cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

Views

Author

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

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

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

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • Python
    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

Formula

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

Views

Author

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

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

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

Keywords

Examples

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

Links

Crossrefs

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

Programs

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

Formula

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

Views

Author

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

Keywords

Examples

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

Links

Crossrefs

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

Programs

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

Formula

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

Views

Author

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

Keywords

Examples

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

Links

Crossrefs

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

Programs

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

Formula

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

A061086 a(n) is the concatenation of n with n^3.

Original entry on oeis.org

11, 28, 327, 464, 5125, 6216, 7343, 8512, 9729, 101000, 111331, 121728, 132197, 142744, 153375, 164096, 174913, 185832, 196859, 208000, 219261, 2210648, 2312167, 2413824, 2515625, 2617576, 2719683, 2821952, 2924389, 3027000, 3129791, 3232768, 3335937, 3439304
Offset: 1

Views

Author

Amarnath Murthy, Apr 19 2001

Keywords

Examples

			a(13) = 132197, where 2197 = 13^3.

References

  • Felice Russo, A set of Smarandache Functions, sequences and conjectures in number theory, page 65.

Links

Crossrefs

Cf. A020338, A053061.

Programs

  • Magma
    [Seqint(Intseq(n^3) cat Intseq(n)): n in [1..40]]; // Vincenzo Librandi, Jan 03 2015
  • Mathematica
    Table[FromDigits[Join[IntegerDigits[n], IntegerDigits[n^3]]],{n, 40}] (* Vincenzo Librandi, Jan 03 2015 *)
  • Python
    def a(n): return int(str(n) + str(n**3))
print([a(n) for n in range(1, 35)]) # Michael S. Branicky, Nov 28 2021

Extensions

Offset corrected by Charles R Greathouse IV, Sep 20 2012
More terms from Vincenzo Librandi, Jan 03 2015

A084854 Triangular array, read by rows: T(n,k) = concatenated decimal representations of n and k, 1<=k<=n.

Original entry on oeis.org

11, 21, 22, 31, 32, 33, 41, 42, 43, 44, 51, 52, 53, 54, 55, 61, 62, 63, 64, 65, 66, 71, 72, 73, 74, 75, 76, 77, 81, 82, 83, 84, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 102, 103, 104, 105, 106, 107, 108, 109, 1010, 111, 112, 113, 114, 115, 116, 117, 118, 119, 1110, 1111
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 09 2003

Keywords

Links

Crossrefs

Cf. A017281, A020338, A055642, A066686, A067574, A084855.

Programs

  • Python
    def T(n, k): return int(str(n) + str(k))
def auptorow(maxrow):
    return [T(n, k) for n in range(1, maxrow+1) for k in range(1, n+1)]
print(auptorow(11)) # Michael S. Branicky, Nov 21 2021

Formula

T(n, k) = n*10^A055642(k) + k.
T(n, 1) = A017281(n); T(n, n) = A020338(n).

A116289 Numbers k such that k*(k+5) gives the concatenation of a number m with itself.

Original entry on oeis.org

6, 96, 385, 429, 567, 611, 814, 996, 4521, 5475, 9996, 90910, 99996, 316832, 683164, 999996, 3636364, 6363632, 9999996, 82352942, 99999996, 331668332, 368421053, 395604391, 442767754, 461538462, 488721800, 511278196, 538461534, 557232242, 604395605, 631578943, 668331664, 700089385, 727272728
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Comments

From Robert Israel, Apr 09 2025: (Start)
Numbers k such that k * (k + 5) = (10^d + 1) * m for some d and m where m has d digits.
Contains 10^d-4 for all d >= 1. (End)

Links

Crossrefs

Cf. A028557, A020338.
Cf. A116158, A116281, A116288, A116290, A116303.

Programs

  • Maple
    q:= proc(d,m) local R,t,a,b,x,q;
   t:= 10^d+1;
   R:= NULL;
   for a in numtheory:-divisors(t) do
     b:= t/a;
     if igcd(a,b) > 1 then next fi;
     for x from chrem([0,-m],[a,b]) by t do
       q:= x*(x+m)/t;
       if q >= 10^d then break fi;
       if q >= 10^(d-1) then R:= R, x fi;
   od od;
   sort(convert({R},list));
end proc:
seq(op(q(d,5)),d=1..10); # Robert Israel, Apr 09 2025

Extensions

More terms from Robert Israel, Apr 09 2025

A336668 Numbers with decimal expansion d_1, ..., d_w such that d_k = d*_{k + d_k} for k = 1..w where d* is the w-periodic sequence with initial terms d_1, ..., d_w.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 20, 22, 24, 26, 28, 33, 40, 42, 44, 46, 48, 55, 60, 62, 64, 66, 68, 77, 80, 82, 84, 86, 88, 99, 111, 222, 300, 303, 306, 309, 330, 333, 336, 339, 360, 363, 366, 369, 390, 393, 396, 399, 444, 555, 600, 603, 606, 609, 630, 633
Offset: 1

Views

Author

Rémy Sigrist, Jul 29 2020

Keywords

Comments

In other words, arranging the decimal digits of a term clockwise around a circle, any digit d, say at position p, appears at position p + d (or equivalently at position p - d).
All repunits (A002275) appear in this sequence, and they are the only terms with a digit 1.
All numbers with repeated digits (A010785) also appear in this sequence.
If m > 0 belongs to the sequence, then A020338(m) and A074842(m) also belong to the sequence.

Examples

			We can arrange the decimal digits of 46064686 around a circle as follows:
.               4
.         6           6
.
.
.      8                 0
.
.
.         6           6
.               4
- moving clockwise:
   - the digit 4 in the north leads to the digit 4 in the south and vice versa,
   - the digit 6 in the northeast leads to the digit 6 in the northwest,
   - the digit 6 in the northwest leads to the digit 6 in the southwest,
   - the digit 6 in the southwest leads to the digit 6 in the southeast,
   - the digit 6 in the southeast leads to the digit 6 in the northeast,
   - the digit 0 leads to itself,
   - the digit 8 leads to itself (after a full turn),
- so 46064686 belongs to this sequence.

Links

Crossrefs

Cf. A002275, A010785, A020338, A074842, A336669.

Programs

  • PARI
    See Links section.
Previous Showing 21-30 of 64 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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