cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A078458 Total number of factors in a factorization of n into Gaussian primes.

Original entry on oeis.org

0, 2, 1, 4, 2, 3, 1, 6, 2, 4, 1, 5, 2, 3, 3, 8, 2, 4, 1, 6, 2, 3, 1, 7, 4, 4, 3, 5, 2, 5, 1, 10, 2, 4, 3, 6, 2, 3, 3, 8, 2, 4, 1, 5, 4, 3, 1, 9, 2, 6, 3, 6, 2, 5, 3, 7, 2, 4, 1, 7, 2, 3, 3, 12, 4, 4, 1, 6, 2, 5, 1, 8, 2, 4, 5, 5, 2, 5, 1, 10, 4, 4, 1, 6, 4, 3, 3, 7, 2, 6, 3, 5, 2, 3, 3, 11, 2, 4, 3, 8, 2, 5, 1, 8
Offset: 1

Views

Author

N. J. A. Sloane, Jan 11 2003

Keywords

Comments

a(n)+1 is also the total number of factors in a factorization of n+n*i into Gaussian primes. - Jason Kimberley, Dec 17 2011
Record high values are at a(2^k) = 2*k for k = 0, 1, 2, ... . - Bill McEachen, Oct 11 2022

Examples

			2 = (1+i)*(1-i), so a(2) = 2; 9 = 3*3, so a(9) = 2.
a(1006655265000) = a(2^3*3^2*5^4*7^5*11^3) = 3*a(2)+2*a(3)+4*a(5)+5*a(7)+3*a(11) = 3*2+2*1+4*2+5*1+3*1 = 24. - _Vladeta Jovovic_, Jan 20 2003

Links

Crossrefs

Cf. A078908-A078911, A007814, A065339, A083025.
Cf. A239626, A239627 (including units).
Equivalent of arithmetic functions in the ring of Gaussian integers (the corresponding functions in the ring of integers are in the parentheses): A062327 ("d", A000005), A317797 ("sigma", A000203), A079458 ("phi", A000010), A227334 ("psi", A002322), A086275 ("omega", A001221), this sequence ("Omega", A001222), A318608 ("mu", A008683).
Equivalent in the ring of Eisenstein integers: A319444.

Programs

  • Mathematica
    Join[{0}, Table[f = FactorInteger[n, GaussianIntegers -> True]; cnt = Total[Transpose[f][[2]]]; If[MemberQ[{-1, I, -I}, f[[1, 1]]], cnt--]; cnt, {n, 2, 100}]] (* T. D. Noe, Mar 31 2014 *)
a[n_]:=PrimeOmega[n, GaussianIntegers -> True]; Array[a,104] (* Stefano Spezia, Sep 29 2024 *)
  • PARI
    a(n)=my(f=factor(n)); sum(i=1,#f~,if(f[i,1]%4==3,1,2)*f[i,2]) \\ Charles R Greathouse IV, Mar 31 2014

Formula

Fully additive with a(p)=2 if p=2 or p mod 4=1 and a(p)=1 if p mod 4=3. - Vladeta Jovovic, Jan 20 2003
a(n) depends on the number of primes of the forms 4k+1 (A083025) and 4k-1 (A065339) and on the highest power of 2 dividing n (A007814): a(n) = 2*A007814(n) + 2*A083025(n) + A065339(n). - T. D. Noe, Jul 14 2003

Extensions

More terms from Vladeta Jovovic, Jan 12 2003

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Jan 11 2003

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

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

Extensions

More terms from Vladeta Jovovic, Jan 11 2003

A078909 Let r+i*s be the sum, with multiplicity, of the first-quadrant Gaussian primes dividing n; sequence gives s values.

Original entry on oeis.org

0, 2, 0, 4, 3, 2, 0, 6, 0, 5, 0, 4, 5, 2, 3, 8, 5, 2, 0, 7, 0, 2, 0, 6, 6, 7, 0, 4, 7, 5, 0, 10, 0, 7, 3, 4, 7, 2, 5, 9, 9, 2, 0, 4, 3, 2, 0, 8, 0, 8, 5, 9, 9, 2, 3, 6, 0, 9, 0, 7, 11, 2, 0, 12, 8, 2, 0, 9, 0, 5, 0, 6, 11, 9, 6, 4, 0, 7, 0, 11, 0, 11, 0, 4, 8, 2, 7, 6, 13, 5, 5, 4, 0, 2, 3, 10, 13, 2, 0
Offset: 1

Views

Author

N. J. A. Sloane, Jan 11 2003

Keywords

Comments

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.
The sequence is fully additive.

Examples

			5 factors into the product of the primes 1+2*i, 1-2*i, but the first-quadrant associate of 1-2*i is i*(1-2*i) = 2+i, so r+i*s = 1+2*i + 2+i = 3+3*i. Therefore a(5) = 3.

Links

Crossrefs

Cf. A078458, A078908, A078910, A078911, A080088, A080089.

Programs

  • Mathematica
    a[n_] := Module[{f = FactorInteger[n, GaussianIntegers->True]}, p = f[[;;,1]]; e = f[[;;,2]]; Im[Plus @@ ((If[Abs[#] == 1, 0, #]& /@ p) * e)]]; Array[a, 100] (* Amiram Eldar, Feb 28 2020 *)

Extensions

More terms and further information from Vladeta Jovovic, Jan 27 2003

A078910 Let r+i*s be the sum of the distinct first-quadrant Gaussian integers dividing n; sequence gives r values.

Original entry on oeis.org

1, 4, 4, 10, 9, 16, 8, 22, 13, 37, 12, 40, 19, 32, 36, 46, 23, 52, 20, 93, 32, 48, 24, 88, 56, 77, 40, 80, 37, 148, 32, 94, 48, 95, 72, 130, 45, 80, 76, 205, 51, 128, 44, 120, 117, 96, 48, 184, 57, 231, 92, 193, 63, 160, 108, 176, 80, 151, 60, 372, 73, 128, 104, 190, 176
Offset: 1

Views

Author

N. J. A. Sloane, Jan 11 2003

Keywords

Comments

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.

Examples

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

Links

Crossrefs

Cf. A000203, A078911, A078458, A078908, A078909, A078930.
Cf. A062327 for the number of first quadrant divisors of n.

Programs

  • Mathematica
    Table[Re[Plus@@Divisors[n, GaussianIntegers -> True]], {n, 65}] (* Alonso del Arte, Jan 24 2012 *)
  • PARI
    A078910(n,S=[])=sigma(n)+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

Formula

a(n) = A078911(n)+A000203(n). - Vladeta Jovovic, Jan 11 2003

Extensions

More terms from Vladeta Jovovic, Jan 11 2003

A080088 Let r+i*s be the sum of the first-quadrant Gaussian primes dividing prime(n); sequence gives r values.

Original entry on oeis.org

2, 3, 3, 7, 11, 5, 5, 19, 23, 7, 31, 7, 9, 43, 47, 9, 59, 11, 67, 71, 11, 79, 83, 13, 13, 11, 103, 107, 13, 15, 127, 131, 15, 139, 17, 151, 17, 163, 167, 15, 179, 19, 191, 19, 15, 199, 211, 223, 227, 17, 21, 239, 19, 251, 17, 263, 23, 271, 23, 21, 283, 19, 307, 311, 25, 25
Offset: 1

Views

Author

Vladeta Jovovic, Jan 27 2003

Keywords

Links

Crossrefs

Cf. A078908, A078909, A080089.

Programs

  • Mathematica
    s[n_] := Module[{f = FactorInteger[n, GaussianIntegers->True]}, p = f[[;;,1]]; e = f[[;;,2]]; Re[Plus @@ ((If[Abs[#]==1, 0, #]& /@ p) * e)]]; s /@ Select[ Range[100], PrimeQ] (* Amiram Eldar, Feb 28 2020 *)

A080089 Let r+i*s be the sum of the first-quadrant Gaussian primes dividing prime(n); sequence gives s values.

Original entry on oeis.org

2, 0, 3, 0, 0, 5, 5, 0, 0, 7, 0, 7, 9, 0, 0, 9, 0, 11, 0, 0, 11, 0, 0, 13, 13, 11, 0, 0, 13, 15, 0, 0, 15, 0, 17, 0, 17, 0, 0, 15, 0, 19, 0, 19, 15, 0, 0, 0, 0, 17, 21, 0, 19, 0, 17, 0, 23, 0, 23, 21, 0, 19, 0, 0, 25, 25, 0, 25, 0, 23, 25, 0, 0, 25, 0, 0, 27, 25, 21, 23, 0, 29, 0, 29, 0, 0, 27, 25
Offset: 1

Views

Author

Vladeta Jovovic, Jan 27 2003

Keywords

Links

Crossrefs

Cf. A078908, A078909, A080088.

Programs

  • Mathematica
    s[n_] := Module[{f = FactorInteger[n, GaussianIntegers->True]}, p = f[[;;,1]]; e = f[[;;,2]]; Im[Plus @@ ((If[Abs[#]==1, 0, #]& /@ p) * e)]]; s /@ Select[ Range[100], PrimeQ] (* Amiram Eldar, Feb 28 2020 *)
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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