A107006 -id:A107006 - cp's OEIS frontend

A139484 Indices of Mersenne primes among primes of the form 24k + 7 (A107006).

1, 2, 5, 129, 1536, 5430, 13138099

Offset: 2

Artur Jasinski, Apr 23 2008

All Mersenne primes with the exception of the first one (3) are of the form 24*k + 7.

Sequence lists indices m where A139483(m) is a Mersenne prime.

			5 is in this sequence, because A107006(5) is a Mersenne prime.

Cf. A107006, A124477, A139479, A139483.

w = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607}; Do[w[[n]] = 2^w[[n]] - 1, {n, 1, Length[w]}]; c = 0; a = {}; Do[k = 24 n + 7; If[PrimeQ[k], c = c + 1; If[MemberQ[w, k], AppendTo[a, c]]], {n, 1, 10000000}]; a
With[{mps=2^#-1&/@MersennePrimeExponent[Range[20]]},Position[ Select[ 24*Range[0,10^8]+7,PrimeQ],?(MemberQ[mps,#]&)]]//Flatten (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Aug 20 2019 *)

Edited name and added example by Dmitry Kamenetsky, Jan 02 2011

a(8) from Charles R Greathouse IV, Mar 22 2011

A135983 a(n)=2^A107006(n)-1.

Original entry on oeis.org

127, 2147483647, 604462909807314587353087, 10141204801825835211973625643007, 170141183460469231731687303715884105727, 2854495385411919762116571938898990272765493247

Offset: 1

Artur Jasinski, Dec 09 2007

nonn

A107006(n) are successive primes of the form 24n+7.

Cf. A107006, A124477, A135657, A135982.

p = Select[24*Range[0, 20] + 7, PrimeQ]; 2^p - 1

A140633 Primes of the form 7x^2+4xy+52y^2.

Original entry on oeis.org

7, 103, 127, 223, 367, 463, 487, 607, 727, 823, 967, 1063, 1087, 1303, 1327, 1423, 1447, 1543, 1567, 1663, 1783, 2143, 2287, 2383, 2503, 2647, 2767, 2887, 3343, 3463, 3583, 3607, 3727, 3823, 3847, 3943, 3967, 4327, 4423, 4447, 4567, 4663

Offset: 1

T. D. Noe, May 19 2008

Discriminant=-1440. Also primes of the forms 7x^2+6xy+87y^2 and 7x^2+2xy+103y^2.
Voight proves that there are exactly 69 equivalence classes of positive definite binary quadratic forms that represent almost the same primes. 48 of those quadratic forms are of the idoneal type discussed in A139827. The remaining 21 begin at A140613 and end here. The cross-references section lists the quadratic forms in the same order as tables 1-6 in Voight's paper. Note that A107169 and A139831 are in the same equivalence class.
In base 12, the sequence is 7, 87, X7, 167, 267, 327, 347, 427, 507, 587, 687, 747, 767, 907, 927, 9X7, X07, X87, XX7, E67, 1047, 12X7, 13X7, 1467, 1547, 1647, 1727, 1807, 1E27, 2007, 20X7, 2107, 21X7, 2267, 2287, 2347, 2367, 2607, 2687, 26X7, 2787, 2847, where X is for 10 and E is for 11. Moreover, the discriminant is X00 and that all elements are {7, 87, X7, 167, 187, 247} mod 260. - Walter Kehowski, May 31 2008

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
John Voight, Quadratic forms that represent almost the same primes, Math. Comp., Vol. 76 (2007), pp. 1589-1617.

Cf. A033205, A007519, A068228, A107135, A107144, A107152, A107151.
Cf. A107181, A139502, A139856, A139854, A139874, A139877, A139897.
Cf. A140613, A140614. A140615, A139923, A140616-A140619, A139988.
Cf. A139993, A140008, A140010, A140620-A140632, A033212, A102273.
Cf. A107007, A107003, A139830, A107169, A139831, A141373, A139847.
Cf. A139850, A139855, A139860, A139879, A139880, A139924, A139990.
Cf. A139998, A139992, A139996, A140003, A140013, A107006, A139858.
Cf. A107008, A140633, A139991, A139857, A139859, A139878, A007645, A002313.

Union[QuadPrimes2[7, 4, 52, 10000], QuadPrimes2[7, -4, 52, 10000]] (* see A106856 *)

A088190 Largest quadratic residue modulo prime(n).

Original entry on oeis.org

1, 1, 4, 4, 9, 12, 16, 17, 18, 28, 28, 36, 40, 41, 42, 52, 57, 60, 65, 64, 72, 76, 81, 88, 96, 100, 100, 105, 108, 112, 124, 129, 136, 137, 148, 148, 156, 161, 162, 172, 177, 180, 184, 192, 196, 196, 209, 220, 225, 228, 232, 232, 240, 249, 256, 258, 268, 268, 276

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu), Sep 22 2003

nonn

Denote a(n) by LQR(p_n). Observations (tested up to 20000 primes): - the sequence of largest QR modulo the primes (LQR(p_n) is 'almost' monotonic, - p_n-LQR(p_n) is either 1 or a prime value (see A088192) - if LQR(p_n)<=LQR(p_{n-1}) then p_n==7 mod 8 (when n>2) (see A088194) - if LQR(p_n)<=LQR(p_{n-1}) then p_n-LQR(p_n) is an odd prime, but never 5 (see A088195) For a similar set of sequences, related to quadratic non-residues, see A088196-A088201.
From Robert Israel, Oct 31 2024: (Start)
a(n) = prime(n)-1 if and only if n is 1 or in A080147.
a(n) = prime(n)-2 if and only if prime(n) is in A007520.
a(n) = prime(n)-3 if and only if prime(n) is in A107006. (End)

Robert Israel, Table of n, a(n) for n = 1..10000
F. Adorjan, The sequence of largest quadratic residues modulo the primes.

Cf. A007520, A080147, A088191, A088192, A088193, A088194, A088195, A088196, A088197, A088198, A088199, A088200, A088201, A107006.

lqr:= proc(p) local k;
  for k from p-1 by -1 do if numtheory:-quadres(k,p) = 1 then return k fi od:
end proc:
seq(lqr(ithprime(i)),i=1..100); # Robert Israel, Oct 31 2024

a[n_] := With[{p = Prime[n]}, SelectFirst[Range[p - 1, 1, -1], JacobiSymbol[#, p] == 1&]]; Array[a, 100] (* Jean-François Alcover, Feb 16 2018 *)

qrp(fr,to)= {/* Sequence of the largest QR modulo the primes */ local(m,p,v=[]); for(i=fr,to,m=1; p=prime(i); j=2; while((j<=(p-1)/2)&&(m

a(n) = max(r, r==j^2 mod p(n)|j=1, 2, ...(p(n)-1)/2)

A033199 Primes of form x^2+6*y^2.

Original entry on oeis.org

7, 31, 73, 79, 97, 103, 127, 151, 193, 199, 223, 241, 271, 313, 337, 367, 409, 433, 439, 457, 463, 487, 577, 601, 607, 631, 673, 727, 751, 769, 823, 919, 937, 967, 991, 1009, 1033, 1039, 1063, 1087, 1129, 1153, 1201, 1231, 1249, 1279, 1297, 1303, 1321, 1327, 1399, 1423, 1447, 1471, 1489, 1543

Offset: 1

N. J. A. Sloane

Appears to also be the primes p such that p mod 6 = 1 and Fibonacci(p) mod 6 = 1. - Gary Detlefs, May 26 2014

N. J. A. Sloane and Vincenzo Librandi, Table of n, a(n) for n = 1..10000 (The first 2000 terms were found by Vincenzo Librandi)
David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989, p. 36.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139643, primes in A002481. Cf. A107006, A107008.

[p: p in PrimesUpTo(1600) | NormEquation(6,p) eq true]; // Bruno Berselli, Jul 03 2016

f[x_, y_] := x^2 + 6*y^2; lst = {}; Do[p = f[x, y]; If[ PrimeQ[ p], AppendTo[ lst, p]], {y, 20}, {x, 50}]; Take[ Union[ lst], 50] (* Vladimir Joseph Stephan Orlovsky, Aug 04 2009 *)

select(n->n%24==1||n%24==7, primes(100)) \\ Charles R Greathouse IV, Nov 09 2012

Same as primes congruent to 1 or 7 mod 24. See e.g. Cox, p. 36.

a(n) ~ 4n log n. - Charles R Greathouse IV, Nov 09 2012

Removed defective Mma program; extended the b-file using Charles R Greathouse's PARI program. - N. J. A. Sloane, Jun 06 2014

A135657 Nonprimes of the form 24n+7.

Original entry on oeis.org

55, 175, 247, 295, 319, 343, 391, 415, 511, 535, 559, 583, 655, 679, 703, 775, 799, 847, 871, 895, 943, 1015, 1111, 1135, 1159, 1183, 1207, 1255, 1351, 1375, 1495, 1519, 1591, 1615, 1639, 1687, 1711, 1735, 1807, 1855, 1903, 1927, 1975, 2023, 2047, 2071

Offset: 1

Artur Jasinski, Nov 25 2007

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

Cf. A107006, A124477.

[a: n in [0..100] | not IsPrime(a) where a is 24*n+7]; // Vincenzo Librandi, Mar 22 2014

a = {}; Do[If[PrimeQ[24n + 7],[null], AppendTo[a, 24n + 7]], {n, 0, 1000}]; a
Select[24 Range[100] + 7, ! PrimeQ@# &] (* Vincenzo Librandi, Mar 22 2014 *)

A135659 a(n) = 24*n + 7.

Original entry on oeis.org

7, 31, 55, 79, 103, 127, 151, 175, 199, 223, 247, 271, 295, 319, 343, 367, 391, 415, 439, 463, 487, 511, 535, 559, 583, 607, 631, 655, 679, 703, 727, 751, 775, 799, 823, 847, 871, 895, 919, 943, 967, 991, 1015, 1039, 1063, 1087, 1111, 1135, 1159, 1183, 1207

Offset: 0

Artur Jasinski, Nov 25 2007

Conjecture: All Mersenne Primes (A000668) > 3 are in this sequence.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A107006, A124477, A135657, A135982.

Table[24n + 7, {n, 0, 100}]
LinearRecurrence[{2,-1},{7,31},60] (* Harvey P. Dale, Jul 14 2013 *)

From Colin Barker, Apr 02 2012: (Start)
a(n) = 2*a(n-1) - a(n-2).
G.f.: (7+17*x)/(1-x)^2. (End)
E.g.f.: (7 + 24*x)*exp(x). - G. C. Greubel, Oct 25 2016

Offset changed to 0 by Omar E. Pol, Oct 25 2016

A135982 a(n) = 2^(24n+7)-1.

Original entry on oeis.org

127, 2147483647, 36028797018963967, 604462909807314587353087, 10141204801825835211973625643007, 170141183460469231731687303715884105727

Offset: 0

Artur Jasinski, Dec 09 2007

Index entries for linear recurrences with constant coefficients, signature (16777217,-16777216).

Cf. A107006, A124477, A135657.

Mathematica
```
Table[2^(24n + 7) - 1, {n, 0, 20}]
```

a(n)=2^(24*n+7)-1 \\ Charles R Greathouse IV, Oct 16 2015

a(n) = 2^A135659(n) - 1.

G.f.: ( 127+16777088*x ) / ( (16777216*x-1)*(x-1) ). - R. J. Mathar, Apr 02 2012

A139483 Numbers n such that 24n+7 is prime.

Original entry on oeis.org

0, 1, 3, 4, 5, 6, 8, 9, 11, 15, 18, 19, 20, 25, 26, 30, 31, 34, 38, 40, 41, 43, 44, 45, 51, 53, 54, 55, 58, 59, 60, 61, 64, 65, 69, 73, 74, 76, 78, 81, 83, 89, 93, 95, 96, 99, 104, 106, 110, 111, 113, 115, 116, 120, 128, 136, 138, 139, 141, 144, 146, 148, 149, 150, 151

Offset: 1

Artur Jasinski, Apr 23 2008

Cf. A107006, A124477, A139479.

a = {}; Do[If[PrimeQ[24 n + 7], AppendTo[a, n]], {n, 0, 200}]; a
Select[Range[0,200],PrimeQ[24#+7]&] (* Harvey P. Dale, Sep 02 2015 *)

is(n)=isprime(24*n+7) \\ Charles R Greathouse IV, Jun 06 2017

A139640 Numbers of the form of 4x^2 - 4xy + 7y^2 (=24k+7) but not of the form 4x^2 + 4xy + 7y^2.

Original entry on oeis.org

151, 487, 727, 751, 1039, 1063, 1399, 1471, 1759, 1783, 2287, 2647, 2671, 2767, 3343, 3463, 3919, 4111, 4423, 4663, 4903, 4999, 5167, 5407, 5791, 6607, 6703, 6823, 7159, 7351, 7639, 7879, 8191, 8263, 8311, 8839, 9127, 9151, 9439, 9967, 10111, 10711

Offset: 1

Artur Jasinski, Apr 28 2008

nonn

Numbers which occur in A107006 but not in A107005.

Cf. A107005, A107006.

More terms from T. D. Noe, May 01 2008

cp's OEIS Frontend

A139484 Indices of Mersenne primes among primes of the form 24k + 7 (A107006).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A135983 a(n)=2^A107006(n)-1.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A140633 Primes of the form 7x^2+4xy+52y^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A088190 Largest quadratic residue modulo prime(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A033199 Primes of form x^2+6*y^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A135657 Nonprimes of the form 24n+7.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

A135659 a(n) = 24*n + 7.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A135982 a(n) = 2^(24n+7)-1.

Views

Author

Keywords

Links

Crossrefs

Programs