A037314 -id:A037314 - cp's OEIS frontend

A208665 Numbers that match odd ternary polynomials; see Comments.

3, 6, 27, 30, 33, 54, 57, 60, 243, 246, 249, 270, 273, 276, 297, 300, 303, 486, 489, 492, 513, 516, 519, 540, 543, 546, 2187, 2190, 2193, 2214, 2217, 2220, 2241, 2244, 2247, 2430, 2433, 2436, 2457, 2460, 2463, 2484, 2487, 2490, 2673, 2676, 2679

Offset: 1

Clark Kimberling, Feb 29 2012

The ternary polynomials (having all coefficients in {0,1,2}) are enumerated at A207966. This sequence shows the numbers n for which p(n,-x)=-p(n,x).

Cf. A037314, A207966, A338086 (ternary digit duplication).

t = Table[IntegerDigits[n, 3], {n, 1, 4000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_, x_] := t[[n]].b[-1 + Length[t[[n]]]]
Table[p[n, x], {n, 1, 30}]
even = {}; Do[n++; If[(p[n, x] /. x -> -x) == p[n, x], AppendTo[even, n]], {n, 1600}];
even     (* A037314 for n >= 2 *)
odd = {}; Do[n++; If[(p[n, x] /. x -> -x) == -p[n, x], AppendTo[odd, n]], {n, 3900}];
odd      (* A208665 *)

a(n) = 3*fromdigits(digits(n,3),9); \\ Kevin Ryde, Oct 17 2020

A163330 Square array A, where entry A(y,x) has the ternary digits of y interleaved with the ternary digits of x, converted back to decimal. Listed by antidiagonals: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

Original entry on oeis.org

0, 3, 1, 6, 4, 2, 27, 7, 5, 9, 30, 28, 8, 12, 10, 33, 31, 29, 15, 13, 11, 54, 34, 32, 36, 16, 14, 18, 57, 55, 35, 39, 37, 17, 21, 19, 60, 58, 56, 42, 40, 38, 24, 22, 20, 243, 61, 59, 63, 43, 41, 45, 25, 23, 81, 246, 244, 62, 66, 64, 44, 48, 46, 26, 84, 82, 249, 247, 245

Offset: 0

Antti Karttunen, Jul 29 2009

Inverse: A163331. a(n) = A163327(A163328(n)). Transpose: A163328. Cf. A007089, A163327, A163332, A163334.

(define (A163330 n) (+ (A037314 (A002262 n)) (* 3 (A037314 (A025581 n)))))

a(n) = 3*A037314(A025581(n)) + A037314(A002262(n))

A163480 Row 0 of A163334 (column 0 of A163336).

Original entry on oeis.org

0, 1, 2, 15, 16, 17, 18, 19, 20, 141, 142, 143, 144, 145, 146, 159, 160, 161, 162, 163, 164, 177, 178, 179, 180, 181, 182, 1275, 1276, 1277, 1278, 1279, 1280, 1293, 1294, 1295, 1296, 1297, 1298, 1311, 1312, 1313, 1314, 1315, 1316, 1437, 1438, 1439

Offset: 0

Antti Karttunen, Jul 29 2009

nonn

Cf. A163481 (Y axis), A037314 (Z-order X axis).

Coordinates: A163528, A163529.

a(n) = my(v=digits(n,3),s=Mod(0,2)); for(i=1,#v, if(s,v[i]+=6); s+=v[i]); fromdigits(v,9); \\ Kevin Ryde, Sep 29 2020

a(n) = A163332(A037314(n)). - Kevin Ryde, Sep 29 2020

A235472 Primes whose base-9 representation also is the base-3 representation of a prime.

Original entry on oeis.org

2, 11, 19, 83, 101, 163, 173, 739, 811, 821, 829, 911, 1549, 1559, 1621, 6563, 6581, 6661, 6733, 8111, 8191, 13933, 14753, 59069, 59141, 59779, 59797, 59951, 60589, 60607, 65629, 65701, 66359, 67079, 67231, 72271, 72353, 72901, 118189, 119557, 119657, 124669, 124823, 125399

Offset: 1

M. F. Hasler, Jan 12 2014

This sequence is part of the two-dimensional array of sequences based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.
For further motivation and cross-references, see sequence A235265 which is the main entry for this whole family of sequences.
Since all digits of the base 9 expansion are less than 3, this is a subsequence of A037314.

			Both 17 = 21_9 and 21_3 = 7 are prime.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235461 - A235482. See the LINK for further cross-references.

is(p,b=3,c=9)=vecmax(d=digits(p,c))

forprime(p=1,2e3,is(p,9,3)&&print1(vector(#d=digits(p,3),i,9^(#d-i))*d~,",")) \\ To produce the terms, this is more efficient than to select them using straightforwardly is(.)=is(.,3,9)

A261691 Change of base from fractional base 3/2 to base 3.

Original entry on oeis.org

0, 1, 2, 6, 7, 8, 21, 22, 23, 63, 64, 65, 69, 70, 71, 192, 193, 194, 207, 208, 209, 213, 214, 215, 579, 580, 581, 621, 622, 623, 627, 628, 629, 642, 643, 644, 1737, 1738, 1739, 1743, 1744, 1745, 1866, 1867, 1868, 1881, 1882, 1883, 1887, 1888, 1889, 1929, 1930

Offset: 0

Tom Edgar, Aug 28 2015

To obtain a(n), we interpret A024629(n) as a base 3 representation (instead of base 3/2). More precisely, if A024629(n) = A007089(m), then a(n) = m.

The digits used in fractional base 3/2 are 0, 1, and 2, which are the same as the digits used in base 3.

			The base 3/2 representation of 7 is (2,1,1); i.e., 7 = 2*(3/2)^2 + 1*(3/2) + 1. Since 2*(3^2) + 1*3 + 1*1 = 22, we have a(7) = 22.

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

Cf. A024629, A007089.
Cf. A005836, A023717, A000695, A037453, A037454, A037455, A037456, A037314.
Cf. A003462, A087088.

a[n_] := a[n] = If[n == 0, 0, 3 * a[2 * Floor[n/3]] + Mod[n, 3]]; Array[a, 100, 0] (* Amiram Eldar, Aug 04 2025 *)

a(n) = { my (v=0, t=1); while (n, v+=t*(n%3); n=(n\3)*2; t*=3); v } \\ Rémy Sigrist, Apr 06 2021

def changebase(n):
    L=[n]
    i=1
    while L[i-1]>2:
        x=L[i-1]
        L[i-1]=x.mod(3)
        L.append(2*floor(x/3))
        i+=1
    return sum([L[i]*3^i for i in [0..len(L)-1]])
[changebase(n) for n in [0..100]]

For n = Sum_{i=0..m} c_i*(3/2)^i with each c_i in {0,1,2}, a(n) = Sum_{i=0..m} c_i*3^i.
From Rémy Sigrist, Apr 06 2021: (Start)
Apparently:
- a(3*n) = a(3*n-1) + A003462(1+A087088(n)) for any n > 0,
- a(3*n+1) = a(3*n) + 1 for any n >= 0,
- a(3*n+2) = a(3*n+1) + 1 for any n >= 0,
(End)

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A208665 Numbers that match odd ternary polynomials; see Comments.

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A163330 Square array A, where entry A(y,x) has the ternary digits of y interleaved with the ternary digits of x, converted back to decimal. Listed by antidiagonals: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

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A163480 Row 0 of A163334 (column 0 of A163336).

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A235472 Primes whose base-9 representation also is the base-3 representation of a prime.

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A261691 Change of base from fractional base 3/2 to base 3.

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