A030102 -id:A030102 - cp's OEIS frontend

A264987 Odd bisection of A263272.

1, 3, 5, 11, 9, 7, 13, 15, 23, 29, 33, 17, 35, 27, 19, 37, 21, 25, 31, 39, 41, 95, 45, 59, 113, 69, 77, 83, 87, 47, 101, 99, 65, 119, 51, 71, 89, 105, 53, 107, 81, 55, 109, 57, 73, 91, 111, 43, 97, 63, 61, 115, 75, 79, 85, 93, 49, 103, 117, 67, 121, 123, 203, 257, 285, 149, 311, 135, 167, 329, 177, 221, 275

Offset: 0

Antti Karttunen, Dec 05 2015

nonn

Cf. A263272, A264986, A264989.

from sympy import factorint
from sympy.ntheory.factor_ import digits
from operator import mul
def a030102(n): return 0 if n==0 else int(''.join(map(str, digits(n, 3)[1:][::-1])), 3)
def a038502(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==3 else i**f[i] for i in f])
def a038500(n): return n/a038502(n)
def a263273(n): return 0 if n==0 else a030102(a038502(n))*a038500(n)
def a(n): return a263273(2*(2*n + 1))/2 # Indranil Ghosh, May 23 2017

(define (A264987 n) (A263272 (+ 1 n n)))

a(n) = A263272((2*n)+1).

A266407 Permutation of natural numbers: a(n) = A064989(A263273((2*n)-1)).

Original entry on oeis.org

1, 2, 5, 3, 4, 17, 11, 10, 9, 7, 6, 19, 13, 8, 21, 31, 34, 71, 29, 22, 61, 25, 20, 59, 41, 18, 73, 23, 14, 33, 43, 12, 53, 37, 38, 35, 15, 26, 67, 47, 16, 157, 107, 42, 145, 55, 62, 197, 69, 68, 179, 113, 142, 129, 39, 58, 191, 137, 44, 45, 49, 122, 227, 101, 50, 199, 151, 40, 121, 57, 118, 211, 89, 82, 111, 149, 36, 91, 85

Offset: 1

Antti Karttunen, Jan 02 2016

Antti Karttunen, Table of n, a(n) for n = 1..9842
Index entries for sequences that are permutations of the natural numbers

Inverse: A266408.
Cf. A064989, A263273.
Cf. also A064216, A266401, A266403.

A030102(n) = { my(r=[n%3]); while(0M. F. Hasler's Nov 04 2011 code in A030102.
A263273 = n -> if(!n,n,A030102(n/(3^valuation(n,3))) * (3^valuation(n, 3))); \\ Taking of the quotient probably unnecessary.
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A266407 = n -> A064989(A263273((2*n)-1));
for(n=1, 9842, write("b266407.txt", n, " ", A266407(n)));

(define (A266407 n) (A064989 (A263273 (+ n n -1))))

a(n) = A064989(A263273((2*n)-1)).

A266408 Permutation of natural numbers: a(n) = (1/2) * (1+A263273(A003961(n))).

Original entry on oeis.org

1, 2, 4, 5, 3, 11, 10, 14, 9, 8, 7, 32, 13, 29, 37, 41, 6, 26, 12, 23, 15, 20, 28, 95, 22, 38, 115, 86, 19, 110, 16, 122, 30, 17, 36, 77, 34, 35, 55, 68, 25, 44, 31, 59, 60, 83, 40, 284, 61, 65, 100, 113, 33, 344, 46, 257, 70, 56, 24, 329, 21, 47, 289, 365, 88, 89, 39, 50, 49, 107, 18, 230, 27, 101, 244, 104, 112, 164, 82, 203, 174, 74

Offset: 1

Antti Karttunen, Jan 02 2016

Antti Karttunen, Table of n, a(n) for n = 1..8191
Index entries for sequences that are permutations of the natural numbers

Inverse: A266407.
Cf. A003961, A263273.
Cf. also A048673, A266401, A266403.

A030102(n) = { my(r=[n%3]); while(0M. F. Hasler's Nov 04 2011 code in A030102.
A263273 = n -> if(!n,n,A030102(n/(3^valuation(n,3))) * (3^valuation(n, 3))); \\ Taking of the quotient probably unnecessary.
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of Michel Marcus
A266408 = n -> (1+A263273(A003961(n)))/2;
for(n=1, 8191, write("b266408.txt", n, " ", A266408(n)));

(define (A266408 n) (/ (+ 1 (A263273 (A003961 n))) 2))

a(n) = (1/2) * (1+A263273(A003961(n))).

A361818 For any number k >= 0, let T_k be the triangle whose base corresponds to the ternary expansion of k (without leading zeros) and other values, say t above u and v, satisfy t = (-u-v) mod 3; this sequence lists the numbers k such that T_k has 3-fold rotational symmetry.

Original entry on oeis.org

0, 1, 2, 4, 8, 13, 26, 34, 40, 46, 59, 65, 80, 112, 121, 130, 224, 233, 242, 304, 364, 424, 518, 578, 728, 772, 862, 925, 1003, 1093, 1183, 1261, 1324, 1414, 1535, 1598, 1688, 1766, 1856, 1919, 2006, 2096, 2186, 2257, 2509, 2734, 3028, 3280, 3532, 3826, 4051

Offset: 1

Rémy Sigrist, Mar 25 2023

We can devise a similar sequence for any fixed base b >= 2; the present sequence corresponds to b = 3, and A334556 corresponds to b = 2.
This sequence is infinite as it contains A048328.
If k belongs to the sequence, then A004488(k) and A030102(k) belong to the sequence.
Empirically, there are 2*3^floor((w-1)/3) positive terms with w ternary digits.
For any k, if t appears above u and v in T_k, then t + u + v = 0 (mod 3) and #{t, u, v} = 1 or 3 (the three values are either equal or all distinct); each value is uniquely determined by the two others in the same way: t = (-u-v) mod 3, u = (-t-v) mod 3, v = (-t-u) mod 3; this means that we can reconstruct T_k from any of its three sides.
If some row of T_k, say r, has w values and corresponds to the ternary expansion of m, then the row above r corresponds to the w-1 rightmost digits of the ternary expansion of A060587(m).
All positive terms belong to A297250 (their most significant digit equals their least significant digit in base 3).

			The ternary expansion of 304 is "102021", and the corresponding triangle is:
             1
            0 2
           2 1 0
          0 1 1 2
         2 1 1 1 0
        1 0 2 0 2 1
As this triangle has 3-fold rotational symmetry, 304 belongs to the sequence.

Rémy Sigrist, Triangles illustrating initial terms
Rémy Sigrist, PARI program
Index entries for sequences related to XOR-triangles

Cf. A004488, A048328, A060587, A297250, A334556.

PARI
```
See Links section.
```

A055946 n + reversal of base 3 digits of n (written in base 10).

Original entry on oeis.org

0, 2, 4, 4, 8, 12, 8, 12, 16, 10, 20, 30, 16, 26, 36, 22, 32, 42, 20, 30, 40, 26, 36, 46, 32, 42, 52, 28, 56, 84, 40, 68, 96, 52, 80, 108, 40, 68, 96, 52, 80, 108, 64, 92, 120, 52, 80, 108, 64, 92, 120, 76, 104, 132, 56, 84, 112, 68, 96, 124, 80, 108, 136, 68, 96, 124, 80

Offset: 0

Henry Bottomley, Jul 18 2000

If n has an even number of digits in base 3 then a(n) is a multiple of 4.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A030102, A056964.

A055946 := proc(n)
    n+A030102(n) ;
end proc: # R. J. Mathar, May 28 2016

Array[#+FromDigits[Reverse[IntegerDigits[#,3]],3]&,70,0] (* Harvey P. Dale, Nov 09 2017 *)

a(n) = n + A030102(n).

A266189 Self-inverse permutation of nonnegative integers: a(n) = A263273(A264985(A263273(n))).

Original entry on oeis.org

0, 1, 3, 2, 4, 10, 6, 9, 12, 7, 5, 11, 8, 13, 37, 24, 28, 31, 21, 19, 57, 18, 27, 30, 15, 36, 39, 22, 16, 34, 23, 17, 35, 69, 29, 32, 25, 14, 38, 26, 40, 118, 78, 109, 112, 75, 46, 100, 72, 82, 91, 51, 85, 94, 66, 64, 192, 20, 73, 219, 60, 171, 138, 63, 55, 165, 54, 81, 84, 33, 90, 111, 48, 58, 174, 45, 108, 93, 42, 117, 120, 67, 49

Offset: 0

Antti Karttunen, Jan 02 2016

Antti Karttunen, Table of n, a(n) for n = 0..6561
Index entries for sequences that are permutations of the natural numbers

Cf. A263273, A264985, A265353, A265354.

Cf. also A265902, A266190.

f[n_] := Block[{g, h}, g[x_] := x/3^IntegerExponent[x, 3]; h[x_] := x/g@ x; If[n == 0, 0, FromDigits[Reverse@ IntegerDigits[#, 3], 3] &@ g[n] h[n]]]; s = Select[f /@ Range@ 5000, OddQ]; t = Table[(s[[n + 1]] - 1)/2, {n, 0, 1000}]; Table[f@ t[[f@ n + 1]], {n, 0, 82}] (* Michael De Vlieger, Jan 04 2016, after Jean-François Alcover at A263273 *)

from sympy import factorint
from sympy.ntheory.factor_ import digits
from operator import mul
def a030102(n): return 0 if n==0 else int(''.join(map(str, digits(n, 3)[1:][::-1])), 3)
def a038502(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==3 else i**f[i] for i in f])
def a038500(n): return n/a038502(n)
def a263273(n): return 0 if n==0 else a030102(a038502(n))*a038500(n)
def a264985(n): return (a263273(2*n + 1) - 1)/2
def a(n): return a263273(a264985(a263273(n))) # Indranil Ghosh, May 22 2017

(define (A266189 n) (A263273 (A264985 (A263273 n))))

a(n) = A263273(A264985(A263273(n))).
As a composition of related permutations:
a(n) = A263273(A265353(n)).
a(n) = A265354(A263273(n)).

A331173 a(n) = min(n, A263273(n)), where A263273 is bijective base-3 reverse.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 11, 20, 15, 14, 23, 24, 17, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 31, 38, 39, 40, 41, 42, 43, 44, 45, 34, 47, 48, 43, 50, 51, 52, 53, 54, 29, 56, 33, 38, 59, 60, 47, 62, 45, 32, 59, 42, 41, 68, 69, 50, 71, 72, 35, 62, 51, 44, 71, 78, 53, 80, 81

Offset: 0

Antti Karttunen, Jan 12 2020

For all i, j:
  a(i) = a(j) => A290094(i) = A290094(j).
For all i, j > 0:
  a(i) = a(j) => A007949(i) = A007949(j).

Antti Karttunen, Table of n, a(n) for n = 0..59048

Cf. A007949, A030102, A263273, A290094.

Cf. also A330740, A331166.

A030102(n) = { my(r=[n%3]); while(0A030102.
A263273 = n -> if(!n,n,A030102(n/(3^valuation(n,3))) * (3^valuation(n, 3)));
A331173(n) = min(n, A263273(n));

A346113 Base-10 numbers k whose number of divisors equals the number of divisors in R(k), where k is written in all bases from base-2 to base-10 and R(k), the digit reversal of k, is read as a number in the same base.

Original entry on oeis.org

1, 9077, 10523, 10838, 30182, 58529, 73273, 77879, 83893, 244022, 303253, 303449, 304853, 329893, 332249, 334001, 334417, 335939, 336083, 346741, 374617, 391187, 504199, 512695, 516982, 595274, 680354, 687142, 758077, 780391, 792214, 854669, 946217, 948539, 995761, 1008487, 1377067, 1389341

Offset: 1

Scott R. Shannon, Jul 05 2021

There are 633 terms below 50 million and 1253 terms below 100 million. All of those have tau(k), the number of divisors of k, equal to 1, 2, 4, 8 or 16. The first term where tau(k) = 2 is n = 93836531, a prime, which is also the first term of A136634. All terms in A136634 will appear in this sequence, as will all terms in A228768(n) for n>=10. The first term with tau(k) = 4 is 9077, the first with tau(k) = 8 is 595274, and the first with tau(k) = 16 is 5170182. It is possible tau(k) must equal 2^i, with i>=0, although this is unknown.

All known terms are squarefree. - Michel Marcus, Jul 07 2021

			9077 is a term as the number of divisors of 9077 = tau(9077) = 4, and this equals the number of divisors of R(9077) when written and then read as a base-j number, with 2 <= j <= 10. See the table below for k = 9077.
.
  base | k_base         | R(k_base)      | R(k_base)_10  | tau(R(k_base)_10)
----------------------------------------------------------------------------------
   2   | 10001101110101 | 10101110110001 | 11185         | 4
   3   | 110110012      | 210011011      | 15421         | 4
   4   | 2031311        | 1131302        | 6002          | 4
   5   | 242302         | 203242         | 6697          | 4
   6   | 110005         | 500011         | 38887         | 4
   7   | 35315          | 51353          | 12533         | 4
   8   | 21565          | 56512          | 23882         | 4
   9   | 13405          | 50431          | 33157         | 4
  10   | 9077           | 7709           | 7709          | 4

Scott R. Shannon, Table of n, a(n) for n = 1..1253

Cf. A000005, A004086.
Cf. A030102, A030103, A030104, A030105, A030106, A030107, A030108.
Cf. A136634 (prime terms), A228768.
Subsequence of A062895.

Select[Range@100000,Length@Union@DivisorSigma[0,Join[{s=#},FromDigits[Reverse@IntegerDigits[s,#],#]&/@Range[2,10]]]==1&] (* Giorgos Kalogeropoulos, Jul 06 2021 *)

isok(k) = {my(t= numdiv(k)); for (b=2, 10, my(d=digits(k, b)); if (numdiv(fromdigits(Vecrev(d), b)) != t, return (0));); return(1);} \\ Michel Marcus, Jul 06 2021

A365803 Dirichlet inverse of bijective base-3 reverse of n (A263273).

Original entry on oeis.org

1, -2, -3, 0, -7, 6, -5, 0, 0, 18, -19, 0, -13, -2, 21, 0, -25, 0, -11, -8, 15, 62, -23, 0, 32, 26, 0, 40, -55, -54, -37, -32, 57, 54, -3, 0, -31, -14, 39, 0, -67, 6, -49, -96, 0, 58, -61, 0, -18, -156, 75, 0, -79, 0, 237, -32, 33, 182, -65, 24, -47, 74, 0, 160, 123, -186, -41, 16, 69, 230, -77, 0, -35, 62, -96, 144

Offset: 1

Antti Karttunen, Sep 19 2023

sign

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A263273, A323239 (parity of terms), A365804.

Cf. also A365711.

A030102(n) = { my(r=[n%3]); while(0A263273 = n -> if(!n,n,A030102(n/(3^valuation(n,3))) * (3^valuation(n, 3)));
memoA365803 = Map();
A365803(n) = if(1==n,1,my(v); if(mapisdefined(memoA365803,n,&v), v, v = -sumdiv(n,d,if(dA263273(n/d)*A365803(d),0)); mapput(memoA365803,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA263273(n/d) * a(d).

A365804 Sum of bijective base-3 reverse of n (A263273) and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 4, 0, 12, 0, 8, 9, 28, 0, 12, 0, 20, 42, 16, 0, 18, 0, 12, 30, 76, 0, 24, 49, 52, 27, 68, 0, -24, 0, 32, 114, 100, 70, 36, 0, 44, 78, 40, 0, 72, 0, -20, 63, 92, 0, 48, 25, -86, 150, 52, 0, 54, 266, 24, 66, 220, 0, 84, 0, 148, 45, 192, 182, -144, 0, 84, 138, 280, 0, 72, 0, 124, -45, 188, 190, 0, 0, 80, 81

Offset: 1

Antti Karttunen, Sep 19 2023

sign

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A263273, A323239, A365803.

Cf. also A365712.

A030102(n) = { my(r=[n%3]); while(0A263273 = n -> if(!n,n,A030102(n/(3^valuation(n,3))) * (3^valuation(n, 3)));
memoA365803 = Map();
A365803(n) = if(1==n,1,my(v); if(mapisdefined(memoA365803,n,&v), v, v = -sumdiv(n,d,if(dA263273(n/d)*A365803(d),0)); mapput(memoA365803,n,v); (v)));
A365804(n) = (A263273(n)+A365803(n));

a(n) = A263273(n) + A365803(n).

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A263273(d) * A365803(n/d).

cp's OEIS Frontend

A264987 Odd bisection of A263272.

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Python

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A266407 Permutation of natural numbers: a(n) = A064989(A263273((2*n)-1)).

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PARI

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A266408 Permutation of natural numbers: a(n) = (1/2) * (1+A263273(A003961(n))).

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A361818 For any number k >= 0, let T_k be the triangle whose base corresponds to the ternary expansion of k (without leading zeros) and other values, say t above u and v, satisfy t = (-u-v) mod 3; this sequence lists the numbers k such that T_k has 3-fold rotational symmetry.

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PARI

A055946 n + reversal of base 3 digits of n (written in base 10).

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Maple

Mathematica

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A266189 Self-inverse permutation of nonnegative integers: a(n) = A263273(A264985(A263273(n))).

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Mathematica

Python

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A331173 a(n) = min(n, A263273(n)), where A263273 is bijective base-3 reverse.

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A346113 Base-10 numbers k whose number of divisors equals the number of divisors in R(k), where k is written in all bases from base-2 to base-10 and R(k), the digit reversal of k, is read as a number in the same base.

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