A007522 -id:A007522 - cp's OEIS frontend

A258149 Triangle of the absolute difference of the two legs (catheti) of primitive Pythagorean triangles.

1, 0, 7, 7, 0, 17, 0, 1, 0, 31, 23, 0, 0, 0, 49, 0, 17, 0, 23, 0, 71, 47, 0, 7, 0, 41, 0, 97, 0, 41, 0, 7, 0, 0, 0, 127, 79, 0, 31, 0, 0, 0, 89, 0, 161, 0, 73, 0, 17, 0, 47, 0, 119, 0, 199, 119, 0, 0, 0, 1, 0, 73, 0, 0, 0, 241

Offset: 2

Wolfdieter Lang, Jun 10 2015

For primitive Pythagorean triangles characterized by certain (n,m) pairs and references see A225949.
Here a(n,m) = 0 for non-primitive Pythagorean triangles, and for primitive Pythagorean triangles a(n,m) = abs(n^2 - m^2 - 2*n*m) = abs((n-m)^2 - 2*m^2).
The number of non-vanishing entries in row n is A055034(n).
D(n,m):= n^2 - m^2 - 2*n*m >= 0 if 1 <= m <= floor(n/(sqrt(2)+1)), and D(n,m) < 0 if n/(sqrt(2)+1)+1 <= m <= n-1, for n >= 2.
The Pell equation (n-m)^2 - 2*m^2 = +/- N is important here to find the representations of +N or -N in the triangle D(n,m). For instance, odd primes N have to be of the +1 (mod 8) (A007519) or -1 (mod 8) (A007522) form, that is, from A001132. See the Nagell reference, Theorem 110, p. 208 with Theorem 111, pp. 210-211. E.g., N = +7 appears for m = 1, 3, 9, 19, 53, ... (A077442) for n = 4, 8, 22, 46, 128, ... (2*A006452).
  N = -7 appears for n = 3, 9, 19, 53, 111, ... (A077442) and m = 2, 4, 8, 22, 46, ... (2*A006452).
For the  signed version 2*n*m - (n^2 - m^2) see A278717. - Wolfdieter Lang, Nov 30 2016

			The triangle a(n,m) begins:
n\m   1  2  3  4  5  6  7   8   9  10  11 ...
2:    1
3:    0  7
4:    7  0 17
5:    0  1  0 31
6:   23  0  0  0 49
7:    0 17  0 23  0 71
8:   47  0  7  0 41  0 97
9:    0 41  0  7  0  0  0 127
10:  79  0 31  0  0  0 89   0 161
11:   0 73  0 17  0 47  0 119   0 199
12: 119  0  0  0  1  0 73   0   0   0 241
...
a(2,1) = |1^2 - 2*1^2| = 1 for the primitive Pythagorean triangle (pPt) [3,4,5] with |3-4| = 1.
a(3,2) = |1^2 - 2*2^2| = 7 for the pPt [5,12,13] with |5 - 12| = 7.
a(4,1) = |3^2 - 2*1^2| = 7 for the pPt [15, 8, 17] with |15 - 8| = 7.

See also A225949.
T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964, pp. 208, 210-211.

Cf. A249866, A222946, A225949, A222951, A258150, A278717 (signed).

a[n_, m_] /; n > m >= 1 && CoprimeQ[n, m] && (-1)^(n+m) == -1 := Abs[n^2 - m^2 - 2*n*m]; a[, ] = 0; Table[a[n, m], {n, 2, 12}, {m, 1, n-1}] // Flatten (* Jean-François Alcover, Jun 16 2015, after given formula *)

a(n,m) = abs(n^2 - m^2 -2*n*m) = abs((n-m)^2 - 2*m^2) if n > m >= 1, gcd(n,m) = 1, and n and m are integers of opposite parity (i.e., (-1)^(n+m) = -1); otherwise a(n,m) = 0.

A269520 Primes 8k + 7 preceding the maximal gaps in A269519.

Original entry on oeis.org

7, 47, 271, 311, 503, 6367, 37223, 42487, 66463, 183527, 259271, 307919, 471007, 1070567, 1801223, 5903687, 6885743, 16936247, 22413319, 38820263, 63977327, 84164447, 147452759, 150334567, 239422639, 300412031, 387154951, 473153959, 539526191, 760400783, 788128039

Offset: 1

Alexei Kourbatov, Feb 28 2016

nonn

Subsequence of A007522.

A269519 lists the corresponding record gap sizes. See more comments there.

			The first two primes of the form 8k + 7 are 7 and 23, so a(1)=7. The next primes of this form are 31, 47; the gaps 31-23 and 47-31 are not records so nothing is added to the sequence. The next prime of this form is 71 and the gap 71-47=24 is a new record, so a(2)=47.

Alexei Kourbatov, Table of n, a(n) for n = 1..39
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

Cf. A007522, A269519, A269521.

re=0; s=7; forprime(p=23, 1e8, if(p%8!=7, next); g=p-s; if(g>re, re=g; print1(s", ")); s=p)

A269521 Primes 8k + 7 at the end of the maximal gaps in A269519.

Original entry on oeis.org

23, 71, 311, 359, 599, 6551, 37423, 42703, 66751, 183823, 259583, 308263, 471391, 1071023, 1801727, 5904247, 6886367, 16936991, 22414079, 38821039, 63978127, 84165271, 147453599, 150335431, 239423519, 300412927, 387155903, 473154943, 539527199, 760401839, 788129191

Offset: 1

Alexei Kourbatov, Feb 28 2016

nonn

Subsequence of A007522.

A269519 lists the corresponding record gap sizes. See more comments there.

			The first two primes of the form 8k + 7 are 7 and 23, so a(1)=23. The next primes of this form are 31, 47; the gaps 31-23 and 47-31 are not records so nothing is added to the sequence. The next prime of this form is 71 and the gap 71-47=24 is a new record, so a(2)=71.

Alexei Kourbatov, Table of n, a(n) for n = 1..39
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

Cf. A007522, A269519, A269520.

re=0; s=7; forprime(p=23, 1e8, if(p%8!=7, next); g=p-s; if(g>re, re=g; print1(p", ")); s=p)

A337145 a(n) is the determinant of the 2 X 2 matrix whose entries (when read by rows) are the n-th primes congruent to 1, 3, 5, 7 mod 8 respectively.

Original entry on oeis.org

104, 800, 1712, 2592, 3760, 4840, 5728, 12848, 15664, 18424, 20888, 23520, 28232, 28560, 25320, 30280, 37248, 50520, 43680, 33664, 61560, 73920, 70544, 57696, 38696, 27408, 79280, 63392, 107328, 109536, 162608, 172296, 187352, 197040, 248064, 228320, 215912, 229152, 255480, 231304, 286408, 256320

Offset: 1

J. M. Bergot and Robert Israel, Jan 27 2021

The first negative term is a(20750) = -58207896.

All terms are divisible by 8.

			The first primes == 1, 3, 5, 7 (mod 8) are 17, 3, 5, 7 respectively, so a(1) = 17*7 - 3*5 = 104.
The second primes == 1, 3, 5, 7 (mod 8) are 41, 11, 13, 23 respectively, so a(2) = 41*23 - 11*13 = 800.
The third primes == 1, 3, 5, 7 (mod 8) are 73, 19, 29, 31 respectively, so a(3) = 73*31 - 19*29 = 1712.

Robert Israel, Table of n, a(n) for n = 1..30000

Cf. A335581, A335592, A337146, A337147.

Cf. A007519, A007520, A007521, A007522.

R:= NULL:
L:= [-7, -5, -3, -1]:
found:= false:
for k from 1 to 100 do
  for i from 1 to 4 do
   for x from L[i]+8 by 8 do until isprime(x);
   L[i]:= x;
  od;
  v:= L[1]*L[4]-L[2]*L[3];
  R:= R,v;
od:
R;

A101789 Safe primes of the form 8k-1: primes of the form 8k-1 such that 4*k-1 is also a prime.

Original entry on oeis.org

7, 23, 47, 167, 263, 359, 383, 479, 503, 719, 839, 863, 887, 983, 1319, 1367, 1439, 1487, 1823, 2039, 2063, 2207, 2447, 2879, 2903, 2999, 3023, 3119, 3167, 3623, 3863, 4007, 4079, 4127, 4679, 4703, 4799, 4919, 5087, 5399, 5639, 5807, 5879, 5927, 6047

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 16 2004

			4*1-1 = 3 and 8*1-1 = 7 are primes, so the first term is 7.

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Intersection of A005385 and A007522.

Cf. A002515.

Select[Prime[Range[800]],Mod[#,8]==7&&PrimeQ[(#-1)/2]&] (* Harvey P. Dale, Jan 31 2012 *)

is(k) = (k % 8 == 7) && isprime(k) && isprime(k\2); \\ Amiram Eldar, May 23 2024

A121068 Numbers k such that 8*k^2 + 7 is prime.

Original entry on oeis.org

0, 3, 27, 30, 33, 48, 57, 60, 72, 75, 78, 108, 117, 123, 135, 150, 162, 192, 198, 207, 228, 243, 270, 300, 303, 312, 342, 345, 390, 408, 417, 423, 435, 480, 498, 507, 510, 513, 543, 552, 555, 573, 618, 633, 642, 645, 657, 675, 705, 723, 732, 738, 747, 750, 780

Offset: 1

Parthasarathy Nambi, Aug 10 2006

All terms are multiples of 3. - Zak Seidov, Aug 11 2006

A201704 is primes of form 8*k^2+7. James C. McMahon, Oct 12 2024

			If k=135 then 8*k^2 + 7 = 145807 (prime).

Cf. A007522, A201704.

[ n: n in [0..1500] | IsPrime(8*n^2 + 7) ]; // Vincenzo Librandi, Jan 31 2011

a:=proc(n) if isprime(8*n^2+7)=true then n else fi end: seq(a(n),n=0..1000); # Emeric Deutsch, Aug 11 2006

Select[Range[0,780],PrimeQ[8#^2+7]&] (* James C. McMahon, Oct 12 2024 *)

is(n)=isprime(8*n^2+7) \\ Charles R Greathouse IV, Jun 13 2017

More terms from Emeric Deutsch and Joshua Zucker, Aug 11 2006

A127597 Least number k such that k 4^n + (4^n-1)/3 is prime.

Original entry on oeis.org

2, 1, 0, 2, 3, 2, 4, 4, 3, 10, 3, 3, 2, 7, 2, 25, 6, 17, 4, 13, 3, 20, 36, 20, 11, 27, 66, 23, 39, 24, 19, 13, 3, 10, 6, 122, 71, 58, 24, 13, 3, 2, 41, 10, 6, 32, 58, 17, 4, 79, 26, 55, 36, 48, 31, 28, 9, 2, 76, 24, 32, 28, 63, 20, 37, 9, 2, 7, 39, 10, 91, 47

Offset: 0

Artur Jasinski, Jan 19 2007

nonn

Michael S. Branicky, Table of n, a(n) for n = 0..2391

Cf. A035050, A007522, A127575, A127576, A127577, A127578, A127580, A127581, A087522, A127586, A127587, A127589, A127590, A127591, A127592, A127593, A127594, A127598.

a = {}; Do[k = 0; While[ !PrimeQ[k 4^n + (4^n - 1)/3], k++ ]; AppendTo[a, k], {n, 0, 50}]; a (*Artur Jasinski*)
lnk[n_]:=Module[{k=0,n4=4^n},While[!PrimeQ[k*n4+(n4-1)/3],k++];k]; Array[ lnk,60,0] (* Harvey P. Dale, May 28 2018 *)

from sympy import isprime
def a(n):
    k, fourn = 0, 4**n
    while not isprime(k*fourn + (fourn-1)//3): k += 1
    return k
print([a(n) for n in range(72)]) # Michael S. Branicky, May 18 2022

Offset corrected and a(51) and beyond from Michael S. Branicky, May 18 2022

A269519 Record (maximal) gaps between primes of the form 8k + 7.

Original entry on oeis.org

16, 24, 40, 48, 96, 184, 200, 216, 288, 296, 312, 344, 384, 456, 504, 560, 624, 744, 760, 776, 800, 824, 840, 864, 880, 896, 952, 984, 1008, 1056, 1152, 1208, 1312, 1384, 1448, 1464, 1472, 1720, 1872

Offset: 1

Alexei Kourbatov, Feb 28 2016

nonn

Dirichlet's theorem on arithmetic progressions suggests that average gaps between primes of the form 8k + 7 below x are about phi(8)*log(x). This sequence shows that the record gap ending at p grows almost as fast as phi(8)*log^2(p). Here phi(n) is A000010, Euler's totient function; phi(8)=4.
Conjecture: a(n) < phi(8)*log^2(A269521(n)) almost always.
A269520 lists the primes preceding the maximal gaps.
A269521 lists the corresponding primes at the end of the maximal gaps.

			The first two primes of the form 8k + 7 are 7 and 23, so a(1)=23-7=16. The next primes of this form are 31, 47; the gaps 31-23 and 47-31 are not records so nothing is added to the sequence. The next prime of this form is 71 and the gap 71-47=24 is a new record, so a(2)=24.

Alexei Kourbatov, On the distribution of maximal gaps between primes in residue classes, arXiv:1610.03340 [math.NT], 2016.
Alexei Kourbatov, On the nth record gap between primes in an arithmetic progression, arXiv:1709.05508 [math.NT], 2017; Int. Math. Forum, 13 (2018), 65-78.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

Cf. A007522, A269520, A269521.

re=0; s=7; forprime(p=23, 1e8, if(p%8!=7, next); g=p-s; if(g>re, re=g; print1(g", ")); s=p)

A367229 Fermat pseudoprimes to base 2 that are products of two Mersenne numbers (not necessarily distinct) that are larger than 1.

Original entry on oeis.org

1905, 15841, 129921, 8322945, 66977281, 4395899025409, 4398012825601, 140735340806145, 36892925197465616385, 2342736497361113055105, 4951750712408555360305545217, 39304596247310823728047193985, 2535301191011725837253847547905, 1298074214624262174166747352924161

Offset: 1

Amiram Eldar, Nov 11 2023

nonn

Without the restriction to Mersenne numbers that are larger than 1 all the composite Mersenne numbers (A065341) will be terms.
Szymiczek (1964) proved that if p is a prime == 7 (mod 8) (A007522) and t = 2^phi((p-1)/2), then M(p)*M(t) is a Fermat pseudoprime to base 2, where phi is the Euler totient function (A000010) and M(n) = 2^n-1 = A000225(n) is the n-th Mersenne number. The smallest pseudoprime that is generated by this rule, for p = 7 and t = 2^phi((7-1)/2) = 4, is M(7) * M(4) = 1905. The next two, corresponding to p = 23 and 31, have 316 and 87 digits, respectively.
Rotkiewicz and Makowski (1966) proved that if p is a prime or a Fermat pseudoprime to base 2 such that o(p), the multiplicative order of 2 modulo p, is odd (A014663 for primes, A367230 for pseudoprimes), then for each positive k <= p/o(o(p)), if t = 2^(k*o(o(p))) then M(p)*M(t) is a Fermat pseudoprime to base 2. For example, for p = 7, p/o(o(7)) = 7/2, so for k = 1, 2 and 3 the resulting pseudoprimes are 1905, 8322945 and 2342736497361113055105, respectively.

			a(1) = 1905 = (2^4-1) * (2^7-1).
a(2) = 15841 = (2^5-1) * (2^9-1).

Amiram Eldar, Table of n, a(n) for n = 1..242
Andrzej Rotkiewicz and Andrzej Makowski, On Pseudoprime Numbers of the Form M_p M_t, Elemente der Mathematik, Vol. 21 (1966), pp. 133-134.
Kazimierz Szymiczek, On prime numbers p,q and r such that pq, pr, and qr are pseudoprimes, Colloquium Mathematicum, Vol. 13 (1964), pp. 259-263; alternative link.

Cf. A000010, A000225, A001567, A007522, A014663, A065341, A367230.

With[{max = 110}, m = 2^Range[2, max] - 1; Sort@ Select[Times @@@ Subsets[m, {2}], # < m[[-1]] && PowerMod[2, # - 1, #] == 1 &]]

A022761 n-th 8k+1 prime plus n-th 8k+7 prime.

Original entry on oeis.org

24, 64, 104, 136, 168, 192, 240, 320, 384, 408, 448, 480, 536, 576, 616, 672, 720, 792, 816, 840, 952, 1008, 1040, 1072, 1088, 1120, 1240, 1280, 1392, 1416, 1528, 1584, 1624, 1680, 1760, 1792, 1840, 1896, 1944, 1968, 2064, 2112, 2144, 2224

Offset: 1

Clark Kimberling

nonn

			The first four primes of the form 8k - 1 are 7, 23, 31, 47. The first four primes of the form 8k + 1 are 17, 41, 73, 89.
Thus a(1) = 7 + 17  = 24.
a(2) = 23 + 41 = 64.
a(3) = 31 + 73 = 104.
a(4) = 47 + 89 = 136.

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

Cf. A007522, A007519.

thresh = 100; A007522 = Select[8Range[thresh] - 1, PrimeQ]; A007519 = Select[8Range[thresh] + 1, PrimeQ]; preExh = Min[Length[A007522], Length[A007519]]; Take[A007522, preExh] + Take[A007519, preExh]
Module[{nn=300,p1,p7,len},p1=Select[Prime[Range[nn]],IntegerQ[(#-1)/8]&];p7=Select[Prime[Range[nn]],IntegerQ[(#-7)/8]&];len=Min[ Length[ p1],Length[ p7]];Total/@Thread[{Take[p1,len],Take[p7,len]}]] (* Harvey P. Dale, May 26 2020 *)

cp's OEIS Frontend

A258149 Triangle of the absolute difference of the two legs (catheti) of primitive Pythagorean triangles.

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A269520 Primes 8k + 7 preceding the maximal gaps in A269519.

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A269521 Primes 8k + 7 at the end of the maximal gaps in A269519.

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A337145 a(n) is the determinant of the 2 X 2 matrix whose entries (when read by rows) are the n-th primes congruent to 1, 3, 5, 7 mod 8 respectively.

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A101789 Safe primes of the form 8*k-1: primes of the form 8*k-1 such that 4*k-1 is also a prime.

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A121068 Numbers k such that 8*k^2 + 7 is prime.

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A127597 Least number k such that k 4^n + (4^n-1)/3 is prime.

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A269519 Record (maximal) gaps between primes of the form 8k + 7.

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A101789 Safe primes of the form 8k-1: primes of the form 8k-1 such that 4*k-1 is also a prime.