A082073 -id:A082073 - cp's OEIS frontend

A082075 First differences of primes of the form 4*k+3 (A002145).

4, 4, 8, 4, 8, 12, 4, 12, 8, 4, 8, 4, 20, 4, 20, 4, 8, 12, 12, 4, 12, 12, 8, 12, 12, 4, 12, 12, 12, 8, 12, 24, 4, 20, 16, 12, 8, 12, 4, 36, 12, 8, 4, 20, 4, 12, 8, 4, 8, 4, 20, 24, 16, 8, 16, 12, 8, 12, 12, 12, 4, 12, 24, 8, 28, 8, 12, 4, 8, 36, 24, 12, 4, 12, 20, 4, 20, 4, 20, 4, 8, 28, 20, 4

Offset: 1

Labos Elemer, Apr 07 2003

nonn

			The first and second primes of the form 4*k+3 are 3 and 7, so a(1) = 7-3 = 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002144, A002145, A082073, A082074, A082075, A082076.

k=0; m=4; r=3; Do[s=Mod[Prime[n], m]; If[Equal[s, r], rp=ep; k=k+1; ep=Prime[n]; Print[ep-rp]; ], {n, 1, 1000}]

a(n) = A002145(n+1) - A002145(n).

A082074 One quarter of first differences of primes of the form 4*k+1 (A002144).

Original entry on oeis.org

2, 1, 3, 2, 1, 3, 2, 3, 4, 2, 1, 2, 1, 6, 3, 2, 4, 2, 3, 1, 8, 1, 2, 4, 3, 2, 1, 3, 5, 1, 5, 3, 1, 5, 4, 2, 1, 2, 3, 3, 4, 2, 1, 12, 3, 5, 4, 3, 2, 4, 2, 3, 1, 6, 3, 2, 3, 1, 6, 2, 6, 6, 1, 2, 1, 6, 3, 3, 2, 6, 1, 5, 1, 12, 2, 1, 3, 6, 5, 3, 1, 2, 3, 4, 3, 2, 6, 1, 3, 2, 3, 6, 7, 3, 2, 3, 1, 3, 2, 3, 7, 3, 2, 1, 5

Offset: 1

Labos Elemer, Apr 07 2003

nonn

			The first and second primes of the form 4*k+1 are 5 and 13, so a(1) = (13-5)/4 = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002144, A002145, A082073, A082075, A082076.

k=0; m=4; r=1; Do[s=Mod[Prime[n], m]; If[Equal[s, r], rp=ep; k=k+1; ep=Prime[n]; Print[(ep-rp)/4]; ], {n, 1, 1000}]
Differences[Select[4*Range[1000]+1,PrimeQ]]/4 (* Harvey P. Dale, Dec 04 2011 *)

a(n) = (A002144(n+1) - A002144(n))/4.

A082076 First differences of primes of the form 4*k+3 (A002145), divided by 4.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 3, 2, 1, 2, 1, 5, 1, 5, 1, 2, 3, 3, 1, 3, 3, 2, 3, 3, 1, 3, 3, 3, 2, 3, 6, 1, 5, 4, 3, 2, 3, 1, 9, 3, 2, 1, 5, 1, 3, 2, 1, 2, 1, 5, 6, 4, 2, 4, 3, 2, 3, 3, 3, 1, 3, 6, 2, 7, 2, 3, 1, 2, 9, 6, 3, 1, 3, 5, 1, 5, 1, 5, 1, 2, 7, 5, 1, 3, 2, 7, 3, 2, 3, 3, 6, 1, 3, 5, 7, 3, 2, 4, 9, 2, 7, 5, 1, 2

Offset: 1

Labos Elemer, Apr 07 2003

nonn

			The first and second primes of the form 4*k+3 are 3 and 7, so a(1) = (7-3)/4 = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002144, A002145, A082073, A082074, A082075.

k=0; m=4; r=3; Do[s=Mod[Prime[n], m]; If[Equal[s, r], rp=ep; k=k+1; ep=Prime[n]; Print[(ep-rp)/4]; ], {n, 1, 1000}]
Differences[Select[Prime[Range[400]],IntegerQ[(#-3)/4]&]]/4 (* Harvey P. Dale, Apr 29 2022 *)

a(n) = (A002145(n+1) - A002145(n))/4.

A082098 a(n) is the smallest prime p of the form 4k+3 such that nextprime(p) - p = 4n.

Original entry on oeis.org

7, 359, 199, 1831, 887, 2179, 2971, 5591, 9551, 33247, 15683, 106543, 25471, 153191, 43331, 288583, 372539, 360091, 873787, 542603, 637939, 544279, 1291691, 860143, 1313467, 1388483, 2238823, 2637799, 6695747, 1895359, 6752623, 3851459

Offset: 1

Labos Elemer, Apr 14 2003

nonn

			a(9) = 9551 since nextprime(9551) - 9551 = 9587 - 9551 = 36 = 4 * 9.

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A002144, A002145, A082073, A082074, A082075, A082076, A082099.

{m=4, r=3}; f[x_] := (Prime[x+1]-Prime[x])/m t=Table[0, {100}]; Do[s=f[n]; s1=Mod[Prime[n+1], m]; If[IntegerQ[s]&&Equal[s1, r]&&s<101&&t[[s]]==0, t[[s]]=Prime[n]], {n, 1, 1000000}]; t

list(len) = {my(v = vector(len), c = 0, p1 = 2, r1 = p1 % 4, r2, i); forprime(p2 = 3, , r2 = p2 % 4; if(r1 == 3 && r2 == 3, i = (p2 - p1) / 4; if(i <= len && v[i] == 0, c++; v[i] = p1; if(c == len, break))); p1 = p2; r1 = r2); v;} \\ Amiram Eldar, Feb 11 2025

A082099 a(n) is the smallest prime p of the form 4k+1 such that nextprime(p) - p = 4n.

Original entry on oeis.org

13, 89, 509, 1933, 1637, 1669, 5953, 27701, 12853, 19333, 36389, 28229, 19609, 82073, 102701, 89689, 134513, 31397, 212701, 815729, 461717, 1155733, 927869, 360653, 396733, 3422813, 3279841, 370261, 5845193, 6085441, 6808273, 9810653

Offset: 1

Labos Elemer, Apr 14 2003

nonn

			a(9) = 12853 since nextprime(12853) - 12853 = 12889 - 12853 = 36 = 4 * 9.

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A002144, A002145, A082073, A082074, A082075, A082076, A082098.

{m=4, r=1}; f[x_] := (Prime[x+1]-Prime[x])/m t=Table[0, {100}]; Do[s=f[n]; s1=Mod[Prime[n+1], m]; If[IntegerQ[s]&&Equal[s1, r]&&s<101&&t[[s]]==0, t[[s]]=Prime[n]], {n, 1, 1000000}]; t

list(len) = {my(v = vector(len), c = 0, p1 = 2, r1 = p1 % 4, r2, i); forprime(p2 = 3, , r2 = p2 % 4; if(r1 == 1 && r2 == 1, i = (p2 - p1) / 4; if(i <= len && v[i] == 0, c++; v[i] = p1; if(c == len, break))); p1 = p2; r1 = r2); v;} \\ Amiram Eldar, Feb 11 2025

A217503 Squared distance between consecutive primes of the form 4k+1 (see below).

Original entry on oeis.org

1, 2, 2, 2, 2, 10, 8, 10, 8, 4, 2, 10, 4, 20, 18, 10, 2, 20, 58, 8, 40, 2, 40, 20, 10, 90, 2, 20, 10, 116, 2, 8, 20, 10, 2, 10, 20, 26, 4, 146, 8, 34, 10, 40, 34, 40, 2, 20, 2, 160, 50, 10, 180, 2, 180, 90, 58, 40, 130, 16, 116, 194, 50, 136, 74, 34, 52, 40

Offset: 1

Thomas Ordowski, Oct 05 2012

nonn

Every prime p of the form 4k+1 has a unique solution p = x^2 + y^2. This sequence gives the squared distance between points (x,y) for consecutive primes of this form.
The squares mutual distance consecutive points with coordinates x(n) = A002331(n) and y(n) = A002330(n), where x(n)^2 + y(n)^2 = A002313(n) is prime.
Theorem: a(n) =/= A082073(n-1) for all n > 1. Generally, it can be shown that there is no pair of primes p = a^2 + b^2 and q = x^2 + y^2 such that (a - x)^2 + (b - y)^2 = |p - q| > 0.

			5 = 1^2 + 2^2 and 13 = 2^2 + 3^2. The squared distance between the points (1,2) and (2,3) is 2, the second term of this sequence.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A002313, A002330, A002331.

nn = 200; p = Select[Prime[Range[nn]], Mod[#, 4] == 1 &]; q = {1, 1}; Table[pp = PowersRepresentations[p[[i]], 2, 2][[1]]; d = pp - q; q = pp; d[[1]]^2 + d[[2]]^2, {i, Length[p] - 1}] (* T. D. Noe, Oct 19 2012 *)

A141614 First differences of A008846.

Original entry on oeis.org

8, 4, 8, 4, 8, 4, 12, 8, 4, 8, 12, 4, 8, 4, 8, 4, 12, 12, 8, 4, 8, 12, 4, 8, 4, 8, 4, 8, 16, 8, 4, 8, 16, 8, 4, 8, 4, 8, 4, 12, 8, 4, 8, 12, 12, 4, 12, 8, 4, 12, 8, 4, 8, 12, 4, 8, 12, 4, 8, 4, 20, 4, 8, 12, 4, 12, 12, 8, 4, 12, 8, 4, 8, 16, 8, 12, 4, 8, 4, 12, 12, 8, 12, 4, 8, 4, 8, 4, 8, 16, 8, 12, 12, 4, 8, 4, 12, 8, 4, 12

Offset: 1

Paul Curtz, Aug 23 2008

Differences between Hypotenuses of primitive Pythagorean triangles.

Cf. A082073, A082075.

a(n)=A008846(n+1)-A008846(n).

Offset set to 1 and extended by R. J. Mathar, Sep 07 2009

A217674 Primes p = a^2 + b^2 such that (x - a)^2 + (y - b)^2 = 2 and q - p = 4, where q = x^2 + y^2 is prime, assuming that a > b > 0 and x > y > 0.

Original entry on oeis.org

13, 313, 613, 3613, 4513, 21013, 52813, 86113, 99013, 148513, 165313, 241513, 255613, 332113, 787513, 800113, 904513, 1073113, 1720513, 2279113, 2679613, 2940313, 3471613, 4307113, 4605613, 4789513, 5168113, 6072613, 6498013, 6716113, 7469113, 8925313

Offset: 1

Thomas Ordowski, Oct 10 2012

nonn

The primes p and q have the smallest difference and the shortest distance on the 2-dimensional plane. Primes p = a^2 + (a-1)^2 == 13 (mod 100) and q = (a+1)^2 + (a-2)^2 == 17 (mod 100), where natural a == 3 (mod 5).

David W. Wilson, Table of n, a(n) for n = 1..1000

Cf. A217503.

isP[p_, p1_List, p2_List] := Module[{q = Sort[Abs[p1 + p2]], s}, s = q[[1]]^2 + q[[2]]^2; q != p1 && s - p == 4 && PrimeQ[s]]; testQ[p_] := Module[{pp = PowersRepresentations[p, 2, 2][[1]]}, isP[p, pp, {-1, -1}] || isP[p, pp, {-1, 1}] || isP[p, pp, {1, -1}] || isP[p, pp, {1, 1}]]; Select[Prime[Range[200000]], Mod[#, 4] == 1 && testQ[#] &] (* T. D. Noe, Oct 19 2012 *)

A002313(n) such that A217503(n) = 2 and A082073(n-1) = 4.
Intersection of A001844 and A023200.
a(n) = A001844(m) for some m == 2 or 12 (mod 15).
a(n) == 13 (mod 300).

Extended by T. D. Noe, Oct 19 2012

A212279 A002144(n+1)^2+1 mod A002144(n), where A002144 are the Pythagorean primes (p=4k+1).

Original entry on oeis.org

0, 0, 0, 28, 17, 39, 4, 72, 79, 65, 17, 65, 17, 29, 145, 65, 84, 65, 145, 17, 109, 17, 65, 0, 145, 65, 17, 145, 88, 17, 64, 145, 17, 28, 257, 65, 17, 65, 145, 145, 257, 65, 17, 269, 145, 401, 257, 145, 65, 257, 65, 145, 17, 577, 145, 65, 145, 17, 577, 65, 577

Offset: 1

M. F. Hasler, May 13 2012

nonn

Motivated by the fact that the first terms are zero (which is of course a coincidence). Other values (17, 65, 145, 257...) occur much more frequently.

Conjecture: a(n) = A082073(n)^2 + 1 for all n > 159. - Charles R Greathouse IV, May 13 2012

			5^2+1 = 2*13, 13^2+1 = 10*17, 17^2=10*29; therefore a(1)=a(2)=a(3)=0.
29^2+1 = 22*37+28, therefore a(4)=28.
Kermit Rose's post in the primenumbers Yahoo group:
  >>> (5**2+1)%13
  0
  >>> (13**2+1)%17
  0
  >>> (17**2+1)%29
  0
  Looks remarkable.
  >>> (29**2+1)%37
  28.
  Oops: Break in the pattern. Another illustration of the law of small numbers. :)

K. Rose, Law of small numbers, primenumbers group, May 2012.

o=5;forprime(p=o+1,900,p%4==1||next;print1((o^2+1)%o=p","))

cp's OEIS Frontend

A082075 First differences of primes of the form 4*k+3 (A002145).

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A082074 One quarter of first differences of primes of the form 4*k+1 (A002144).

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A082076 First differences of primes of the form 4*k+3 (A002145), divided by 4.

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A082098 a(n) is the smallest prime p of the form 4k+3 such that nextprime(p) - p = 4n.

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A082099 a(n) is the smallest prime p of the form 4k+1 such that nextprime(p) - p = 4n.

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A217503 Squared distance between consecutive primes of the form 4k+1 (see below).

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A141614 First differences of A008846.

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A217674 Primes p = a^2 + b^2 such that (x - a)^2 + (y - b)^2 = 2 and q - p = 4, where q = x^2 + y^2 is prime, assuming that a > b > 0 and x > y > 0.

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