A020331 -id:A020331 - cp's OEIS frontend

A020332 Numbers whose base-4 representation is the juxtaposition of two identical strings.

5, 10, 15, 68, 85, 102, 119, 136, 153, 170, 187, 204, 221, 238, 255, 1040, 1105, 1170, 1235, 1300, 1365, 1430, 1495, 1560, 1625, 1690, 1755, 1820, 1885, 1950, 2015, 2080, 2145, 2210, 2275, 2340, 2405, 2470, 2535, 2600, 2665, 2730, 2795, 2860, 2925, 2990

Offset: 1

David W. Wilson, Melia Aldridge (ma38(AT)spruce.evansville.edu)

			102_10 = 1212_4. - _Jon E. Schoenfield_, Feb 11 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A020330, A020331, A020333, A020334, A020335, A020336, A020337, A020338.

a[n_] := n + n*4^Floor[Log[4, n] + 1]; Array[a, 50] (* Amiram Eldar, Apr 06 2021 *)
b4jQ[n_]:=Module[{idn4=IntegerDigits[n,4],len},len=Length[idn4];EvenQ[len] && Take[ idn4,len/2]==Take[idn4,-len/2]]; Select[Range[3000],b4jQ] (* or *) Table[If[ #[[1]] == 0,Nothing,FromDigits[#,4]]&/@(Flatten[Join[{#,#}]]&/@Tuples[ {0,1,2,3},n]),{n,3}]//Flatten(* Harvey P. Dale, Sep 02 2022 *)

a(n) = n*4^floor(log_4(n)+1) + n. - Ilya Gutkovskiy, Jan 26 2018

A020333 Numbers whose base-5 representation is the juxtaposition of two identical strings.

Original entry on oeis.org

6, 12, 18, 24, 130, 156, 182, 208, 234, 260, 286, 312, 338, 364, 390, 416, 442, 468, 494, 520, 546, 572, 598, 624, 3150, 3276, 3402, 3528, 3654, 3780, 3906, 4032, 4158, 4284, 4410, 4536, 4662, 4788, 4914, 5040, 5166, 5292, 5418, 5544, 5670, 5796, 5922, 6048

Offset: 1

David W. Wilson, Melia Aldridge (ma38(AT)spruce.evansville.edu)

			182_10 = 1212_5. - _Jon E. Schoenfield_, Feb 11 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A020330, A020331, A020332, A020334, A020335, A020336, A020337, A020338.

tis5Q[n_]:=Module[{idn=IntegerDigits[n,5],len},len=Length[idn];EvenQ[len] && Take[idn,len/2]==Take[idn,-len/2]]; Select[Range[6500],tis5Q]  (* or *) Flatten[Table[FromDigits[#,5]&/@Select[(Flatten[{#,#}]&/@Tuples[ Range[ 0,4],n]),#[[1]]!=0&],{n,3}]] (* The second program is significantly faster than the first. *) (* Harvey P. Dale, Apr 08 2013 *)
a[n_] := n + n*5^Floor[Log[5, n] + 1]; Array[a, 50] (* Amiram Eldar, Apr 06 2021 *)

from itertools import count, product
def agen():
    for d in count(1):
        for first in "1234":
            for p in product("01234", repeat=d-1):
                yield int((first+"".join(p))*2, 5)
g = agen()
print([next(g) for n in range(1, 49)]) # Michael S. Branicky, Jun 12 2021

a(n) = n*5^floor(log_5(n)+1) + n. - Ilya Gutkovskiy, Jan 26 2018

A020334 Numbers whose base-6 representation is the juxtaposition of two identical strings.

Original entry on oeis.org

7, 14, 21, 28, 35, 222, 259, 296, 333, 370, 407, 444, 481, 518, 555, 592, 629, 666, 703, 740, 777, 814, 851, 888, 925, 962, 999, 1036, 1073, 1110, 1147, 1184, 1221, 1258, 1295, 7812, 8029, 8246, 8463, 8680, 8897, 9114, 9331, 9548, 9765, 9982, 10199, 10416

Offset: 1

David W. Wilson, Melia Aldridge (ma38(AT)spruce.evansville.edu)

			296_10 = 1212_6. - _Jon E. Schoenfield_, Feb 11 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A020330, A020331, A020332, A020333, A020335, A020336, A020337, A020338.

jtiQ[n_]:=Module[{idn6=IntegerDigits[n,6],len},len=Length[idn6];EvenQ[ len] && Take[idn6,len/2]==Take[idn6,(-len/2)]]; Select[ Range[ 11000], jtiQ] (* Harvey P. Dale, May 29 2016 *)
a[n_] := n + n*6^Floor[Log[6, n] + 1]; Array[a, 50] (* Amiram Eldar, Apr 06 2021 *)

a(n) = n*6^floor(log_6(n)+1) + n. - Ilya Gutkovskiy, Jan 26 2018

A020335 Numbers whose base-7 representation is the juxtaposition of two identical strings.

Original entry on oeis.org

8, 16, 24, 32, 40, 48, 350, 400, 450, 500, 550, 600, 650, 700, 750, 800, 850, 900, 950, 1000, 1050, 1100, 1150, 1200, 1250, 1300, 1350, 1400, 1450, 1500, 1550, 1600, 1650, 1700, 1750, 1800, 1850, 1900, 1950, 2000, 2050, 2100, 2150, 2200, 2250, 2300, 2350

Offset: 1

David W. Wilson, Melia Aldridge (ma38(AT)spruce.evansville.edu)

			450_10 = 1212_7. - _Jon E. Schoenfield_, Feb 12 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A020330, A020331, A020332, A020333, A020334, A020336, A020337, A020338.

a[n_] := n + n*7^Floor[Log[7, n] + 1]; Array[a, 50] (* Amiram Eldar, Apr 06 2021 *)

a(n) = n*7^floor(log_7(n)+1) + n. - Ilya Gutkovskiy, Jan 26 2018

A020336 Numbers whose base-8 representation is the juxtaposition of two identical strings.

Original entry on oeis.org

9, 18, 27, 36, 45, 54, 63, 520, 585, 650, 715, 780, 845, 910, 975, 1040, 1105, 1170, 1235, 1300, 1365, 1430, 1495, 1560, 1625, 1690, 1755, 1820, 1885, 1950, 2015, 2080, 2145, 2210, 2275, 2340, 2405, 2470, 2535, 2600, 2665, 2730, 2795, 2860, 2925, 2990, 3055

Offset: 1

David W. Wilson, Melia Aldridge (ma38(AT)spruce.evansville.edu)

			650_10 = 1212_8. - _Jon E. Schoenfield_, Feb 12 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A020330, A020331, A020332, A020333, A020334, A020335, A020337, A020338.

a[n_] := n + n*8^Floor[Log[8, n] + 1]; Array[a, 50] (* Amiram Eldar, Apr 06 2021 *)

a(n) = n*8^floor(log_8(n)+1) + n. - Ilya Gutkovskiy, Jan 26 2018

A020337 Numbers whose base-9 representation is the juxtaposition of two identical strings.

Original entry on oeis.org

10, 20, 30, 40, 50, 60, 70, 80, 738, 820, 902, 984, 1066, 1148, 1230, 1312, 1394, 1476, 1558, 1640, 1722, 1804, 1886, 1968, 2050, 2132, 2214, 2296, 2378, 2460, 2542, 2624, 2706, 2788, 2870, 2952, 3034, 3116, 3198, 3280, 3362, 3444, 3526, 3608, 3690, 3772

Offset: 1

David W. Wilson, Melia Aldridge (ma38(AT)spruce.evansville.edu)

			902_10 = 1212_9. - _Jon E. Schoenfield_, Feb 12 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A020330, A020331, A020332, A020333, A020334, A020335, A020336, A020338.

a[n_] := n + n*9^Floor[Log[9, n] + 1]; Array[a, 50] (* Amiram Eldar, Apr 06 2021 *)

a(n) = n*9^floor(log_9(n)+1) + n. - Ilya Gutkovskiy, Jan 26 2018

A338086 Duplicate the ternary digits of n, so each 0, 1 or 2 becomes 00, 11 or 22 respectively.

Original entry on oeis.org

0, 4, 8, 36, 40, 44, 72, 76, 80, 324, 328, 332, 360, 364, 368, 396, 400, 404, 648, 652, 656, 684, 688, 692, 720, 724, 728, 2916, 2920, 2924, 2952, 2956, 2960, 2988, 2992, 2996, 3240, 3244, 3248, 3276, 3280, 3284, 3312, 3316, 3320, 3564, 3568, 3572, 3600, 3604

Offset: 0

Kevin Ryde, Oct 09 2020

Also, numbers whose ternary digit runs are all even lengths (including 0 reckoned as no digits at all).  Also, change ternary digits 0,1,2 to base 9 digits 0,4,8, and hence numbers which can be written in base 9 using only digits 0,4,8.
Digit duplication 00,11,22 can be compared to A037314 which is 0 above each so 00,01,02, or A208665 which is 0 below each so 00,10,20.  Duplication is the sum of these, or any one is a suitable multiple of another (*3, *4, etc).
This sequence is the points on the X=Y diagonal of the ternary Z-order curve (see example table in A163328).  The Z-order curve takes a point number p and splits its ternary digits alternately to X and Y coordinates so X(p) = A163325(p) and Y(p) = A163326(p).  Duplicate digits in a(n) are the diagonal X(a(n)) = Y(a(n)) = n.

			n=73 is ternary 2201 which duplicates to 22220011 ternary = 8804 base 9 = 6484 decimal.

Kevin Ryde, Table of n, a(n) for n = 0..6560

Cf. A037314, A208665, A163325, A163326, A163328.
Cf. A020331 (ternary concatenation).
Digit duplication in other bases: A001196, A338754.

PARI
```
a(n) = fromdigits(digits(n,3),9)<<2;
```

from gmpy2 import digits
def A338086(n): return int(''.join(d*2 for d in digits(n,3)),3) # Chai Wah Wu, May 07 2022

a(n) = A037314(n) + A208665(n) = 4*A037314(n) = (4/3)*A208665(n).

a(n) = 4*Sum_{i=0..k} d[i]*9^i where the ternary expansion of n is n = Sum_{i=0..k} d[i]*3^i with digits d[i]=0,1,2.

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A020332 Numbers whose base-4 representation is the juxtaposition of two identical strings.

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A020333 Numbers whose base-5 representation is the juxtaposition of two identical strings.

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A020334 Numbers whose base-6 representation is the juxtaposition of two identical strings.

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A020335 Numbers whose base-7 representation is the juxtaposition of two identical strings.

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A020336 Numbers whose base-8 representation is the juxtaposition of two identical strings.

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A020337 Numbers whose base-9 representation is the juxtaposition of two identical strings.

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A338086 Duplicate the ternary digits of n, so each 0, 1 or 2 becomes 00, 11 or 22 respectively.

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