511, A081759, A017281, A010051, A004615 (multiplicative closure).
Cf. A001583 (subsequence).
Union of A132230 and A132232. - Ray Chandler, Apr 07 2009

a030430 n = a030430_list !! (n-1)
a030430_list = filter ((== 1) . a010051) a017281_list
-- Reinhard Zumkeller, Apr 16 2012

Select[Prime@Range[210], Mod[ #, 10] == 1 &] (* Ray Chandler, Dec 06 2006 *)
Select[Range[11,1291,10],PrimeQ] (*Zak Seidov, Aug 14 2011*)

is(n)=n%10==1 && isprime(n) \\ Charles R Greathouse IV, Sep 06 2012

lista(nn) = forprime(p=11, nn, if(p%10==1, print1(p, ", "))) \\ Iain Fox, Dec 30 2017

a(n) = 10*A024912(n)+1 = 5*A081759(n)+6.
A104146(floor(a(n)/10)) = 1.
Union of A132230 and A132232. - Ray Chandler, Apr 07 2009
a(n) ~ 4n log n. - Charles R Greathouse IV, Sep 06 2012
Intersection of A000040 and A017281. - Iain Fox, Dec 30 2017

A031363 Positive numbers of the form x^2 + xy - y^2; or, of the form 5x^2 - y^2.

Original entry on oeis.org

1, 4, 5, 9, 11, 16, 19, 20, 25, 29, 31, 36, 41, 44, 45, 49, 55, 59, 61, 64, 71, 76, 79, 80, 81, 89, 95, 99, 100, 101, 109, 116, 121, 124, 125, 131, 139, 144, 145, 149, 151, 155, 164, 169, 171, 176, 179, 180, 181, 191, 196, 199, 205, 209, 211, 220, 225, 229, 236

Offset: 1

N. J. A. Sloane

5x^2 - y^2 has discriminant 20, x^2 + xy - y^2 has discriminant 5. - N. J. A. Sloane, May 30 2014
Representable as x^2 + 3xy + y^2 with 0 <= x <= y. - Benoit Cloitre, Nov 16 2003
Numbers k such that x^2 - 3xy + y^2 + k = 0 has integer solutions. - Colin Barker, Feb 04 2014
Numbers k such that x^2 - 7xy + y^2 + 9k = 0 has integer solutions. - Colin Barker, Feb 10 2014
Also positive numbers of the form x^2 - 5y^2. - Jon E. Schoenfield, Jun 03 2022

M. Baake, "Solution of coincidence problem ...", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.

Robert Israel and Vincenzo Librandi, Table of n, a(n) for n = 1..10000
M. Baake, Solution of the coincidence problem in dimensions d <= 4, arxiv:math/0605222 [math.MG], 2006.
M. Baake and R. V. Moody, Similarity submodules and semigroups in Quasicrystals and Discrete Geometry, ed. J. Patera, Fields Institute Monographs, vol. 10 AMS, Providence, RI (1998) pp. 1-13.
J. H. Conway, E. M. Rains and N. J. A. Sloane, On the existence of similar sublattices, Canad. J. Math. 51 (1999), 1300-1306 (Abstract, pdf, ps).
Norbert Hungerbühler and Maciej Smela, Geometric approach to the Diophantine equation x^2 + x*y - y^2 = m, hal-04835410, 2024. See p. 18.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Numbers representable as x^2 + k*x*y + y^2 with 0 <= x <= y, for k=0..9: A001481(k=0), A003136(k=1), A000290(k=2), this sequence, A084916(k=4), A243172(k=5), A242663(k=6), A243174(k=7), A243188(k=8), A316621(k=9).
See A035187 for number of representations.
Primes in this sequence: A038872, also A141158.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
See also the related sequence A263849 based on a theorem of Maass.

select(t -> nops([isolve(5*x^2-y^2=t)])>0, [$1..1000]); # Robert Israel, Jun 12 2014

ok[n_] := Resolve[Exists[{x, y}, Element[x|y, Integers], n == 5*x^2-y^2]]; Select[Range[236], ok]
(* or, for a large number of terms: *)
max = 60755 (* max=60755 yields 10000 terms *); A031363 = {}; xm = 1;
While[T = A031363; A031363 = Table[5*x^2 - y^2, {x, 1, xm}, {y, 0, Floor[ x*Sqrt[5]]}] // Flatten // Union // Select[#, # <= max&]&; A031363 != T, xm = 2*xm]; A031363  (* Jean-François Alcover, Mar 21 2011, updated Mar 17 2018 *)

select(x -> x, direuler(p=2,101,1/(1-(kronecker(5,p)*(X-X^2))-X)), 1) \\ Fixed by Andrey Zabolotskiy, Jul 30 2020, after hints by Colin Barker, Jun 18 2014, and Michel Marcus

is(n)=#bnfisintnorm(bnfinit(z^2-z-1),n) \\ Ralf Stephan, Oct 18 2013

seq(M,k=3) = { \\ assume k >= 0
setintersect([1..M], setbinop((x,y)->x^2 + k*x*y + y^2, [0..1+sqrtint(M)]));
};
seq(236) \\ Gheorghe Coserea, Jul 29 2018

from itertools import count, islice
from sympy import factorint
def A031363_gen(): # generator of terms
    return filter(lambda n:all(not((1 < p % 5 < 4) and e & 1) for p, e in factorint(n).items()),count(1))
A031363_list = list(islice(A031363_gen(),30)) # Chai Wah Wu, Jun 28 2022

Consists exactly of numbers in which primes == 2 or 3 mod 5 occur with even exponents.

Indices of the nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m, p)+1)*p^(-s)+Kronecker(m, p)*p^(-2s))^(-1) for m = 5.

More terms from Erich Friedman

b-file corrected and extended by Robert Israel, Jun 12 2014

A141373 Primes of the form 3x^2+16y^2. Also primes of the form 4x^2+4xy-5y^2 (as well as primes the form 4x^2+12xy+3y^2).

Original entry on oeis.org

3, 19, 43, 67, 139, 163, 211, 283, 307, 331, 379, 499, 523, 547, 571, 619, 643, 691, 739, 787, 811, 859, 883, 907, 1051, 1123, 1171, 1291, 1459, 1483, 1531, 1579, 1627, 1699, 1723, 1747, 1867, 1987, 2011, 2083, 2131, 2179, 2203, 2251, 2347, 2371, 2467, 2539

Offset: 1

T. D. Noe, May 13 2005; Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 28 2008

nonn

The discriminant is -192 (or 96, or ...), depending on which quadratic form is used for the definition. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1. See A107132 for more information.

Except for 3, also primes of the forms 4x^2 + 4xy + 19y^2 and 16x^2 + 8xy + 19y^2. See A140633. - T. D. Noe, May 19 2008

			19 is a member because we can write 19=4*2^2+4*2*1-5*1^2 (or 19=4*1^2+12*1*1+3*1^2).

Z. I. Borevich and I. R. Shafarevich, Number Theory.

Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
William C. Jagy and Irving Kaplansky, Positive definite binary quadratic forms that represent the same primes [Cached copy] See Table II.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A107132, A139827, A107003, A141375, A141376 (d=96).
See also A038872 (d=5),
A038873 (d=8),
A068228, A141123 (d=12),
A038883 (d=13),
A038889 (d=17),
A141158 (d=20),
A141159, A141160 (d=21),
A141170, A141171 (d=24),
A141172, A141173 (d=28),
A141174, A141175 (d=32),
A141176, A141177 (d=33),
A141178 (d=37),
A141179, A141180 (d=40),
A141181 (d=41),
A141182, A141183 (d=44),
A033212, A141785 (d=45),
A068228, A141187 (d=48),
A141188 (d=52),
A141189 (d=53),
A141190, A141191 (d=56),
A141192, A141193 (d=57),
A107152, A141302, A141303, A141304 (d=60),
A141215 (d=61),
A141111, A141112 (d=65),
A141336, A141337 (d=92),
A141338, A141339 (d=93),
A141161, A141163 (d=148),
A141165, A141166 (d=229),
A141167, A141167, A141167 (d=257).

[3] cat [ p: p in PrimesUpTo(3000) | p mod 24 in {19 } ]; // Vincenzo Librandi, Jul 24 2012

QuadPrimes2[3, 0, 16, 10000] (* see A106856 *)

list(lim)=my(v=List(),w,t); for(x=1, sqrtint(lim\3), w=3*x^2; for(y=0, sqrtint((lim-w)\16), if(isprime(t=w+16*y^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 09 2017

Except for 3, the primes are congruent to 19 (mod 24). - T. D. Noe, May 02 2008

More terms from Colin Barker, Apr 05 2015

Edited by N. J. A. Sloane, Jul 14 2019, combining two identical entries both with multiple cross-references.

A154939 Primes p such that (p-1)*(p+1)-+p are primes.

Original entry on oeis.org

3, 5, 11, 31, 101, 131, 149, 181, 241, 331, 419, 449, 709, 1051, 1061, 1171, 1409, 1549, 1579, 1699, 1759, 1831, 2069, 3229, 3449, 3761, 3911, 4159, 4951, 5821, 6029, 6481, 6661, 6679, 6899, 7079, 7151, 7229, 7369, 8101, 8219, 8629, 8861, 9091, 9161, 9521

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 17 2009

nonn

That is, primes p such that p^2+p-1 and p^2-p-1 are both primes: intersection of A053184 and A091567. - Michel Marcus, Jul 10 2016

			2*4=8-+3 -> primes, 4*6=24-+5 -> primes,...

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

Cf. A053184, A038872, A141158, A091567, A098058, A038936, A089270, A140559.

[p: p in PrimesUpTo(10000) | IsPrime(p^2+p-1) and IsPrime(p^2-p-1)]; // Vincenzo Librandi, Jul 10 2016

lst={};Do[p=Prime[n];If[PrimeQ[(p-1)*(p+1)-p]&&PrimeQ[(p-1)*(p+1)+p],AppendTo[lst,p]],{n,7!}];lst
Select[Prime[Range[1500]], And@@PrimeQ/@{#^2 - # - 1, #^2 + # - 1} &] (* Vincenzo Librandi, Jul 10 2016 *)
Select[Prime[Range[1500]],AllTrue[(#-1)(#+1)+{#,-#},PrimeQ]&] (* Harvey P. Dale, Sep 21 2023 *)

A155006 Primes p such that (p-2)(p+2)-+2p are primes.

Original entry on oeis.org

5, 7, 13, 23, 37, 43, 73, 167, 233, 263, 433, 557, 587, 593, 607, 727, 857, 1153, 1597, 1627, 1753, 2143, 2663, 2713, 3433, 3607, 3863, 3947, 4027, 4363, 4423, 4673, 5147, 5477, 5623, 5807, 5903, 6277, 7237, 7333, 7577, 8287, 8647, 8837, 8887, 9043, 10067

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 18 2009

nonn

3*7-10=11, 3*7+10=31,...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A125272, A053184, A038872, A141158, A038615, A098058, A038936, A089270, A140559, A154939

lst={};Do[p=Prime[n];If[PrimeQ[(p-2)*(p+2)-2*p]&&PrimeQ[(p-2)*(p+2)+2*p],AppendTo[lst,p]],{n,7!}];lst
Select[Prime[Range[1500]],AllTrue[(#-2)(#+2)+{2#,-2#},PrimeQ]&] (* Harvey P. Dale, Jan 01 2025 *)

A002381 Numbers of the form (p^2 - 1)/120 where p is 1 or prime.

Original entry on oeis.org

0, 1, 3, 7, 8, 14, 29, 31, 42, 52, 66, 85, 99, 143, 161, 185, 190, 267, 273, 304, 330, 371, 437, 476, 484, 525, 603, 612, 658, 806, 913, 1015, 1074, 1197, 1261, 1340, 1394, 1463, 1477, 1548, 1606, 1680, 1771, 1912, 2009, 2075, 2159, 2262, 2439, 2698, 2717

Offset: 1

N. J. A. Sloane

nonn

For n>1, primes p corresponding to a(n) are in A038872(n) = A045468(n-1) = A141158(n). - Ray Chandler, Jul 29 2019

H. Gupta, On a conjecture of Chowla, Proc. Indian Acad. Sci., 5A (1937), 381-384.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ray Chandler, Table of n, a(n) for n = 1..10000
H. Gupta, On a conjecture of Chowla, Proc. Indian Acad. Sci., 5A (1937), 381-384. [Annotated scanned copy]

Cf. A002382, A002855, A038872, A045468, A141158, subsequence of A093722.

Join[{0},Select[Table[(p^2-1)/120,{p,Prime[Range[200]]}],IntegerQ]] (* Harvey P. Dale, Jan 14 2020 *)

j=[]; for(n=0,150,x=prime(n)^2-1; if(Mod(x,120)==0,j=concat(j,(x/120)))); j

More terms from Jason Earls, Jul 29 2001

A045453 Primes congruent to {0, 1} mod 5.

Original entry on oeis.org

5, 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 211, 241, 251, 271, 281, 311, 331, 401, 421, 431, 461, 491, 521, 541, 571, 601, 631, 641, 661, 691, 701, 751, 761, 811, 821, 881, 911, 941, 971, 991, 1021, 1031, 1051, 1061, 1091, 1151, 1171, 1181, 1201, 1231

Offset: 1

N. J. A. Sloane

Being a subset of A141158, this is also a subset of the primes of form x^2 - 5y^2. - Tito Piezas III, Dec 28 2008

			a(1) = 5 is the first primes that is congruent to 0 or 1, modulo 5.
a(2) = 11 is the first prime congruent to 1 modulo 5, and therefore (since there is no other prime congruent to 0 mod 5) the second term of this sequence.
a(10^k) = (181, 2791, 38201, 479771, 5803381, 67881871, 776580131, ...) for k = 1, 2, 3, ...

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

Same as A030430 with addition of the only prime congruent to 0 (mod 5), a(1) = 5.

Cf. A000040.

[ p: p in PrimesUpTo(1300) | p mod 5 in {0, 1} ]; // Vincenzo Librandi, Aug 13 2012

Select[Prime@Range[210], MemberQ[{0, 1}, Mod[ #, 5]] &] (* Ray Chandler, Dec 06 2006 *)

A045453_vec(Nmax)=select( p->p%5<2, primes([1,Nmax])) \\ or:
A045453(n)=forprime(p=1,,p%5>1||n--||return(p)) \\ M. F. Hasler, Jan 15 2018

a(n) = A030430(n-1) for all n >= 2. - M. F. Hasler, Jan 15 2018

Extended by Ray Chandler, Nov 28 2003
Checked by Neven Juric (neven.juric(AT)apis-it.hr), Feb 04 2008
Edited and a(1000) double checked by M. F. Hasler, Jan 15 2018

A155007 Primes p such that (p-3)(p+3)-+3p are primes.

Original entry on oeis.org

7, 17, 37, 113, 157, 227, 283, 293, 313, 347, 443, 587, 787, 883, 1063, 1097, 1237, 1303, 1327, 1427, 1567, 1723, 1933, 1973, 2087, 2347, 2467, 2687, 2777, 3457, 3593, 4447, 4703, 4793, 4967, 5737, 5827, 6317, 6607, 6793, 6857, 8297, 8563, 8803, 9433

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 18 2009

nonn

4*10-3*7=19, 4*10+3*7=61, ...

Cf. A125272, A053184, A038872, A141158, A038615, A098058, A038936, A089270, A140559, A154939, A155006

lst={};Do[p=Prime[n];If[PrimeQ[(p-3)*(p+3)-3*p]&&PrimeQ[(p-3)*(p+3)+3*p],AppendTo[lst,p]],{n,7!}];lst

A045457 Primes congruent to {0, 4} mod 5.

Original entry on oeis.org

5, 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, 349, 359, 379, 389, 409, 419, 439, 449, 479, 499, 509, 569, 599, 619, 659, 709, 719, 739, 769, 809, 829, 839, 859, 919, 929, 1009, 1019, 1039, 1049, 1069, 1109, 1129, 1229, 1249, 1259, 1279

Offset: 1

N. J. A. Sloane

Being a subset of A141158, this is also a subset of the primes of form x^2 - 5*y^2. - Tito Piezas III, Dec 28 2008

Apparently primes p such that 5 | tau(p), where Ramanujan's tau is A000594. - Charles R Greathouse IV, Jul 08 2013

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

Same as A030433 prefixed with 5.

Cf. A000040.

[ p: p in PrimesUpTo(1300) | p mod 5 in {0, 4} ]; // Vincenzo Librandi, Aug 13 2012

Select[Prime@Range[210], MemberQ[{0, 4}, Mod[ #, 5]] &] (* Ray Chandler, Nov 07 2006 *)

is(n)=n==5 || (n%5==4 && isprime(n)) \\ Charles R Greathouse IV, Jul 08 2013

Extended by Ray Chandler, Nov 07 2006

A141750 Primes of the form 4x^2 + 3x*y - 4y^2 (as well as of the form 2x^2 + 9xy + y^2).

Original entry on oeis.org

2, 3, 19, 23, 37, 41, 61, 67, 71, 73, 79, 89, 97, 109, 127, 137, 149, 173, 181, 211, 223, 227, 251, 257, 269, 283, 293, 311, 317, 347, 349, 353, 359, 367, 373, 383, 389, 397, 401, 419, 439, 457, 461, 463, 479, 487, 499, 503, 509, 523, 547, 557, 587, 593, 607

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (sergarmor(AT)yahoo.es), Jul 03 2008

nonn

Discriminant = 73. Class = 1. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2-4ac.

Is this the same as A038957? - R. J. Mathar, Jul 04 2008. Answer: almost certainly - see the Tunnell notes in A033212. - N. J. A. Sloane, Oct 18 2014

			a(2) = 3 because we can write 3 = 4*1^2 + 3*1*1 - 4*1^2.

Z. I. Borevich and I. R. Shafarevich, Number Theory.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

See also A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141158 (d=20). A141159, A141160 (d=21). A141170, A141171 (d=24). A141172, A141173 (d=28). A141174, A141175 (d=32). A141176, A141177 (d=33). A141178 (d=37). A141179, A141180 (d=40). A141181 (d=41). A141182, A141183 (d=44). A033212, A141785 (d=45). A068228, A141187 (d=48). A141188 (d=52). A141189 (d=53). A141190, A141191 (d=56). A141192, A141193 (d=57). A107152, A141302, A141303, A141304 (d=60). A141215 (d=61). A141111, A141112 (d=65). A141161, A141163 (d=148). A141165, A141166 (d=229). A141167, A141168 (d=257).

cp's OEIS Frontend

A030430 Primes of the form 10*n+1.

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A031363 Positive numbers of the form x^2 + xy - y^2; or, of the form 5x^2 - y^2.

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A141373 Primes of the form 3*x^2+16*y^2. Also primes of the form 4*x^2+4*x*y-5*y^2 (as well as primes the form 4*x^2+12*x*y+3*y^2).

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A154939 Primes p such that (p-1)*(p+1)-+p are primes.

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Magma

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A155006 Primes p such that (p-2)*(p+2)-+2*p are primes.

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Mathematica

A002381 Numbers of the form (p^2 - 1)/120 where p is 1 or prime.

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A045453 Primes congruent to {0, 1} mod 5.

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A141373 Primes of the form 3x^2+16y^2. Also primes of the form 4x^2+4xy-5y^2 (as well as primes the form 4x^2+12xy+3y^2).

A155006 Primes p such that (p-2)(p+2)-+2p are primes.

A155007 Primes p such that (p-3)(p+3)-+3p are primes.

A141750 Primes of the form 4x^2 + 3x*y - 4y^2 (as well as of the form 2x^2 + 9xy + y^2).