A002145 -id:A002145 - cp's OEIS frontend

A282036 a(n) is the sum of quadratic nonresidues of A002145(n) (the n-th prime == 3 mod 4).

2, 14, 33, 95, 161, 279, 473, 658, 944, 1139, 1491, 1738, 1826, 2884, 2996, 4318, 4585, 5004, 6191, 6683, 7849, 8413, 10314, 10746, 11394, 13157, 13393, 16013, 16566, 18936, 19783, 20376, 23946, 27057, 27804, 30883, 35541, 35232, 36384, 39832, 45671, 50858, 51363, 50059, 55097, 56040

Offset: 1

N. J. A. Sloane, Feb 20 2017

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Christian Aebi, Grant Cairns, Sums of Quadratic residues and nonresidues, arXiv:1512.00896 [math.NT], 2015.

Sums of residues, nonresidues, and their differences, for p == 1 mod 4, p == 3 mod 4, and all p: A171555; A282035, A282036, A282037; A076409, A125615, A282038.

Cf. A002145.

with(numtheory):
a:=[]; m:=[]; d:=[];
for i1 from 1 to 200 do
p:=ithprime(i1);
if (p mod 4) = 3 then
sp:=0; sm:=0;
for j from 1 to p-1 do
if legendre(j,p)=1 then sp:=sp+j; else sm:=sm+j; fi; od;
a:=[op(a),sp]; m:=[op(m),sm]; d:=[op(d),sm-sp];
fi;
od:
a; m; d; # A282035, A282036, A282037
# Alternative:
f:= p -> add(-k^2 mod p, k=1..(p-1)/2)::
map(f, select(isprime, [seq(p,p=3..1000,4)])); # Robert Israel, Nov 09 2020

f[p_] := Total[Range[p-1] ~Complement~ Table[Mod[k^2, p], {k, (p-1)/2}] ]; f /@ Select[Range[3, 1000, 4], PrimeQ] (* Jean-François Alcover, Feb 16 2018, after Robert Israel *)

lista(nn) = forprime(p=2, nn, if(p%4==3, print1(sum(k=1, p-1, if (!issquare(Mod(k, p)), k)), ", "))); \\ Michel Marcus, Nov 09 2020

a(n) = Sum_{k=1..(A002145(n)-1)/2} (-k^2) mod A002145(n). - J. M. Bergot and Robert Israel, Nov 09 2020

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.

A222298 Length of the Gaussian prime spiral beginning at the n-th positive real Gaussian prime (A002145).

Original entry on oeis.org

12, 12, 260, 12, 236, 28, 28, 28, 28, 236, 20, 44, 44, 20, 20, 36, 76, 12, 12, 4, 12, 4, 36, 36, 36, 3276, 76, 36, 36, 3276, 84, 20, 12, 12, 20, 36, 36, 2444, 2444, 36, 44, 1356, 156, 28, 12, 220, 12, 12, 84, 12, 132, 28, 68, 36, 36, 1044, 20, 20, 28, 1044, 20

Offset: 1

T. D. Noe, Feb 25 2013

nonn

The Gaussian prime spiral is described in the short note by O'Rourke and Wagon. It is not known if every iteration is a closed loop. See A222299 for the number of distinct primes on the spiral. See A222300 for the length of the spiral (which is the same as the number of numbers tested for primality, without memory).

This idea can be extended to any Gaussian prime. Sequences A222594, A222595, and A222596 show the results for first-quadrant Gaussian primes. - T. D. Noe, Feb 27 2013

			The loop beginning with 31 is {31, 43, 43 - 8i, 37 - 8i, 37 - 2i, 45 - 2i, 45 - 8i, 43 - 8i, 43, 47, 47 - 2i, 45 - 2i, 45 + 2i, 47 + 2i, 47, 43, 43 + 8i, 45 + 8i, 45 + 2i, 37 + 2i, 37 + 8i, 43 + 8i, 43, 31, 31 + 4i, 41 + 4i, 41 - 4i, 31 - 4i, 31}. The first and last numbers are the same. So only one is counted.

Joseph O'Rourke and Stan Wagon, Gaussian prime spirals, Mathematics Magazine, vol. 86, no. 1 (2013), p. 14.

T. D. Noe, Table of n, a(n) for n = 1..1000
T. D. Noe, Plot beginning with 11 (similar to the cover of Mathematics Magazine, vol. 86, no. 1 (2013))
Joseph O'Rourke, MathOverflow: Gaussian prime spirals

loop[n_] := Module[{p = n, direction = 1}, lst = {n}; While[While[p = p + direction; ! PrimeQ[p, GaussianIntegers -> True]]; direction = direction*(-I); AppendTo[lst, p]; ! (p == n && direction == 1)]; Length[lst]]; cp = Select[Range[1000], PrimeQ[#, GaussianIntegers -> True] &]; Table[loop[p]-1, {p, cp}]

A334447 Decimal expansion of Product_{k>=1} (1 + 1/A002145(k)^4).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Apr 30 2020

In general, for s>1, Product_{k>=1} (1 + 1/A002145(k)^s)/(1 - 1/A002145(k)^s) = 2^s * (2^s - 1) * zeta(s) / (zeta(s, 1/4) - zeta(s, 3/4)) = 1 / (2 * (-1)^s * PolyGamma(s-1, 1/4) / (2^s * (2^s - 1) * Gamma(s) * zeta(s)) - 1).

			1.01284973750365824105373880963011203968450421655386945092221...

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. A002145, A243381, A334426, A334451.

A334447 / A334448 = 1/(PolyGamma(3, 1/4)/(8*Pi^4) - 1).

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

More digits from Vaclav Kotesovec, Jun 27 2020

A334451 Decimal expansion of Product_{k>=1} (1 + 1/A002145(k)^5).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Apr 30 2020

In general, for s>0, Product_{k>=1} (1 + 1/A002145(k)^(2*s+1))/(1 - 1/A002145(k)^(2*s+1)) = (2*s)! * (2^(2*s + 2) - 2) * zeta(2*s+1) / (Pi^(2*s+1) * A000364(s)). - Dimitris Valianatos, May 01 2020

In general, for s>1, Product_{k>=1} (1 + 1/A002145(k)^s)/(1 - 1/A002145(k)^s) = 2^s * (2^s - 1) * zeta(s) / (zeta(s, 1/4) - zeta(s, 3/4)).

			1.0041818167708566903887269765658569606315819506367432882834249768697794496439...

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. A002145, A243381, A334426, A334447.

A334451 / A334452 = 1488*zeta(5)/(5*Pi^5).

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

More digits from Vaclav Kotesovec, Jun 27 2020

A334452 Decimal expansion of Product_{k>=1} (1 - 1/A002145(k)^5).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Apr 30 2020

In general, for s>0, Product_{k>=1} (1 + 1/A002145(k)^(2*s+1))/(1 - 1/A002145(k)^(2*s+1)) = (2*s)! * (2^(2*s + 2) - 2) * zeta(2*s+1) / (Pi^(2*s+1) * A000364(s)). - Dimitris Valianatos, May 01 2020

In general, for s>1, Product_{k>=1} (1 + 1/A002145(k)^s)/(1 - 1/A002145(k)^s) = 2^s * (2^s - 1) * zeta(s) / (zeta(s, 1/4) - zeta(s, 3/4)).

			0.99581872986808059594338516164316597187434727318491056639835771469803963967031...

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 3 5 = 1/A334452).

Cf. A002145, A243379, A334427, A334448.

A334451 / A334452 = 1488*zeta(5)/(5*Pi^5).

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

More digits from Vaclav Kotesovec, Jun 27 2020

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).

A352400 a(n) is the left Aurifeuillian factor of p^p + 1 for A002145(n), where A002145 lists the primes congruent to 3 (mod 4).

Original entry on oeis.org

1, 113, 58367, 113631466919, 348275601426959, 8403855868042458448127, 7248206084007410402911299180581471, 105318477338066161993242388018074119617, 830220061043693789623432394289631761145130727636121

Offset: 1

Patrick A. Thomas, Jun 08 2022

nonn

For prime factorizations of p^p + 1 see A125136.

			105318477338066161993242388018074119617 is the smaller Aurifeuillian factor of 47^47 + 1, and 47 is the 8th term of A002145, so it is a(8).

Patrick A. Thomas, Table of n, a(n) for n = 1..65
Calculators, Aurifeuillian LMs
Wikipedia, Aurifeuillean factorization.

Cf. A002145, A125136, A230377, A352711, A352732, A352401.

If R is (p^p+1)/(p+1), where p == 3 (mod 4) and p > 7, then an approximation of the left Aurifeuillian factor of R is (1/e) * sqrt(R/(1+z)), where z =
   2/(3p) + 28/(45p^2) + 1706/(2835p^3) if p=1,79,109,121,151  or 169 (mod 210),
   2/(3p) + 28/(45p^2) +   86/(2835p^3) if p=19,31,61,139,181  or 199 (mod 210),
   2/(3p) -  8/(45p^2) +  194/(2835p^3) if p=37,43,67,127,163  or 193 (mod 210),
   2/(3p) -  8/(45p^2) - 1426/(2835p^3) if p=13,73,97,103,157  or 187 (mod 210),
  -2/(3p) -  8/(45p^2) + 1426/(2835p^3) if p=23,53,107,113,137 or 197 (mod 210),
  -2/(3p) -  8/(45p^2) -  194/(2835p^3) if p=17,47,83,143,167  or 173 (mod 210),
  -2/(3p) + 28/(45p^2) -   86/(2835p^3) if p=11,29,71,149,179  or 191 (mod 210),
  -2/(3p) + 28/(45p^2) - 1706/(2835p^3) if p=41,59,89,101,131  or 209 (mod 210).

A385166 Let p = A002145(n) be the n-th prime == 3 (mod 4); a(n) = (p+1) * ord(5,p) / ord(2+-i,p) = (p+1) * ord(5,p) / A385165(n). Here ord(a,m) is the multiplicative order of a modulo m.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 2, 5, 1, 1, 1, 1, 3, 2, 2, 1, 1, 2, 2, 1, 1, 1, 11, 1, 4, 3, 10, 1, 1, 1, 3, 1, 2, 1, 1, 1, 6, 1, 6, 1, 3, 1, 1, 1, 4, 3, 24, 1, 1, 1, 1, 1, 3, 1, 4, 2, 1, 1, 1, 1, 1, 2, 1, 1, 8, 1, 27, 1, 1, 1, 1, 20, 3, 1, 4, 1, 1

Offset: 1

Jianing Song, Jun 20 2025

Of course if a and m are integers, it doesn't matter if the base ring is Z or Z[i] for ord(a,m).

			The multiplicative order of 2+-i modulo A002145(3) = 11 is A385165(3) = 30, so a(3) = (12*ord(5,11))/30 = 2.
The multiplicative order of 2+-i modulo A002145(13) = 83 is A385165(13) = 2296, so a(13) = (84*ord(5,83))/2296 = 3.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A002145, A385165. Primes corresponding to special terms: A385168 (>1), A385167 (even), A385180 (divisible by 4).

Cf. A211450.

quot(p) = my(z = znorder(Mod(5,p)), d = divisors((p+1)*z)); for(i=1, #d, if(Mod([2,-1;1,2],p)^d[i] == 1, return((p+1)*z/d[i]))) \\ for a prime p == 3 (mod 4), returns (p+1) * ord(5,p) / ord(2+-i, p)
forprime(p=3, 1e3, if(p%4==3, print1(quot(p), ", ")))

A095104 Diving index of the n-th 4k+3 prime (A002145(n)).

Original entry on oeis.org

0, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 0, 0, 7, 7, 0, 3, 0, 3, 0, 11, 0, 0, 3, 13, 61, 0, 0, 0, 0, 3, 3, 0, 3, 7, 0, 45, 3, 0, 0, 0, 0, 7, 7, 35, 0, 7, 35, 3, 0, 3, 3, 0, 3, 15, 0, 0, 3, 15, 3, 0, 0, 7, 3, 0, 45, 3, 0, 0, 3, 3, 7, 7, 0, 3, 0, 3, 0, 3, 0, 0, 7, 7, 0, 0, 0, 67, 0, 0, 3, 0

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

Diving index of an odd number n is the first integer u > 1 where Sum_{i=1..u} J(i/n) results -1 and zero if never. Here J(i/n) is Jacobi symbol of i and n, which reduces to a Legendre symbol L(i/n) when n is a prime.

A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

a(n)=A095105(n)+1 modulo A002145(n). Cf. A095106, A095108 (same sequence with zeros removed), A095269.

cp's OEIS Frontend

A282036 a(n) is the sum of quadratic nonresidues of A002145(n) (the n-th prime == 3 mod 4).

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

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A222298 Length of the Gaussian prime spiral beginning at the n-th positive real Gaussian prime (A002145).

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A334447 Decimal expansion of Product_{k>=1} (1 + 1/A002145(k)^4).

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A334451 Decimal expansion of Product_{k>=1} (1 + 1/A002145(k)^5).

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A334452 Decimal expansion of Product_{k>=1} (1 - 1/A002145(k)^5).

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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.

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A352400 a(n) is the left Aurifeuillian factor of p^p + 1 for A002145(n), where A002145 lists the primes congruent to 3 (mod 4).

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