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 1-5 of 5 results.

A078908 Let r+i*s be the sum, with multiplicity, of the first-quadrant Gaussian primes dividing n; sequence gives r values (with a(1) = 0).

Original entry on oeis.org

0, 2, 3, 4, 3, 5, 7, 6, 6, 5, 11, 7, 5, 9, 6, 8, 5, 8, 19, 7, 10, 13, 23, 9, 6, 7, 9, 11, 7, 8, 31, 10, 14, 7, 10, 10, 7, 21, 8, 9, 9, 12, 43, 15, 9, 25, 47, 11, 14, 8, 8, 9, 9, 11, 14, 13, 22, 9, 59, 10, 11, 33, 13, 12, 8, 16, 67, 9, 26, 12, 71, 12, 11, 9, 9, 23, 18, 10, 79, 11, 12, 11
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, A078909, A078910, A078911, A080088, A080089.

Programs

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

Extensions

More terms and information from Vladeta Jovovic, Jan 27 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

A078930 Let r+i*s be the sum of the distinct first-quadrant Gaussian integers dividing n; sequence gives r+s values.

Original entry on oeis.org

1, 5, 4, 13, 12, 20, 8, 29, 13, 56, 12, 52, 24, 40, 48, 61, 28, 65, 20, 144, 32, 60, 24, 116, 81, 112, 40, 104, 44, 224, 32, 125, 48, 136, 96, 169, 52, 100, 96, 320, 60, 160, 44, 156, 156, 120, 48, 244, 57, 369, 112, 288, 72, 200, 144, 232, 80, 212, 60, 576, 84, 160, 104
Offset: 1

Views

Author

Vladeta Jovovic, Jan 11 2003

Keywords

Comments

A078910(n)-A078911(n) = A000203(n).
Also equal to the sum of all divisors of n (in Z[i]) having positive real part. - M. F. Hasler, Nov 22 2007

Links

Programs

  • PARI
    A078930(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) = A078910(n)+A078911(n).
a(n) = A000203(n)+2*A078911(n). - M. F. Hasler, Nov 22 2007

A132994 Partial sums of A078930.

Original entry on oeis.org

1, 6, 10, 23, 35, 55, 63, 92, 105, 161, 173, 225, 249, 289, 337, 398, 426, 491, 511, 655, 687, 747, 771, 887, 968, 1080, 1120, 1224, 1268, 1492, 1524, 1649, 1697, 1833, 1929, 2098, 2150, 2250, 2346, 2666, 2726, 2886, 2930, 3086, 3242, 3362, 3410, 3654, 3711
Offset: 1

Views

Author

M. F. Hasler, Nov 22 2007

Keywords

Comments

a(10^5)=17924657155 ; a(10^8)=? (Problem 153 on projecteuler.net).

Examples

			a(5) = 35 = A078930(1)+A078930(2)+A078930(3)+A078930(4)+A078930(5).

Links

Crossrefs

Cf. A078930, A078910, A078911.

Programs

Showing 1-5 of 5 results.

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

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