A065620 -id:A065620 - cp's OEIS frontend

A325572 Numbers n that have divisor d > 1 such that A048720(A065621(d),n/d) = n.

2, 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 21, 22, 24, 26, 28, 30, 32, 33, 34, 35, 36, 38, 40, 42, 44, 45, 46, 48, 49, 50, 52, 54, 56, 58, 60, 62, 64, 65, 66, 68, 70, 72, 74, 75, 76, 78, 80, 82, 84, 86, 88, 90, 92, 93, 94, 96, 98, 100, 102, 104, 105, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 129, 130, 132, 133

Offset: 1

Antti Karttunen, May 10 2019

nonn

Equally, numbers n such that there exists natural numbers t > 1 and u >= 1, for which A048720(t,u) = n and A065620(t)*u = n.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A048720, A065620, A065621, A325570 (complement).

Union of A005843 (without zero) and A325573 (odd terms).

A048720(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A065621(n) = bitxor(n-1,n+n-1);
isA325572(n) = fordiv(n,d,if(A048720(A065621(n/d),d)==n,return(d

A348690 For any nonnegative number n with binary expansion Sum_{k >= 0} b_k * 2^k, a(n) is the real part of f(n) = Sum_{k >= 0} b_k * (i^Sum_{j = 0..k-1} b_j) * (1+i)^k (where i denotes the imaginary unit); sequence A348691 gives the imaginary part.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Oct 29 2021

The function f defines a bijection from the nonnegative integers to the Gaussian integers.
The function f has similarities with A065620; here the nonzero digits in base 1+i cycle through powers of i, there nonzero digits in base 2 cycle through powers of -1.
If we replace 1's in binary expansions by powers of i from left to right (rather than right to left as here), then we obtain the Lévy C curve (A332251, A332252).

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Chandler Davis and Donald Knuth, Number representations and Dragon Curves I, Journal of Recreational Mathematics, volume 3, number 2 (April 1970), pages 66-81. Reprinted in Donald E. Knuth, Selected Papers on Fun and Games, CSLI Publications, 2011, pages 571-614.
Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 66-81, and number 3, July 1970, pages 133-149. [Cached copy, with permission]
Rémy Sigrist, Colored representation of f(n) for n < 2^18 in the complex plane (the hue is function of n)
Rémy Sigrist, Colored representation of f(n) for n < 2^18 in the complex plane (the color is function of A000120(n) mod 4)
Rémy Sigrist, Colored representation of f(n) for n < 2^18 in the complex plane (the color is function of the binary length of n, A070939(n))

See A332251 for a similar sequence.

Cf. A000120, A065620, A146559, A348691, A332251, A332252.

a(n) = { my (v=0, o=0, x); while (n, n-=2^x=valuation(n, 2); v+=I^o * (1+I)^x; o++); real(v) }

a(2^k) = A146559(k) for any k >= 0.

A348691 For any nonnegative number n with binary expansion Sum_{k >= 0} b_k * 2^k, a(n) is the imaginary part of f(n) = Sum_{k >= 0} b_k * (i^Sum_{j = 0..k-1} b_j) * (1+i)^k (where i denotes the imaginary unit); sequence A348690 gives the real part.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Oct 29 2021

The function f defines a bijection from the nonnegative integers to the Gaussian integers.
The function f has similarities with A065620; here the nonzero digits in base 1+i cycle through powers of i, there nonzero digits in base 2 cycle through powers of -1.
If we replace 1's in binary expansions by powers of i from left to right (rather than right to left as here), then we obtain the Lévy C curve (A332251, A332252).

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Chandler Davis and Donald Knuth, Number representations and Dragon Curves I, Journal of Recreational Mathematics, volume 3, number 2 (April 1970), pages 66-81. Reprinted in Donald E. Knuth, Selected Papers on Fun and Games, CSLI Publications, 2011, pages 571-614.
Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 66-81, and number 3, July 1970, pages 133-149. [Cached copy, with permission]
Rémy Sigrist, Colored representation of f(n) for n < 2^18 in the complex plane (the hue is function of n)
Rémy Sigrist, Colored representation of f(n) for n < 2^18 in the complex plane (the color is function of A000120(n) mod 4)
Rémy Sigrist, Colored representation of f(n) for n < 2^18 in the complex plane (the color is function of the binary length of n, A070939(n))

See A332251, A332252 for a similar sequence.

Cf. A000120, A009545, A065620, A070939, A348690.

a(n) = { my (v=0, o=0, x); while (n, n-=2^x=valuation(n, 2); v+=I^o * (1+I)^x; o++); imag(v) }

a(2^k) = A009545(k) for any k >= 0.

A333773 Replace 2's with (-1)'s in ternary representation of n and sum nonzero terms with alternating signs.

Original entry on oeis.org

0, 1, -1, 3, 2, 4, -3, -4, -2, 9, 8, 10, 6, 7, 5, 12, 13, 11, -9, -10, -8, -12, -11, -13, -6, -5, -7, 27, 26, 28, 24, 25, 23, 30, 31, 29, 18, 19, 17, 21, 20, 22, 15, 14, 16, 36, 37, 35, 39, 38, 40, 33, 32, 34, -27, -28, -26, -30, -29, -31, -24, -23, -25, -36

Offset: 0

Rémy Sigrist, Apr 05 2020

This sequence is a variant of A117966, and shares features with A065620.

Every integer appears exactly once in this sequence.

			For n = 97:
- 97 = 3^4 + 3^2 + 2*3^1 + 3^0,
- hence a(97) = 3^4 - 3^2 + (-1)*3^1 - 3^0 = 68.

Rémy Sigrist, Table of n, a(n) for n = 0..6561

Cf. A004488, A024023, A065620, A117966, A132141, A157671, A160384, A333780.

a(n) = { my (v=0, t=Vecrev(digits(n,3))); for (k=1, #t, if (t[k]==1, v=+3^(k-1)-v, t[k]==2, v=-3^(k-1)-v)); v }

a(3*n)   = 3*a(n).
a(3*n+1) = 3*a(n) + (-1)^A160384(n).
a(3*n+2) = 3*a(n) - (-1)^A160384(n).
Sum_{k=0..n} a(k) >= 0 with equality iff n belongs to A024023.
a(n) > 0 iff n belongs to A132141.
a(n) < 0 iff n belongs to A157671.
a(A004488(n)) = -a(n).

A355624 a(0) = 0, and for any n > 0, a(3n) = 3a(n), a(3n+1) = 1-3a(n), a(3n+2) = 2-3a(n).

Original entry on oeis.org

0, 1, 2, 3, -2, -1, 6, -5, -4, 9, -8, -7, -6, 7, 8, -3, 4, 5, 18, -17, -16, -15, 16, 17, -12, 13, 14, 27, -26, -25, -24, 25, 26, -21, 22, 23, -18, 19, 20, 21, -20, -19, 24, -23, -22, -9, 10, 11, 12, -11, -10, 15, -14, -13, 54, -53, -52, -51, 52, 53, -48, 49

Offset: 0

Rémy Sigrist, Jul 14 2022

This sequence establishes a bijection from the nonnegative integers (N) to the integers (Z).
This sequence is to base 3 what A065620 is to base 2.
To compute a(n): write n as a sum of terms of A038754 with distinct 3-adic valuations and take the alternating sum.

			For n = 107:
  107 = 3^4 + 2*3^2 + 2*3^1 + 2*3^0,
  so a(107) = -3^4 + 2*3^2 - 2*3^1 + 2*3^0 = -67.

Rémy Sigrist, Table of n, a(n) for n = 0..6561

Cf. A038754, A065620, A355675.

a(n, base=3) = { my (d=digits(n, base), s=1); forstep (k=#d, 1, -1, if (d[k], d[k]*=s; s=-s)); return (fromdigits(d, base)) }

a(n) = n iff n = 0 or n belongs to A038754.

A355675 a(0) = 0, and for any n > 0 and d = 1..9, a(10n) = 10a(n), a(10n + d) = d - 10a(n).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 20, -19, -18, -17, -16, -15, -14, -13, -12, -11, 30, -29, -28, -27, -26, -25, -24, -23, -22, -21, 40, -39, -38, -37, -36, -35, -34, -33, -32, -31, 50, -49, -48, -47, -46, -45, -44, -43, -42

Offset: 0

Rémy Sigrist, Jul 14 2022

This sequence establishes a bijection from the nonnegative integers (N) to the integers (Z).
This sequence is to base 10 what A065620 is to base 2.
To compute a(n): write n a sum of terms of A037124 with distinct 10-adic valuations and take the alternating sum.
This sequence has similarities with A073835; they first differ in absolute value for n = 101: a(101) = 99 whereas A073835(101) = 101.
This sequence has similarities with A334387; they first differ in absolute value for n = 111: a(111) = 91 whereas A334387(111) = 81.

			For n = 17039:
  17039 = 10000 + 7000 + 30 + 9,
  so a(17039) = -10000 + 7000 - 30 + 9 = -3021.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Index entries for sequences related to decimal expansion of n

Cf. A037124, A065620, A073835, A334387, A355624.

a(n, base=10) = { my (d=digits(n, base), s=1); forstep (k=#d, 1, -1, if (d[k], d[k]*=s; s=-s)); return (fromdigits(d, base)) }

a(n) = 0 iff n = 0 or n belongs to A037124.

A355678 For any nonnegative number n with factorial base expansion Sum_{k > 0} d_k * k!, a(n) = Sum_{k > 0} d_k * k! * (-1)^(Sum_{i < k} sign(d_i)).

Original entry on oeis.org

0, 1, 2, -1, 4, -3, 6, -5, -4, 5, -2, 3, 12, -11, -10, 11, -8, 9, 18, -17, -16, 17, -14, 15, 24, -23, -22, 23, -20, 21, -18, 19, 20, -19, 22, -21, -12, 13, 14, -13, 16, -15, -6, 7, 8, -7, 10, -9, 48, -47, -46, 47, -44, 45, -42, 43, 44, -43, 46, -45, -36, 37

Offset: 0

Rémy Sigrist, Jul 14 2022

This sequence establishes a bijection from the nonnegative integers (N) to the integers (Z).
This sequence is to factorial base what A065620 is to base 2.
To compute a(n): write n as a minimal sum of terms of A051683 and take the alternating sum.

			For n = 28:
  28 = 4! + 2*2!,
  so a(28) = -4! + 2*2! = -20.

Cf. A051683, A065620, A355679.

a(n) = { my (v=0, f=1, s=1, d); for (r=2, oo, if (n==0, return (v), d=n%r; if (d, v+=d*f*s; s=-s); n\=r; f*=r)) }

a(n) = n iff n = 0 or n belongs to A051683.

A355679 For any nonnegative number n with primorial base expansion Sum_{k >= 0} d_k * A002110(k), a(n) = Sum_{k >= 0} d_k * A002110(k) * (-1)^(Sum_{i < k} sign(d_i)).

Original entry on oeis.org

0, 1, 2, -1, 4, -3, 6, -5, -4, 5, -2, 3, 12, -11, -10, 11, -8, 9, 18, -17, -16, 17, -14, 15, 24, -23, -22, 23, -20, 21, 30, -29, -28, 29, -26, 27, -24, 25, 26, -25, 28, -27, -18, 19, 20, -19, 22, -21, -12, 13, 14, -13, 16, -15, -6, 7, 8, -7, 10, -9, 60, -59

Offset: 0

Rémy Sigrist, Jul 14 2022

This sequence establishes a bijection from the nonnegative integers (N) to the integers (Z).
This sequence is to primorial base what A065620 is to base 2.
To compute a(n): write n as a minimal sum of terms of A060735 and take the alternating sum.

			For n = 13:
  13 = 2*6 + 1,
  so a(13) = -2*6 + 1 = -11.

Rémy Sigrist, Table of n, a(n) for n = 0..2310
Index entries for sequences related to primorial base

Cf. A002110, A060735, A065620, A355678.

a(n) = { my (v=0, f=1, s=1, d); forprime (r=2, oo, if (n==0, return (v), d=n%r; if (d, v+=d*f*s; s=-s); n\=r; f*=r)) }

a(n) = n iff n = 0 or n belongs to A060735.

cp's OEIS Frontend

A325572 Numbers n that have divisor d > 1 such that A048720(A065621(d),n/d) = n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A348690 For any nonnegative number n with binary expansion Sum_{k >= 0} b_k * 2^k, a(n) is the real part of f(n) = Sum_{k >= 0} b_k * (i^Sum_{j = 0..k-1} b_j) * (1+i)^k (where i denotes the imaginary unit); sequence A348691 gives the imaginary part.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A348691 For any nonnegative number n with binary expansion Sum_{k >= 0} b_k * 2^k, a(n) is the imaginary part of f(n) = Sum_{k >= 0} b_k * (i^Sum_{j = 0..k-1} b_j) * (1+i)^k (where i denotes the imaginary unit); sequence A348690 gives the real part.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A333773 Replace 2's with (-1)'s in ternary representation of n and sum nonzero terms with alternating signs.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A355624 a(0) = 0, and for any n > 0, a(3*n) = 3*a(n), a(3*n+1) = 1-3*a(n), a(3*n+2) = 2-3*a(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A355675 a(0) = 0, and for any n > 0 and d = 1..9, a(10*n) = 10*a(n), a(10*n + d) = d - 10*a(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A355678 For any nonnegative number n with factorial base expansion Sum_{k > 0} d_k * k!, a(n) = Sum_{k > 0} d_k * k! * (-1)^(Sum_{i < k} sign(d_i)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A355679 For any nonnegative number n with primorial base expansion Sum_{k >= 0} d_k * A002110(k), a(n) = Sum_{k >= 0} d_k * A002110(k) * (-1)^(Sum_{i < k} sign(d_i)).

Views

Author

Keywords

Comments

Examples

A355624 a(0) = 0, and for any n > 0, a(3n) = 3a(n), a(3n+1) = 1-3a(n), a(3n+2) = 2-3a(n).

A355675 a(0) = 0, and for any n > 0 and d = 1..9, a(10n) = 10a(n), a(10n + d) = d - 10a(n).