A002144 -id:A002144 - cp's OEIS frontend

A059321 Smallest number m such that m^2+1 is divisible by A002144(n)^2 (= squares of primes congruent to 1 mod 4).

7, 70, 38, 41, 117, 378, 500, 682, 776, 3861, 4052, 515, 5744, 1710, 6613, 1744, 11018, 13241, 3458, 5099, 1393, 16610, 26884, 15006, 2072, 13637, 31361, 4443, 26508, 7850, 37520, 31152, 39922, 37107, 6072, 4005, 32491, 4030, 43211, 12238

Offset: 1

Marc LeBrun, Jan 26 2001

a(2) = 70 since A002144(2)=13, 70^2+1 = 4091 = 13^2 * 29 and for no k<70 does 13^2 divide k^2+1. Related to period-1 continued fractions.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..2000 from Klaus Brockhaus)

Cf. A002144, A059591, A059592.

from itertools import islice
from sympy import nextprime, sqrt_mod_iter
def A059321_gen(): # generator of terms
    p = 1
    while (p:=nextprime(p)):
        if p&3==1:
            yield min(sqrt_mod_iter(-1,p**2))
A059321_list = list(islice(A059321_gen(),20)) # Chai Wah Wu, May 04 2024

A348746 Fully multiplicative with a(2) = 3, a(3) = 5, a(A002144(n)) = A002144(1+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, 3, 5, 9, 13, 15, 7, 27, 25, 39, 11, 45, 17, 21, 65, 81, 29, 75, 19, 117, 35, 33, 23, 135, 169, 51, 125, 63, 37, 195, 31, 243, 55, 87, 91, 225, 41, 57, 85, 351, 53, 105, 43, 99, 325, 69, 47, 405, 49, 507, 145, 153, 61, 375, 143, 189, 95, 111, 59, 585, 73, 93, 175, 729, 221, 165, 67, 261, 115, 273, 71, 675, 89, 123

Offset: 1

Antti Karttunen, Nov 02 2021

Permutation of odd numbers. Preserves the prime signature.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000720, A002144, A002145, A348744, A348747 (left inverse).

Cf. also A003961, A332818 for similar maps.

A348746(n) = { my(f=factor(n)); for(k=1,#f~, if(2==f[k,1], f[k,1]=3, if(3==f[k,1], f[k,1]=5, if(1==(f[k,1]%4), for(i=1+primepi(f[k,1]),oo,if(1==(prime(i)%4), f[k,1]=prime(i); break)))))); factorback(f); };

Fully multiplicative with a(p) = A348744(A000720(p)), where A348744 is the lexicographically earliest bijection from primes to odd primes where each prime of the form 4k+1 is mapped to the next larger prime of the same form.

A080175 Fourth power of primes of the form 4k+1 (A002144).

Original entry on oeis.org

625, 28561, 83521, 707281, 1874161, 2825761, 7890481, 13845841, 28398241, 62742241, 88529281, 104060401, 141158161, 163047361, 352275361, 492884401, 607573201, 895745041, 1073283121, 1387488001, 1506138481, 2750058481

Offset: 1

Cino Hilliard, Mar 16 2003

a(n) is the hypotenuse of four and only four right triangles with integral legs (Fermat). See the Dickson reference, (A) on p. 227.

In 1640 Fermat generalized the 3,4,5 Pythagorean triangle with the theorem: A prime of the form 4k+1 is the hypotenuse of one and only one right triangle with integral legs. The square of a prime of the form 4k+1 is the hypotenuse of two and only two... The cube of three and only three...

			625 is the hypotenuse of triangles 175, 600, 625; 220, 585, 625; 336, 527, 625; 375, 500, 625.

L. E. Dickson, History of the Theory of Numbers, Volume II, Diophantine Analysis. Carnegie Institution Publication No. 256, Vol II, Washington, DC, 1920, p. 227.
Morris Kline, Mathematical Thought from Ancient to Modern Times, 1972, pp. 275-276.

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

Cf. A002144, A080109, A334446.

[a^4: n in [0..40] | IsPrime(a) where a is 4*n + 1 ]; // Vincenzo Librandi, Jun 24 2015

seq(p^4, p = select(isprime,[seq(4*k+1,k=1..100)])); # Robert Israel, Jan 14 2015

Select[4 Range[100] + 1, PrimeQ[#] &]^4 (* Vincenzo Librandi, Jun 24 2015 *)

fermat(n) = { for(x=1,n, y=4*x+1; if(isprime(y),print1(y^4, " ")) ) }

a(n) = A002144(n)^4 = A080109(n)^2, n >= 1.

Product_{n>=1} (1 - 1/a(n)) = A334446. - Amiram Eldar, Dec 02 2022

Edited: name shortened, part of old name as a comment, comment changed, Dickson reference, formula and cross references added. - Wolfdieter Lang, Jan 14 2015

A094178 Numbers n such that 4n+1 is divisible only by primes of form 4m+1 (i.e., by the Pythagorean primes A002144).

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 10, 13, 15, 16, 18, 21, 22, 24, 25, 27, 28, 31, 34, 36, 37, 39, 42, 43, 45, 46, 48, 49, 51, 55, 57, 58, 60, 64, 66, 67, 69, 70, 72, 73, 76, 78, 79, 81, 84, 87, 88, 91, 93, 94, 97, 99, 100, 102, 105, 106, 108, 111, 112, 114, 115, 120, 121, 123, 126, 127

Offset: 1

Lekraj Beedassy, May 06 2004

nonn

For the actual numbers 4n+1, see A008846(n).

Complement of A124934; A125203(a(n)) = 0; A000290 and A000217 are subsequences. - Reinhard Zumkeller, Nov 24 2006

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

import Data.List (elemIndices)
a094178 n = a094178_list !! (n-1)
a094178_list = map (+ 1) $ elemIndices 0 a125203_list
-- Reinhard Zumkeller, Jan 02 2013

More terms from Ray Chandler, Jun 20 2004

A145299 Smallest k such that k^2+1 is divisible by A002144(n)^6.

Original entry on oeis.org

1068, 1999509, 390112, 253879357, 756360062, 2363588163, 5041394261, 9435321777, 41865466758, 102666405913, 197177418061, 316411915250, 171829799914, 625667121807, 182312430890, 1095001339019, 6390289199260

Offset: 1

Klaus Brockhaus, Oct 17 2008

nonn

			a(1) = 1068 since A002144(1) = 5, 1068^2+1 = 1140625 = 5^6*73 and for no k < 1068 does 5^6 divide k^2+1. a(11) = 197177418061 since A002144(11) = 97, 197177418061^2+1 = 38878934193202368999722 = 2*97^6*23337479509 and for no k < 197177418061 does 97^6 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, A145298.

{ e=6; forprime(p=2, 1000, if(p%4==1, k=lift(sqrt(-1+O(p^e))); if(k>p^e/2,k=p^e-k); print1(k, ", "))) }

from itertools import islice
from sympy import nextprime, sqrt_mod_iter
def A145299_gen(): # generator of terms
    p = 1
    while (p:=nextprime(p)):
        if p&3==1:
            yield min(sqrt_mod_iter(-1,p**6))
A145299_list = list(islice(A145299_gen(),20)) # Chai Wah Wu, May 04 2024

More terms and efficient PARI program from. - Max Alekseyev, Oct 28 2008

A267101 2 followed by permutation of odd primes, where each n-th prime of the form 4k+1 (A002144) has been replaced with the n-th prime of the form 4k+3 (A002145) and vice versa.

Original entry on oeis.org

2, 5, 3, 13, 17, 7, 11, 29, 37, 19, 41, 23, 31, 53, 61, 43, 73, 47, 89, 97, 59, 101, 109, 67, 71, 79, 113, 137, 83, 103, 149, 157, 107, 173, 127, 181, 131, 193, 197, 139, 229, 151, 233, 163, 167, 241, 257, 269, 277, 179, 191, 281, 199, 293, 211, 313, 223, 317, 227, 239, 337, 251, 349, 353, 263, 271, 373, 283, 389

Offset: 1

Antti Karttunen, Feb 01 2016

nonn

After 2, for each n >= 1, swap the places of primes A002144(n) and A002145(n) in A000040.

			For n=2, for which A000040(2) = 3, the first prime of the form 4k+3, we select the first prime of the form 4k+1, which is 5, thus a(2) = 5.
For n=3, for which A000040(3) = 5, the first prime of the form 4k+1, we select the first prime of the form 4k+3, which is 3, thus a(3) = 3.
For n=4, for which A000040(4) = 7, the second prime of the form 4k+3, we select the second prime of the form 4k+1, which is 13, thus a(4) = 13.
For n=5, for which A000040(5) = 11, the third prime of the form 4k+3, we select the third prime of the form 4k+1, which is 17, thus a(5) = 17.

Antti Karttunen, Table of n, a(n) for n = 1..10001

Cf. A000040, A002144, A002145, A267097, A267098, A267099, A267100.

Cf. also A108546.

(define (A267101 n) (cond ((= 1 n) 2) ((= 1 (modulo (A000040 n) 4)) (A002145 (A267097 n))) (else (A002144 (A267098 n)))))

a(1) = 2; after which, if prime(n) modulo 4 = 1, a(n) = A002145(A267097(n)), otherwise a(n) = A002144(A267098(n)).
a(n) = A000040(A267100(n)).
a(n) = A267099(A000040(n)).

A334446 Decimal expansion of Product_{k>=1} (1 - 1/A002144(k)^4).

Original entry on oeis.org

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

Offset: 0

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)).

For s>1, zeta(s, 1/4) - zeta(s, 3/4) = (-1)^s*(PolyGamma(s-1, 1/4) - PolyGamma(s-1, 3/4))/(s-1)! = 2*(-1)^s * PolyGamma(s-1, 1/4) / Gamma(s) - 2^s*(2^s - 1)*zeta(s) = 4^s * DirichletBeta(s).

			0.998350495723200406499905517565541629191539407019605795046314...

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.
X. Gourdon and P. Sebah, Some Constants from Number theory
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 4 = 1/A334446).

Cf. A002144, A088539, A334425, A334450.

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

A334446 * A334448 = 96/Pi^4.

More digits from Vaclav Kotesovec, Jun 27 2020

A121387 Semiprimes p*q with p and q primes of the form 4k+1 (A002144).

Original entry on oeis.org

25, 65, 85, 145, 169, 185, 205, 221, 265, 289, 305, 365, 377, 445, 481, 485, 493, 505, 533, 545, 565, 629, 685, 689, 697, 745, 785, 793, 841, 865, 901, 905, 949, 965, 985, 1037, 1073, 1145, 1157, 1165, 1189, 1205, 1241, 1261, 1285, 1313, 1345, 1369, 1385, 1405, 1417

Offset: 1

Alford Arnold, Jul 26 2006, corrected Jun 24 2007

p and q can be the same. [Harvey P. Dale, Jan 15 2012]

The terms are semiprimes of the form 4k + 1, and comprise only a portion of all such semiprimes, see A108181. - Richard R. Forberg, Aug 27 2013

			65 = 5 * 13. Note that 5 mod 4 = 1 and 13 mod 4 = 1, so 65 is a term.

François Huppé, Table of n, a(n) for n = 1..10000 (First 1000 terms from T. D. Noe).

Cf. A001358, A002144.
Fifth row of A121388.
Union of A080109 and A131574.
Cf. A107978, A080774, A108181.

With[{prs=Select[Prime[Range[150]],Mod[#,4]==1&]},Take[Union[Times @@@ Tuples[prs,2]],60]] (* Harvey P. Dale, Jan 15 2012 *)

Better definition from T. D. Noe, Sep 25 2007

A145296 Smallest k such that k^2 + 1 is divisible by A002144(n)^3.

Original entry on oeis.org

57, 239, 1985, 10133, 9466, 11389, 27590, 51412, 153765, 344464, 107551, 296344, 172078, 432436, 931837, 753090, 676541, 2321221, 2027724, 3394758, 1706203, 4841182, 1438398, 2947125, 398366, 5657795, 4942017, 9400802, 11906503

Offset: 1

Klaus Brockhaus, Oct 08 2008

nonn

			a(3) = 1985 since A002144(3) = 17, 1985^2 + 1 = 3940226 = 2*17^3*401 and for no k < 1985 does 17^3 divide k^2+1.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..150 from Klaus Brockhaus)

Cf. A002144 (primes of form 4n+1), A002313 (-1 is a square mod p), A059321, A145297, A145298, A145299.

{m=12000000; pmax=300; z=70; v=vector(z); for(n=1, m, fac=factor(n^2+1); for(j=1, #fac[, 1], if(fac[j, 2]>=3&&fac[j, 1]<=pmax, q=primepi(fac[j, 1]); if(q<=z&&v[q]==0, v[q]=n)))); t=1; j=0; while(t&&j

{e=3; forprime(p=2, 300, if(p%4==1, q=p^e; m=q; while(!ispower(m-1,2,&n), m=m+q); print1(n, ",")))} \\ Klaus Brockhaus, Oct 09 2008

from itertools import islice
from sympy import nextprime, sqrt_mod_iter
def A145296_gen(): # generator of terms
    p = 1
    while (p:=nextprime(p)):
        if p&3==1:
            yield min(sqrt_mod_iter(-1,p**3))
A145296_list = list(islice(A145296_gen(),20)) # Chai Wah Wu, May 04 2024

A145297 Smallest k such that k^2+1 is divisible by A002144(n)^4.

Original entry on oeis.org

182, 239, 27493, 34522, 800982, 1251967, 623098, 6304056, 6459524, 20099637, 22709274, 35764191, 40317977, 54397650, 166206108, 187800003, 165728858, 152475014, 282599844, 312923750, 154613663, 485200742, 912190662, 548850444

Offset: 1

Klaus Brockhaus, Oct 11 2008

nonn

			a(1) = 182 since A002144(1) = 5, 182^2+1 = 33125 = 5^4*53 and for no k < 182 does 5^4 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, A145298, A145299.

{e=4; forprime(p=2, 250, 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 A145297_gen(): # generator of terms
    p = 1
    while (p:=nextprime(p)):
        if p&3==1:
            yield min(sqrt_mod_iter(-1,p**4))
A145297_list = list(islice(A145297_gen(),20)) # Chai Wah Wu, May 04 2024

cp's OEIS Frontend

A059321 Smallest number m such that m^2+1 is divisible by A002144(n)^2 (= squares of primes congruent to 1 mod 4).

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A348746 Fully multiplicative with a(2) = 3, a(3) = 5, a(A002144(n)) = A002144(1+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.

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A080175 Fourth power of primes of the form 4k+1 (A002144).

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A094178 Numbers n such that 4n+1 is divisible only by primes of form 4m+1 (i.e., by the Pythagorean primes A002144).

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A145299 Smallest k such that k^2+1 is divisible by A002144(n)^6.

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A267101 2 followed by permutation of odd primes, where each n-th prime of the form 4k+1 (A002144) has been replaced with the n-th prime of the form 4k+3 (A002145) and vice versa.

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A334446 Decimal expansion of Product_{k>=1} (1 - 1/A002144(k)^4).

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A121387 Semiprimes p*q with p and q primes of the form 4k+1 (A002144).

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