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 1-3 of 3 results.

A289310 Let f be the multiplicative function satisfying f(p^k) = (1 + p*I)^k for any prime p and k > 0 (where I^2 = -1); a(n) = the real part of f(n).

Original entry on oeis.org

1, 1, 1, -3, 1, -5, 1, -11, -8, -9, 1, -15, 1, -13, -14, -7, 1, -20, 1, -23, -20, -21, 1, -5, -24, -25, -26, -31, 1, -30, 1, 41, -32, -33, -34, 0, 1, -37, -38, -1, 1, -40, 1, -47, -38, -45, 1, 65, -48, -44, -50, -55, 1, 10, -54, 3, -56, -57, 1, 10, 1, -61, -50
Offset: 1

Views

Author

Rémy Sigrist, Jul 02 2017

Keywords

Comments

See A289311 for the imaginary part of f.
See A289320 for the square of the norm of f.
a(p) = 1 for any prime p.
If a(n) = 0, then a(n^(2*k-1)) = 0 and A289311(n^(2*k)) = 0 for any k > 0.
a(n) = 0 iff Sum_{i=1..k} ( arctan(p_i) * e_i ) = Pi/2 * (2*j + 1) for some integer j (where Product_{i=1..k} p_i^e_i is the prime factorization of n).
a(n) = 0 for n = 36, 3969, 13608, 46656, 1500282, 5143824, 6718383, ...
As a(36) = 0 and 36 = 2^2 * 3^3, we have arctan(2)*2 + arctan(3)*2 = Pi/2 * (2*j + 1) (with j = 1).
If |a(n)| = |A289311(n)|, then |a(n^(2k-1))| = |A289311(n^(2k-1))| for any k > 0.
|a(n)| = |A289311(n)| iff Sum_{i=1..k} ( arctan(p_i) * e_i ) = Pi/4 * (2*j + 1) for some integer j (where Product_{i=1..k} p_i^e_i is the prime factorization of n).
|a(n)| = |A289311(n)| for n = 6, 63, 216, 2268, 7776, 23814, 81648, 106641, 250047, 279936, 312273, 857304, ...
As |a(63)| = |A289311(63)| and 63 = 3^2 * 7, we have arctan(3)*2 + arctan(7) = Pi/4 * (2*j + 1) (with j=1).
The scatterplot of this sequence vs A289311 is interesting (see Links section).

Examples

			f(12) = f(2^2 * 3) = (1 + 2*I)^2 * (1 + 3*I) = -15 - 5*I, hence a(12) = -15.

Links

Crossrefs

Cf. A289311, A289320.

Programs

  • Mathematica
    Array[Re[Times @@ Map[(1 + #1 I)^#2 & @@ # &, FactorInteger@ #]] &, 63] (* Michael De Vlieger, Jul 03 2017 *)
  • PARI
    a(n) = my (f=factor(n)); real (prod(i=1, #f~, (1 + f[i,1]*I) ^ f[i,2]))

A289311 Let f be the multiplicative function satisfying f(p^k) = (1 + p*I)^k for any prime p and k > 0 (where I^2 = -1); a(n) = the imaginary part of f(n).

Original entry on oeis.org

0, 2, 3, 4, 5, 5, 7, -2, 6, 7, 11, -5, 13, 9, 8, -24, 17, -10, 19, -11, 10, 13, 23, -35, 10, 15, -18, -17, 29, -20, 31, -38, 14, 19, 12, -50, 37, 21, 16, -57, 41, -30, 43, -29, -34, 25, 47, -45, 14, -38, 20, -35, 53, -70, 16, -79, 22, 31, 59, -80, 61, 33, -50
Offset: 1

Views

Author

Rémy Sigrist, Jul 02 2017

Keywords

Comments

See A289310 for the real part of f and additional comments.
See A289320 for the square of the norm of f.
a(p) = p for any prime p.
The numbers 4 and 2700 are composite fixed points.
If a(n) = 0, then a(n^k) = 0 for any k > 0.
a(n) = 0 iff Sum_{i=1..k} ( arctan(p_i) * e_i ) = Pi * j for some integer j (where Product_{i=1..k} p_i^e_i is the prime factorization of n).
a(n) = 0 for n = 1, 378, 1296, 142884, 489888, 639846, 1679616, 1873638, ...
As a(378) = 0 and 378 = 2 * 3^3 * 7, we have arctan(2) + arctan(3)*3 + arctan(7) = j * Pi (with j = 2).

Examples

			f(12) = f(2^2 * 3) = (1 + 2*I)^2 * (1 + 3*I) = -15 - 5*I, hence a(12) = -5.

Links

Crossrefs

Cf. A289310, A289320.

Programs

  • Mathematica
    Array[Im[Times @@ Map[(1 + #1 I)^#2 & @@ # &, FactorInteger@ #]] - Boole[# == 1] &, 63] (* Michael De Vlieger, Jul 03 2017 *)
  • PARI
    a(n) = my (f=factor(n)); imag (prod(i=1, #f~, (1 + f[i,1]*I) ^ f[i,2]))

A351475 Multiplicative with a(prime(k)^e) = k^2 + e^2 for any k, e > 0.

Original entry on oeis.org

1, 2, 5, 5, 10, 10, 17, 10, 8, 20, 26, 25, 37, 34, 50, 17, 50, 16, 65, 50, 85, 52, 82, 50, 13, 74, 13, 85, 101, 100, 122, 26, 130, 100, 170, 40, 145, 130, 185, 100, 170, 170, 197, 130, 80, 164, 226, 85, 20, 26, 250, 185, 257, 26, 260, 170, 325, 202, 290, 250
Offset: 1

Views

Author

Rémy Sigrist, Feb 12 2022

Keywords

Comments

This sequence gives the norm of the function f defined in A351464-A351465.

Examples

			For n = 42:
- 42 = 2 * 3 * 7 = prime(1)^1 * prime(2)^1 * prime(4)^1,
- a(42) = (1^2 + 1^2) * (2^2 + 1^2) * (4^2 + 1^2) = 170.

Crossrefs

Cf. A351464, A351465, A289320.

Programs

  • Maple
    a:= proc(n) option remember; uses numtheory;
      mul(pi(i[1])^2+i[2]^2, i=ifactors(n)[2])
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Feb 15 2022
  • Mathematica
    f[p_, e_] := PrimePi[p]^2 + e^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 15 2022 *)
  • PARI
    a(n) = { my (f=factor(n), p=f[, 1]~, e=f[, 2]~); prod (k=1, #p, primepi(p[k])^2 + e[k]^2) }

Formula

a(n) = A351464(n)^2 + A351465(n)^2.
Showing 1-3 of 3 results.

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