A002144 -id:A002144 - cp's OEIS frontend

A145298 Smallest k such that k^2+1 is divisible by A002144(n)^5.

1068, 143044, 390112, 7745569, 6423465, 46464143, 23048345, 144762466, 404034898, 2153335831, 331407850, 1108900220, 2581164875, 760839155, 10734466938, 6595297216, 773302059, 61063137802, 31915893786, 112699451831

Offset: 1

Klaus Brockhaus, Oct 14 2008

nonn

			a(4) = 7745569 since A002144(4) = 29, 7745569^2+1 = 59993839133762 = 2*29^5*97*15077 and for no k < 7745569 does 29^5 divide k^2+1.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A002144 (primes of form 4n+1), A002313 (-1 is a square mod p), A059321, A145296, A145297, A145299.

{e=5; forprime(p=2, 200, if(p%4==1, q=p^e; m=q; while(!ispower(m-1,2,&n), m=m+q); print1(n, ",")))}

from itertools import islice
from sympy import nextprime, sqrt_mod_iter
def A145298_gen(): # generator of terms
    p = 1
    while (p:=nextprime(p)):
        if p&3==1:
            yield min(sqrt_mod_iter(-1,p**5))
A145298_list = list(islice(A145298_gen(),20)) # Chai Wah Wu, May 04 2024

A152680 a(n) = 4*A005098(n) = A002144(n) - 1.

Original entry on oeis.org

4, 12, 16, 28, 36, 40, 52, 60, 72, 88, 96, 100, 108, 112, 136, 148, 156, 172, 180, 192, 196, 228, 232, 240, 256, 268, 276, 280, 292, 312, 316, 336, 348, 352, 372, 388, 396, 400, 408, 420, 432, 448, 456, 460, 508, 520, 540, 556, 568, 576, 592, 600, 612, 616

Offset: 1

Artur Jasinski, Dec 10 2008

nonn

If we take the 4 numbers 1, A002314(n), A152676(n), A152680(n) then the multiplication table modulo A002144(n) is isomorphic with the Latin square
1 2 3 4
2 4 1 3
3 1 4 2
4 3 2 1
and isomorphic with the multiplication table of {1,I,-I,-1} where I is sqrt(-1), A152680(n) is isomorphic with -1, A002314(n) with I or -I and A152676(n) vice versa -I or I.
1, A002314(n), A152676(n), A152680(n) are subfields of the Galois Field [A002144(n)].
Numbers n such that A172019(n) + 1 = primes - 1. - Giovanni Teofilatto, Feb 02 2010

Cf. A002314, A152676, A152680.

aa = {}; Do[If[Mod[Prime[n], 4] == 1, AppendTo[aa, Prime[n] - 1]], {n, 1, 200}]; aa

A334445 Decimal expansion of Product_{k>=1} (1 + 1/A002144(k)^4).

Original entry on oeis.org

1, 0, 0, 1, 6, 4, 9, 6, 6, 4, 0, 3, 3, 0, 0, 0, 4, 2, 5, 3, 7, 8, 5, 7, 8, 0, 7, 1, 9, 2, 9, 3, 9, 0, 8, 8, 8, 2, 7, 3, 9, 8, 4, 4, 0, 4, 3, 8, 6, 6, 9, 9, 3, 0, 0, 0, 8, 9, 8, 3, 7, 4, 0, 9, 6, 6, 7, 9, 2, 0, 4, 8, 0, 8, 2, 3, 6, 3, 4, 3, 4, 4, 1, 9, 2, 9, 8, 6, 5, 3, 3, 1, 1, 7, 8, 9, 9, 7, 0, 6, 1, 5, 7, 0, 9

Offset: 1

Vaclav Kotesovec, Apr 30 2020

In general, for s>1, Product_{k>=1} (1 + 1/A002144(k)^s)/(1 - 1/A002144(k)^s) = (zeta(s, 1/4) - zeta(s, 3/4)) * zeta(s) / (2^s * (2^s + 1) * zeta(2*s)).

			1.001649664033000425378578071929390888273984404386699300089837...

B. C. Berndt, Ramanujan's notebook part IV, Springer-Verlag, 1994, p. 64-65.

Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants, Feb 18 1996, p. 7-8.

Cf. A002144, A243380, A334424, A334449.

A334445 / A334446 = 35*(PolyGamma(3, 1/4)/(8*Pi^4) - 1)/34.

A334445 * A334447 = 1680 / (17*Pi^4).

More digits from Vaclav Kotesovec, Jun 27 2020

A334449 Decimal expansion of Product_{k>=1} (1 + 1/A002144(k)^5).

Original entry on oeis.org

1, 0, 0, 0, 3, 2, 3, 4, 7, 5, 1, 4, 8, 0, 7, 1, 6, 3, 8, 6, 0, 3, 6, 8, 6, 4, 2, 7, 3, 3, 9, 9, 4, 2, 3, 6, 9, 2, 6, 5, 2, 4, 6, 5, 5, 2, 2, 0, 2, 7, 3, 7, 9, 8, 0, 4, 0, 7, 5, 0, 7, 1, 6, 4, 8, 5, 9, 9, 6, 3, 8, 1, 1, 3, 7, 4, 6, 8, 0, 4, 2, 2, 4, 4, 0, 6, 0, 5, 6, 3, 2, 9, 6, 0, 0, 1, 4, 1, 9, 1, 2, 7, 9, 3, 2

Offset: 1

Vaclav Kotesovec, Apr 30 2020

In general, for s>0, Product_{k>=1} (1 + 1/A002144(k)^(2*s+1))/(1 - 1/A002144(k)^(2*s+1)) = Pi^(2*s+1) * A000364(s) * zeta(2*s+1) / ((2^(2*s+2) + 2) * (2*s)! * zeta(4*s+2)). - Dimitris Valianatos, May 01 2020

In general, for s>1, Product_{k>=1} (1 + 1/A002144(k)^s)/(1 - 1/A002144(k)^s) = (zeta(s, 1/4) - zeta(s, 3/4)) * zeta(s) / (2^s * (2^s + 1) * zeta(2*s)).

			1.0003234751480716386036864273399423692652465522027379804075071648599638113746...

B. C. Berndt, Ramanujan's notebook part IV, Springer-Verlag, 1994, p. 64-65.

Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants, Feb 18 1996, p. 7-8.

Cf. A002144, A243380, A334424, A334445.

A334449 / A334450 = 4725*zeta(5)/(16*Pi^5).

A334449 * A334451 = 90720*zeta(5)/Pi^10.

More digits from Vaclav Kotesovec, Jun 27 2020

A334450 Decimal expansion of Product_{k>=1} (1 - 1/A002144(k)^5).

Original entry on oeis.org

9, 9, 9, 6, 7, 6, 5, 2, 7, 0, 7, 9, 6, 2, 6, 6, 6, 2, 0, 1, 8, 2, 4, 6, 1, 8, 0, 8, 7, 3, 0, 8, 3, 7, 0, 1, 5, 0, 0, 7, 5, 1, 5, 7, 4, 3, 7, 9, 5, 5, 4, 4, 3, 0, 5, 6, 8, 4, 3, 2, 8, 4, 0, 4, 2, 4, 9, 7, 5, 9, 8, 1, 9, 2, 1, 2, 1, 9, 1, 3, 2, 9, 9, 7, 0, 4, 0, 0, 3, 0, 2, 9, 1, 9, 3, 0, 4, 4, 5, 3, 7, 5, 2, 8, 3, 9

Offset: 0

Vaclav Kotesovec, Apr 30 2020

In general, for s>0, Product_{k>=1} (1 + 1/A002144(k)^(2*s+1))/(1 - 1/A002144(k)^(2*s+1)) = Pi^(2*s+1) * A000364(s) * zeta(2*s+1) / ((2^(2*s+2) + 2) * (2*s)! * zeta(4*s+2)). - Dimitris Valianatos, May 01 2020

In general, for s>1, Product_{k>=1} (1 + 1/A002144(k)^s)/(1 - 1/A002144(k)^s) = (zeta(s, 1/4) - zeta(s, 3/4)) * zeta(s) / (2^s * (2^s + 1) * zeta(2*s)).

			0.999676527079626662018246180873083701500751574379554430568432840424975981921219...

B. C. Berndt, Ramanujan's notebook part IV, Springer-Verlag, 1994, p. 64-65.

Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants, Feb 18 1996, p. 7-8.
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, p. 26 (case 4 1 5 = 1/A334450).

Cf. A002144, A088539, A334446, A334450.

A334449 / A334450 = 4725*zeta(5)/(16*Pi^5).

A334450 * A334452 = 32/(31*zeta(5)).

More digits from Vaclav Kotesovec, Jun 27 2020

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.

A152676 a(n) = A002144(n) - A002314(n).

Original entry on oeis.org

3, 8, 13, 17, 31, 32, 30, 50, 46, 55, 75, 91, 76, 98, 100, 105, 129, 93, 162, 112, 183, 122, 144, 177, 241, 187, 217, 228, 155, 288, 203, 189, 213, 311, 269, 274, 334, 381, 266, 392, 254, 382, 348, 413, 301, 286, 489, 439, 483, 553, 516, 476, 578, 423, 487, 504

Offset: 1

Artur Jasinski, Dec 10 2008

nonn

For the four numbers {1, A002314(n), A152676(n), A152680(n)}, the multiplication table modulo A002144(n) is isomorphic with the Latin square
  1 2 3 4
  2 4 1 3
  3 1 4 2
  4 3 2 1
and is isomorphic with the multiplication table for {1,i,-i,-1} where i = sqrt(-1), A152680(n) is isomorphic with -1, A002314(n) with i or -i and A152676(n) vice versa -i or i.
1, A002314(n), A152676(n), A152680(n) are a subfield of the Galois Field [A002144(n)].
Let p = A002144(n), the n-th prime of the form 4k+1. Then a(n) and A002314(n) are the two square roots of -1 (mod p). Note that a(n) is also the multiplicative inverse of A002314(n) (mod p). - T. D. Noe, Feb 18 2010

T. D. Noe, Table of n, a(n) for n = 1..1000
A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. See Vol. 1, pp. 1-21.

Cf. A002144, A002314, A152680.

aa = {}; Do[If[Mod[Prime[n], 4] == 1, k = 1; While[ ! Mod[k^2 + 1, Prime[n]] == 0, k++ ]; AppendTo[aa, Prime[n] - k]], {n, 1, 200}]; aa

A173331 Second of two intermediate sequences for integral solution of A002144(n)=x^2+y^2.

Original entry on oeis.org

2, 2, 13, 2, 31, 4, 2, 55, 8, 81, 4, 91, 99, 105, 133, 10, 6, 2, 10, 181, 183, 227, 8, 237, 16, 10, 14, 265, 2, 301, 303, 16, 18, 8, 355, 379, 6, 381, 389, 14, 421, 429, 453, 451, 487, 20, 531, 543, 20, 24, 585, 24, 18, 16, 637, 631, 655, 12, 651, 675, 22, 731, 26, 741, 757

Offset: 1

Reinhard Zumkeller, Feb 16 2010

nonn

a(n) = A173330(n)*A010050(A005098(n)) mod A002144(n);

A002973(n) = MIN(a(n), A002144(n) - a(n)) / 2.

			n=7: A002144(7) = 53 = 4*13 + 1,
a(7) = A173330(7) * 26! mod 53 = 7*403291461126605635584000000 mod 53 = 2,
A002973(7) = MIN(2, 53 - 2) / 2 = 1;
n=8: A002144(8) = 61 = 4*15 + 1,
a(8) = A173330(8) * 30! mod 61 = 5*265252859812191058636308480000000 mod 61 = 55,
A002973(8) = MIN(55, 61 - 55) / 2 = 3.

H. Davenport, The Higher Arithmetic (Cambridge University Press 7th ed., 1999), ch. V.3, p.122.

Cf. A123072, A000142.

a(n) = ((2k)! / 2(k!))^2 mod p, where p = 4*k+1 = A002144(n).

A348747 Fully multiplicative with a(2) = 1, a(3) = 2, a(5) = 3, a(A002144(1+n)) = A002144(n) and a(A002145(1+n)) = a(A002145(1+n)) for all n >= 1, where A002144 and A002145 give the primes of the form 4k+1 and 4k+3 respectively.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 7, 1, 4, 3, 11, 2, 5, 7, 6, 1, 13, 4, 19, 3, 14, 11, 23, 2, 9, 5, 8, 7, 17, 6, 31, 1, 22, 13, 21, 4, 29, 19, 10, 3, 37, 14, 43, 11, 12, 23, 47, 2, 49, 9, 26, 5, 41, 8, 33, 7, 38, 17, 59, 6, 53, 31, 28, 1, 15, 22, 67, 13, 46, 21, 71, 4, 61, 29, 18, 19, 77, 10, 79, 3, 16, 37, 83, 14, 39, 43, 34, 11, 73

Offset: 1

Antti Karttunen, Nov 02 2021

Left inverse of A348746.
Cf. A000265, A000720, A002144, A002145, A348745.
Cf. also A064989, A332819 for similar maps.

A348747(n) = { my(f=factor(n)); for(k=1,#f~, if(f[k,1]<=3, f[k,1]--, if(5==f[k,1], f[k,1]=3, if(1==(f[k,1]%4), forstep(i=primepi(f[k,1])-1,0,-1,if(1==(prime(i)%4), f[k,1]=prime(i); break)))))); factorback(f); };

Fully multiplicative with a(p) = A348745(A000720(p)).
a(A348746(n)) = n.
a(2n) = a(A000265(n)) = a(n).

A096029 Values (x+y-1)/2, where x^2+y^2=p runs over the Pythagorean primes A002144.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 5, 6, 7, 7, 8, 8, 7, 9, 9, 7, 8, 10, 9, 8, 11, 11, 10, 9, 12, 12, 12, 11, 12, 12, 13, 12, 10, 11, 14, 14, 13, 12, 14, 13, 15, 15, 16, 16, 12, 15, 14, 17, 17, 14, 17, 15, 17, 13, 15, 18, 14, 17, 19, 18, 19, 18, 16, 19, 18, 20, 16, 17, 20, 21, 21, 19, 20

Offset: 1

Lekraj Beedassy, Jun 16 2004

nonn

Cf. A005098, A096030.

a(n)=(A079886(n) - 1)/2

More terms from Ray Chandler, Jun 26 2004

cp's OEIS Frontend

A145298 Smallest k such that k^2+1 is divisible by A002144(n)^5.

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A152680 a(n) = 4*A005098(n) = A002144(n) - 1.

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A334445 Decimal expansion of Product_{k>=1} (1 + 1/A002144(k)^4).

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A334449 Decimal expansion of Product_{k>=1} (1 + 1/A002144(k)^5).

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A334450 Decimal expansion of Product_{k>=1} (1 - 1/A002144(k)^5).

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

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A152676 a(n) = A002144(n) - A002314(n).

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A173331 Second of two intermediate sequences for integral solution of A002144(n)=x^2+y^2.

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