A004614 -id:A004614 - cp's OEIS frontend

A078911 Let r+i*s be the sum of the distinct first-quadrant Gaussian integers dividing n; sequence gives s values.

0, 1, 0, 3, 3, 4, 0, 7, 0, 19, 0, 12, 5, 8, 12, 15, 5, 13, 0, 51, 0, 12, 0, 28, 25, 35, 0, 24, 7, 76, 0, 31, 0, 41, 24, 39, 7, 20, 20, 115, 9, 32, 0, 36, 39, 24, 0, 60, 0, 138, 20, 95, 9, 40, 36, 56, 0, 61, 0, 204, 11, 32, 0, 63, 92, 48, 0, 113, 0, 152, 0, 91, 11, 71, 100, 60, 0, 140

Offset: 1

N. J. A. Sloane, Jan 11 2003

A Gaussian integer z = x+iy is in the first quadrant if x > 0, y >= 0. Just one of the 4 associates z, -z, i*z, -i*z is in the first quadrant.

a(A004614(n)) = 0; a(n) = A078910(n)-A000203(n). - Vladeta Jovovic, Jan 11 2003

			The distinct first-quadrant divisors of 4 are 1, 1+i, 2, 2+2*i, 4, with sum 10+3*i, so a(4) = 3.

M. F. Hasler, Table of n, a(n) for n = 1..1000.
Michael Somos, PARI program for finding prime decomposition of Gaussian integers
Index entries for Gaussian integers and primes

Cf. A078910, A078458, A078908, A078909, A078930.

Table[Im[Plus@@Divisors[n, GaussianIntegers -> True]], {n, 65}] (* Alonso del Arte, Jan 24 2012; typo fixed by Virgile Andreani, Jul 10 2016 *)

A078911(n,S=[])=sumdiv(n*I,d,if(real(d)&imag(d)&!setsearch(S,d=vecsort(abs([real(d),imag(d)]))),S=setunion(S,[d]);(d[1]+d[2])>>(d[1]==d[2]))) \\ M. F. Hasler, Nov 22 2007

More terms from Vladeta Jovovic, Jan 11 2003

A103223 Imaginary part of the totient function phi(n) for Gaussian integers. See A103222 for the real part and A103224 for the norm.

Original entry on oeis.org

0, 1, 0, 2, 2, 2, 0, 4, 0, 4, 0, 4, 4, 6, 4, 8, 4, 6, 0, 8, 0, 10, 0, 8, 10, 12, 0, 12, 6, 8, 0, 16, 0, 16, 12, 12, 6, 18, 8, 16, 8, 12, 0, 20, 12, 22, 0, 16, 0, 20, 8, 24, 8, 18, 20, 24, 0, 28, 0, 16, 10, 30, 0, 32, 24, 20, 0, 32, 0, 24, 0, 24, 10, 36, 20, 36, 0, 24, 0, 32, 0, 40, 0, 24, 32, 42

Offset: 1

T. D. Noe, Jan 26 2005

nonn

Note that a(n)=0 when n is in A004614, the product of real Gaussian primes. It appears that all terms are nonnegative.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A103222, A103224.

phi[z_] := Module[{f, k, prod}, If[Abs[z]==1, z, f=FactorInteger[z, GaussianIntegers->True]; If[Abs[f[[1, 1]]]==1, k=2; prod=f[[1, 1]], k=1; prod=1]; Do[prod=prod*(f[[i, 1]]-1)f[[i, 1]]^(f[[i, 2]]-1), {i, k, Length[f]}]; prod]]; Im[Table[phi[n], {n, 100}]]

A185321 Carmichael numbers congruent to 3 modulo 4.

Original entry on oeis.org

8911, 1024651, 1152271, 5481451, 10267951, 14913991, 64377991, 67902031, 139952671, 178482151, 368113411, 395044651, 612816751, 652969351, 743404663, 1419339691, 1588247851, 2000436751, 2199931651, 2560600351, 3102234751, 3215031751, 3411338491, 4340265931

Offset: 1

José María Grau Ribas, Jan 27 2012

nonn

Most Carmichael numbers are congruent to 1 modulo 4.
This is a subsequence of A167181: if a prime p | a(n), (p-1) | (a(n)-1) by Korselt's criterion. But a(n)-1 is 2 mod 4, so p-1 cannot be 0 mod 4. Hence all primes dividing a(n) are 3 mod 4. - Charles R Greathouse IV, Jan 27 2012
Pinch call the intersection of A007304 with this sequence C3, which are precisely those numbers which pass a Rabin-Miller test to a random base with probability 1/4. The first member of this sequence not in C3 is a(16) = 7 * 11 * 19 * 103 * 9419. - Charles R Greathouse IV, Jan 27 2012
Wright proves that this sequence is infinite, and in particular there are more than x^(k/(log log log x)^2) terms up to x for some k and large enough x. - Charles R Greathouse IV, Nov 09 2015

Donovan Johnson and Charles R Greathouse IV, Table of n, a(n) for n = 1..15447 (first 6838 terms from Johnson)
Charles R Greathouse IV, GP script to compute terms
Charles R Greathouse IV, Alternate GP script to compute terms
R. G. E. Pinch, The Carmichael numbers up to 10^15, Mathematics of Computation 61:203 (1993), pp. 381-391.
Thomas Wright, Infinitely many Carmichael numbers in arithmetic progressions, Bull. London Math. Soc. 45:5 (2013), pp. 943-952.

Subsequence of A002997, A167181 (and hence A004614), A026424, and A177884.

Select[4Range[10^4] + 3, (!PrimeQ[#] && IntegerQ[(# - 1)/CarmichaelLambda[#]]) &]

Korselt(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]>1||(n-1)%(f[i, 1]-1), return(0))); 1
p=5;forprime(q=7,1e7,forstep(n=if(p%4==3,p+4,p+2),q-2,4,if(Korselt(n),print1(n", ")));p=q) \\ Charles R Greathouse IV, Jan 27 2012

a(7)-a(40) from Charles R Greathouse IV, Jan 27 2012

A369105 Primes p such that p+2 has only prime factors congruent to -1 modulo 4.

Original entry on oeis.org

5, 7, 17, 19, 29, 31, 41, 47, 61, 67, 79, 97, 101, 127, 131, 137, 139, 149, 197, 199, 211, 229, 241, 251, 269, 277, 281, 307, 359, 379, 397, 421, 439, 461, 467, 487, 499, 521, 569, 571, 587, 601, 617, 619, 631, 641, 647, 691, 709, 719, 727, 751, 757, 787, 809, 811

Offset: 1

Stefano Spezia, Jan 13 2024

nonn

Jones and Zvonkin call these primes BCC primes, where BCC stands for Bujalance, Cirre, and Conder.

Amiram Eldar, Table of n, a(n) for n = 1..10000
E. Bujalance, F. J. Cirre, and M. D. E. Conder, Bounds on the orders of groups of automorphisms of a pseudo-real surface of given genus, Journal of the London Mathematical Society, Volume 101, Issue 2, p. 877-906, (2019).
Gareth A. Jones and Alexander K. Zvonkin, A number-theoretic problem concerning pseudo-real Riemann surfaces, arXiv:2401.00270 [math.NT], 2023. See page 1.

Cf. A001221, A004614, A004767, A027748, A369107, A369108, A369109, A369111.

Select[Prime[Range[150]], PrimeQ[f=First/@FactorInteger[#+2]] == Table[True,{j,PrimeNu[#+2]}] && Mod[f,4] == Table[3, {m,PrimeNu[#+2]}] &]

is1(n) = {my(p = factor(n)[, 1]); for(i = 1, #p, if(p[i] % 4 == 1, return(0))); 1;};
lista(pmax) = forprime(p = 3, pmax, if(is1(p+2), print1(p, ", "))); \\ Amiram Eldar, Jun 03 2024

A094179 Numbers congruent to 3 mod 4 which are divisible only by primes congruent to 3 mod 4.

Original entry on oeis.org

3, 7, 11, 19, 23, 27, 31, 43, 47, 59, 63, 67, 71, 79, 83, 99, 103, 107, 127, 131, 139, 147, 151, 163, 167, 171, 179, 191, 199, 207, 211, 223, 227, 231, 239, 243, 251, 263, 271, 279, 283, 307, 311, 331, 343, 347, 359, 363, 367, 379, 383, 387, 399, 419, 423, 431

Offset: 1

Lekraj Beedassy, May 06 2004

nonn

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

Intersection of A004614 and A004767; intersection of A004614 and A026424.

Cf. A094180.

Select[Range[3,431,4],Union[Mod[Transpose[FactorInteger[#]][[1]],4]] == {3}&] (* Harvey P. Dale, Apr 28 2012 *)

{forstep(n=3,440,4,fac=factor(n);if(vecmin(vector(matsize(fac)[1],j,fac[j,1])%4)==3,print1(n,",")))} \\ Klaus Brockhaus, May 08 2004

is(n)=if(n%4<3, return(0)); Set(factor(n)[,1]%4)==[3] \\ Charles R Greathouse IV, Jan 20 2022

More terms from Klaus Brockhaus, May 08 2004

A242822 Decimal expansion of B. Davis' constant Pi^2/(8*G), a Riesz-Kolmogorov constant, where G is Catalan's constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, May 23 2014

			1.3468852519994065951820075554411...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 7.7 Riesz-Kolmogorov Constants, p. 474.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A001221, A002145, A004614, A006752, A111003, A330890, A377753.

SetDefaultRealField(RealField(100)); R:=RealField(); Pi(R)^2/(8*Catalan(R)); // G. C. Greubel, Aug 25 2018

s:= convert(evalf(Pi^2/(8*Catalan), 140), string):
map(parse, subs("."=NULL, [seq(i, i=s)]))[]; # Alois P. Heinz, May 23 2014

RealDigits[Pi^2/(8*Catalan), 10, 100] // First

default(realprecision, 100); Pi^2/(8*Catalan) \\ G. C. Greubel, Aug 25 2018

(Sum_{n>=0} 1/(2*n + 1)^2) / (Sum_{n>=0} (-1)^n/(2*n + 1)^2) = A111003/A006752.
Equals Product_{k>=1} (1 + 1/A002145(k)^2)/(1 - 1/A002145(k)^2) = A243381 / A243379. - Vaclav Kotesovec, Apr 30 2020
Equals Sum_{q in A004614} 2^A001221(q)/q^2. - R. J. Mathar, Jan 27 2021
Equals 1/A377753. - Hugo Pfoertner, Nov 22 2024

A369107 a(n) is the number of numbers less than or equal to 10^n that are divisible only by primes congruent to 3 mod 4.

Original entry on oeis.org

4, 26, 201, 1680, 14902, 135124, 1243370, 11587149, 108941388, 1031330156, 9816605847

Offset: 1

Stefano Spezia, Jan 13 2024

Gareth A. Jones and Alexander K. Zvonkin, A number-theoretic problem concerning pseudo-real Riemann surfaces, arXiv:2401.00270 [math.NT], 2023. See Table 1 at page 6 and Table 2 at page 7.

Cf. A004614, A004767, A011557, A369105, A369108, A369109, A369111.

a[n_] := Length[Join[{1}, Select[Range[10^n], PrimeQ[f = First/@FactorInteger[#]] == Table[True, {j,PrimeNu[#]}] && Mod[f,4] == Table[3, {m,PrimeNu[#]}] && #<=10^n &]]]; Array[a, 10]

is1(n) = {my(p = factor(n)[, 1]); for(i = 1, #p, if(p[i] % 4 == 1, return(0))); 1;};
lista(nmax) = {my(c = 0, pow = 10, n = 1, nm = nmax + 1); forstep(k = 1, 10^nmax + 1, 2, if(k > pow, print1(c, ", "); pow *= 10; n++; if(n == nm, break)); if(is1(k), c++));} \\ Amiram Eldar, Jun 03 2024

a(11) from Amiram Eldar, Jun 03 2024

A369111 a(n) is the number of primes p less than or equal to 10^n such that p+2 has only prime factors congruent to -1 modulo 4.

Original entry on oeis.org

2, 12, 65, 388, 2708, 19969, 155369, 1250182, 10345920, 87545946, 753285178, 6571105993

Offset: 1

Stefano Spezia, Jan 13 2024

			a(2) = 12 since there are 12 primes p less than or equal to 10^2 such that p+2 has only prime factors congruent to -1 modulo 4 (cf. A369105): 5, 7, 17, 19, 29, 31, 41, 47, 61, 67, 79, 97.

Gareth A. Jones and Alexander K. Zvonkin, A number-theoretic problem concerning pseudo-real Riemann surfaces, arXiv:2401.00270 [math.NT], 2023. See Table 5 at page 14.

Cf. A001221, A004614, A004767, A027748, A369105, A369107, A369108, A369109.

a[n_] := Length[Select[Prime[Range[10^n]], PrimeQ[f=First/@FactorInteger[#+2]] == Table[True, {j,PrimeNu[#+2]}] && Mod[f,4] == Table[3, {m,PrimeNu[#+2]}] && #<=10^n &]]; Array[a,10]

is1(n) = {my(p = factor(n)[, 1]); for(i = 1, #p, if(p[i] % 4 == 1, return(0))); 1;};
lista(nmax) = {my(c = 0, pow = 10, n = 1, nm = nmax + 1); forprime(p = 3, , if(p > pow, print1(c, ", "); pow *= 10; n++; if(n == nm, break)); if(is1(p+2), c++));} \\ Amiram Eldar, Jun 03 2024

a(11)-a(12) from Amiram Eldar, Jun 03 2024

A369564 Powerful numbers whose prime factors are all of the form 4*k + 3.

Original entry on oeis.org

1, 9, 27, 49, 81, 121, 243, 343, 361, 441, 529, 729, 961, 1089, 1323, 1331, 1849, 2187, 2209, 2401, 3087, 3249, 3267, 3481, 3969, 4489, 4761, 5041, 5929, 6241, 6561, 6859, 6889, 8649, 9261, 9747, 9801, 10609, 11449, 11907, 11979, 12167, 14283, 14641, 16129, 16641

Offset: 1

Amiram Eldar, Jan 26 2024

nonn

Closed under multiplication.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to powerful numbers.

Intersection of A001694 and A004614.
Similar sequence: A352492, A369563, A369565, A369566.
Cf. A002145, A013661, A175647, A334426.

q[n_] := n == 1 || AllTrue[FactorInteger[n], Mod[First[#], 4] == 3 && Last[#] > 1 &]; Select[Range[20000], q]

is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 1]%4 != 3 || f[i, 2] == 1, return(0))); 1;}

Sum_{n>=1} 1/a(n) = Product_{primes p == 3 (mod 4)} (1 + 1/(p*(p-1))) = 3*A013661*A334426/(4*A175647) = 1.2161513254... .

A081339 Numbers n such that sigma(n^2) modulo 4 = 1.

Original entry on oeis.org

1, 3, 7, 9, 10, 11, 19, 20, 21, 23, 25, 26, 27, 30, 31, 33, 34, 40, 43, 47, 49, 52, 57, 58, 59, 60, 63, 65, 67, 68, 69, 70, 71, 74, 75, 77, 78, 79, 80, 81, 82, 83, 85, 90, 93, 99, 102, 103, 104, 106, 107, 110, 116, 120, 121, 122, 127, 129, 131, 133, 136, 139, 140, 141, 145

Offset: 1

Benoit Cloitre, Apr 20 2003

nonn

Numbers n such that the sum of exponents of primes == 1 (mod 4) in the prime factorization of n is not congruent to n mod 2. - Robert Israel, Jan 22 2017

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000203, A065764, A081325.

Contains A004614.

filter:= proc(n) local F, t;
  F:= select(t -> t[1] mod 4 = 1, ifactors(n)[2]);
  (add(t[2],t=F) - n) mod 2 = 1;
end proc:
select(filter, [$1..1000]); # Robert Israel, Jan 22 2017

Select[Range[150],Mod[DivisorSigma[1,#^2],4]==1&] (* Harvey P. Dale, Apr 07 2012 *)

isok(n) = (sigma(n^2) % 4) == 1; \\ Michel Marcus, Jan 22 2017

cp's OEIS Frontend

A078911 Let r+i*s be the sum of the distinct first-quadrant Gaussian integers dividing n; sequence gives s values.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A103223 Imaginary part of the totient function phi(n) for Gaussian integers. See A103222 for the real part and A103224 for the norm.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A185321 Carmichael numbers congruent to 3 modulo 4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A369105 Primes p such that p+2 has only prime factors congruent to -1 modulo 4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A094179 Numbers congruent to 3 mod 4 which are divisible only by primes congruent to 3 mod 4.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A242822 Decimal expansion of B. Davis' constant Pi^2/(8*G), a Riesz-Kolmogorov constant, where G is Catalan's constant.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A369107 a(n) is the number of numbers less than or equal to 10^n that are divisible only by primes congruent to 3 mod 4.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A369111 a(n) is the number of primes p less than or equal to 10^n such that p+2 has only prime factors congruent to -1 modulo 4.

Views