A316658 -id:A316658 - cp's OEIS frontend

A316657 For any n >= 0 with base-5 expansion Sum_{k=0..w} d_k * 5^k, let f(n) = Sum_{k=0..w} [d_k > 0] * (2 + i)^k * i^(d_k - 1) (where [] is an Iverson bracket and i denotes the imaginary unit); a(n) equals the real part of f(n).

0, 1, 0, -1, 0, 2, 3, 2, 1, 2, -1, 0, -1, -2, -1, -2, -1, -2, -3, -2, 1, 2, 1, 0, 1, 3, 4, 3, 2, 3, 5, 6, 5, 4, 5, 2, 3, 2, 1, 2, 1, 2, 1, 0, 1, 4, 5, 4, 3, 4, -4, -3, -4, -5, -4, -2, -1, -2, -3, -2, -5, -4, -5, -6, -5, -6, -5, -6, -7, -6, -3, -2, -3, -4, -3

Offset: 0

Rémy Sigrist, Jul 09 2018

See A316658 for the imaginary part of f.
See A316707 for the square of the modulus of f.
The function f has nice fractal features (see scatterplot in Links section).
It appears that f defines a bijection from the nonnegative integers to the Gaussian integers.

Rémy Sigrist, Table of n, a(n) for n = 0..15624
Rémy Sigrist, Colored scatterplot of (a(n), A316658(n)) for n=0..5^8-1 (where the hue is function of n)
Wikipedia, Gaussian integer

Cf. A139011, A316658, A316707.

a[n_] := Module[{d, z}, d = IntegerDigits[n, 5] // Reverse; z = Sum[ If[d[[i]]>0, (2+I)^(i-1)*I^(d[[i]]-1), 0], {i, 1, Length[d]}]; Re[z]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 06 2021, after PARI code *)

a(n) = my (d=Vecrev(digits(n, 5)), z=sum(i=1, #d, if (d[i], (2+I)^(i-1) * I^(d[i]-1), 0))); real(z)

a(5^n) = A139011(n) for any n >= 0.

a(3 * 5^n) = -A139011(n) for any n >= 0.

A334493 a(n) is the "w" part of f(n) = Sum_{k>=0, d_k>0} (1+w)^(d_k-1) * (3+w)^k where Sum_{k>=0} d_k * 7^k is the base 7 representation of n and w = -1/2 + sqrt(-3)/2 is a primitive cube root of unity; sequence A334492 gives "real" parts.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, May 03 2020

For any Eisenstein integer z = u + v*w (where u and v are integers), we call u the "real" part of z and v the "w" part of z.
This sequence has connections with A316658; here we work with Eisenstein integers, there with Gaussian integers.
It appears that f defines a bijection from the nonnegative integers to the Eisenstein integers.

			The following diagram depicts f(n) for n = 0..13:
            "w" axis
                \
           .     .     .     .     .     .     .     .
                  \              10     9
                   \
        .     .     .     .     .     .     .     .
                   3 \   2    11     7     8
                      \
           ._____._____._____._____._____._____._____. "real" axis
                4     0 \   1    12    13
                         \
        .     .     .     .     .     .     .     .
                   5     6 \
- f(9) = 4 + 2*w, hence a(9) = 2.

Rémy Sigrist, Table of n, a(n) for n = 0..16806
Rémy Sigrist, PARI program for A334493
Wikipedia, Eisenstein integer

Cf. A307012 (equivalent coordinate for a counterclockwise spiral), A316658, A334492.

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A348653 For any nonnegative number n with base-13 expansion Sum_{k >= 0} d_k13^k, a(n) is the imaginary part of Sum_{k >= 0} g(d_k)(3+2i)^k where g(0) = 0, and g(1+u+3v) = (1+ui)i^v for any u = 0..2 and v = 0..3 (where i denotes the imaginary unit); see A348652 for the real part.

Original entry on oeis.org

0, 0, 1, 2, 1, 1, 1, 0, -1, -2, -1, -1, -1, 2, 2, 3, 4, 3, 3, 3, 2, 1, 0, 1, 1, 1, 5, 5, 6, 7, 6, 6, 6, 5, 4, 3, 4, 4, 4, 8, 8, 9, 10, 9, 9, 9, 8, 7, 6, 7, 7, 7, 3, 3, 4, 5, 4, 4, 4, 3, 2, 1, 2, 2, 2, 1, 1, 2, 3, 2, 2, 2, 1, 0, -1, 0, 0, 0, -1, -1, 0, 1, 0, 0

Offset: 0

Rémy Sigrist, Oct 27 2021

The function f defines a bijection from the nonnegative integers to the Gaussian integers.
The following diagram depicts g(d) for d = 0..12:
                      |
                      |    +
                      |     3
                      |
            +    +    +    +
             6    5   |4    2
                      |
         --------+----+----+-------
                  7   |0    1
                      |
                 +    +    +    +
                  8   |10   11   12
                      |
                 +    |
                  9   |

Rémy Sigrist, Table of n, a(n) for n = 0..2197
Stephen K. Lucas, Base 2 + i with digit set {0, +/-1, +/-i}, ResearchGate (October 2021).
Rémy Sigrist, Colored representation of f for n = 0..13^5-1 in the complex plane (the hue is function of n)

See A316658 for a similar sequence.

Cf. A188982, A348652.

g(d) = { if (d==0, 0, (1+I*((d-1)%3))*I^((d-1)\3)) }
a(n) = imag(subst(Pol([g(d)|d<-digits(n, 13)]), 'x, 3+2*I))

abs(a(13^k)) = A188982(k).

A316707 For any n >= 0 with base-5 expansion Sum_{k=0..w} d_k * 5^k, let f(n) = Sum_{k=0..w} [d_k > 0] * (2 + i)^k * i^(d_k - 1) (where [] is an Iverson bracket and i denotes the imaginary unit); a(n) equals the square of the modulus of f(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 10, 8, 2, 4, 5, 4, 10, 8, 2, 5, 2, 4, 10, 8, 5, 8, 2, 4, 10, 25, 32, 34, 20, 18, 50, 61, 61, 41, 41, 40, 45, 53, 37, 29, 10, 13, 17, 9, 5, 20, 29, 25, 13, 17, 25, 18, 32, 34, 20, 20, 17, 29, 25, 13, 50, 41, 61, 61, 41, 40, 29, 45, 53, 37, 10

Offset: 0

Rémy Sigrist, Jul 11 2018

See A316657 for the real part of f and additional comments.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A316657, A316658.

a(n) = my (d=Vecrev(digits(n, 5)), z=sum(i=1, #d, if (d[i], (2+I)^(i-1) * I^(d[i]-1), 0))); real(z)^2 + imag(z)^2

a(n) = A316657(n)^2 + A316658(n)^2.
a(5 * n) = 5 * a(n) for any n >= 0.
a(5^k) = 5^k for any k >= 0.

A348354 The base-5 expansion of a(n) is obtained by replacing 1's, 2's, 3's and 4's by 3's, 4's, 1's and 2's, respectively, in the base-5 expansion of n.

Original entry on oeis.org

0, 3, 4, 1, 2, 15, 18, 19, 16, 17, 20, 23, 24, 21, 22, 5, 8, 9, 6, 7, 10, 13, 14, 11, 12, 75, 78, 79, 76, 77, 90, 93, 94, 91, 92, 95, 98, 99, 96, 97, 80, 83, 84, 81, 82, 85, 88, 89, 86, 87, 100, 103, 104, 101, 102, 115, 118, 119, 116, 117, 120, 123, 124, 121

Offset: 0

Rémy Sigrist, Oct 14 2021

This sequence is a self-inverse permutation of the nonnegative integers.
It is possible to build a similar sequence for any fixed base b > 1 and any permutation p of {1, ..., b-1}.
This sequence is interesting as it satisfies f(a(n)) = -f(n), where f(n) = (A316657(n), A316658(n)).

			The first terms, in decimal and in base 5, are:
  n   a(n)  q(n)  q(a(n))
  --  ----  ----  -------
   0     0     0        0
   1     3     1        3
   2     4     2        4
   3     1     3        1
   4     2     4        2
   5    15    10       30
   6    18    11       33
   7    19    12       34
   8    16    13       31
   9    17    14       32
  10    20    20       40

See A004488, A048647 and A348355 for similar sequences.

Cf. A316657, A316658.

a[n_] := With[{d = {0, 3, 4, 1, 2}}, FromDigits[d[[IntegerDigits[n, 5] + 1]], 5]]; Array[a, 64, 0] (* Amiram Eldar, Oct 16 2021 *)

a(n, p=[3,4,1,2]) = fromdigits(apply(d -> if (d, p[d], 0), digits(n, #p+1)), #p+1)

A316657(n) + A316657(a(n)) = 0.

A316658(n) + A316658(a(n)) = 0.

cp's OEIS Frontend

A316657 For any n >= 0 with base-5 expansion Sum_{k=0..w} d_k * 5^k, let f(n) = Sum_{k=0..w} [d_k > 0] * (2 + i)^k * i^(d_k - 1) (where [] is an Iverson bracket and i denotes the imaginary unit); a(n) equals the real part of f(n).

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A334493 a(n) is the "w" part of f(n) = Sum_{k>=0, d_k>0} (1+w)^(d_k-1) * (3+w)^k where Sum_{k>=0} d_k * 7^k is the base 7 representation of n and w = -1/2 + sqrt(-3)/2 is a primitive cube root of unity; sequence A334492 gives "real" parts.

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A348653 For any nonnegative number n with base-13 expansion Sum_{k >= 0} d_k*13^k, a(n) is the imaginary part of Sum_{k >= 0} g(d_k)*(3+2*i)^k where g(0) = 0, and g(1+u+3*v) = (1+u*i)*i^v for any u = 0..2 and v = 0..3 (where i denotes the imaginary unit); see A348652 for the real part.

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A316707 For any n >= 0 with base-5 expansion Sum_{k=0..w} d_k * 5^k, let f(n) = Sum_{k=0..w} [d_k > 0] * (2 + i)^k * i^(d_k - 1) (where [] is an Iverson bracket and i denotes the imaginary unit); a(n) equals the square of the modulus of f(n).

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A348354 The base-5 expansion of a(n) is obtained by replacing 1's, 2's, 3's and 4's by 3's, 4's, 1's and 2's, respectively, in the base-5 expansion of n.

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A348653 For any nonnegative number n with base-13 expansion Sum_{k >= 0} d_k13^k, a(n) is the imaginary part of Sum_{k >= 0} g(d_k)(3+2i)^k where g(0) = 0, and g(1+u+3v) = (1+ui)i^v for any u = 0..2 and v = 0..3 (where i denotes the imaginary unit); see A348652 for the real part.