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.

Showing 1-9 of 9 results.

A108090 Numbers of the form (11^i)*(13^j).

Original entry on oeis.org

1, 11, 13, 121, 143, 169, 1331, 1573, 1859, 2197, 14641, 17303, 20449, 24167, 28561, 161051, 190333, 224939, 265837, 314171, 371293, 1771561, 2093663, 2474329, 2924207, 3455881, 4084223, 4826809, 19487171, 23030293, 27217619
Offset: 1

Views

Author

Douglas Winston (douglas.winston(AT)srupc.com), Jun 03 2005

Keywords

Links

Crossrefs

Cf. A003586, A003592, A003593, A003591, A003594, A003595, A003596, A003597, A003598, A003599, A107326, A107364, A107466, A108056.

Programs

  • Haskell
    import Data.Set (singleton, deleteFindMin, insert)
a108090 n = a108090_list !! (n-1)
a108090_list = f $ singleton (1,0,0) where
   f s = y : f (insert (11 * y, i + 1, j) $ insert (13 * y, i, j + 1) s')
         where ((y, i, j), s') = deleteFindMin s
-- Reinhard Zumkeller, May 15 2015
  • Magma
    [n: n in [1..10^7] | PrimeDivisors(n) subset [11, 13]]; // Vincenzo Librandi, Jun 27 2016
  • Mathematica
    mx = 3*10^7; Sort@ Flatten@ Table[ 11^i*13^j, {i, 0, Log[11, mx]}, {j, 0, Log[13, mx/11^i]}] (* Robert G. Wilson v, Aug 17 2012 *)
fQ[n_]:=PowerMod[143, n, n] == 0; Select[Range[2 10^7], fQ] (* Vincenzo Librandi, Jun 27 2016 *)
  • PARI
    list(lim)=my(v=List(),t); for(j=0,logint(lim\=1,13), t=13^j; while(t<=lim, listput(v,t); t*=11)); Set(v) \\ Charles R Greathouse IV, Aug 29 2016
  • Python
    from sympy import integer_log
def A108090(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(integer_log(x//13**i,11)[0]+1 for i in range(integer_log(x,13)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Mar 25 2025

Formula

Sum_{n>=1} 1/a(n) = (11*13)/((11-1)*(13-1)) = 143/120. - Amiram Eldar, Sep 23 2020
a(n) ~ exp(sqrt(2*log(11)*log(13)*n)) / sqrt(143). - Vaclav Kotesovec, Sep 23 2020

A108761 Numbers of the form (10^i)*(13^j), with i, j >= 0.

Original entry on oeis.org

1, 10, 13, 100, 130, 169, 1000, 1300, 1690, 2197, 10000, 13000, 16900, 21970, 28561, 100000, 130000, 169000, 219700, 285610, 371293, 1000000, 1300000, 1690000, 2197000, 2856100, 3712930, 4826809, 10000000, 13000000, 16900000
Offset: 1

Views

Author

Douglas Winston (douglas.winston(AT)srupc.com), Jun 24 2005

Keywords

Links

Crossrefs

Cf. A107326, A107364, A107462, A107466, A107710, A108056, A107764, A108748, A108090.

Programs

  • Haskell
    import Data.Set (singleton, deleteFindMin, insert)
a108761 n = a108761_list !! (n-1)
a108761_list = f $ singleton (1,0,0) where
   f s = y : f (insert (10 * y, i + 1, j) $ insert (13 * y, i, j + 1) s')
         where ((y, i, j), s') = deleteFindMin s
-- Reinhard Zumkeller, May 15 2015
  • Mathematica
    n = 10^7; Flatten[Table[10^i*13^j, {i, 0, Log10[n]}, {j, 0, Log[13, n/10^i]}]] // Sort (* Amiram Eldar, Sep 25 2020 *)

Formula

Sum_{n>=1} 1/a(n) = (10*13)/((10-1)*(13-1)) = 65/54. - Amiram Eldar, Sep 25 2020
a(n) ~ exp(sqrt(2*log(10)*log(13)*n)) / sqrt(130). - Vaclav Kotesovec, Sep 25 2020

A108056 Numbers of the form (7^i)*(13^j).

Original entry on oeis.org

1, 7, 13, 49, 91, 169, 343, 637, 1183, 2197, 2401, 4459, 8281, 15379, 16807, 28561, 31213, 57967, 107653, 117649, 199927, 218491, 371293, 405769, 753571, 823543, 1399489, 1529437, 2599051, 2840383, 4826809, 5274997, 5764801, 9796423, 10706059, 18193357, 19882681
Offset: 1

Views

Author

Douglas Winston (douglas.winston(AT)srupc.com), Jun 02 2005

Keywords

Links

Crossrefs

Cf. A003586, A003592, A003593, A003591, A003594, A003595, A003596, A003597, A003598, A003599, A107326, A107364, A107466.

Programs

  • Mathematica
    n = 10^7; Flatten[Table[7^i*13^j, {i, 0, Log[7, n]}, {j, 0, Log[13, n/7^i]}]] // Sort (* Amiram Eldar, Sep 23 2020 *)
  • PARI
    list(lim)=my(v=List(),N);for(n=0,log(lim)\log(13),N=13^n;while(N<=lim,listput(v,N);N*=7));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011
  • Python
    from sympy import integer_log
def A108056(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(integer_log(x//13**i,7)[0]+1 for i in range(integer_log(x,13)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Oct 22 2024

Formula

Sum_{n>=1} 1/a(n) = (7*13)/((7-1)*(13-1)) = 91/72. - Amiram Eldar, Sep 23 2020
a(n) ~ exp(sqrt(2*log(7)*log(13)*n)) / sqrt(91). - Vaclav Kotesovec, Sep 23 2020

Extensions

More terms from Amiram Eldar, Sep 23 2020

A107462 Numbers of the form (4^i)*(13^j), with i, j >= 0.

Original entry on oeis.org

1, 4, 13, 16, 52, 64, 169, 208, 256, 676, 832, 1024, 2197, 2704, 3328, 4096, 8788, 10816, 13312, 16384, 28561, 35152, 43264, 53248, 65536, 114244, 140608, 173056, 212992, 262144, 371293, 456976, 562432, 692224, 851968, 1048576, 1485172
Offset: 1

Views

Author

Douglas Winston (douglas.winston(AT)srupc.com), Jun 09 2005

Keywords

Links

Crossrefs

Subsequence of A107326.
Cf. A025617, A025618, A025619, A025620, A025621, A107364, A107466, A108056, A108090.

Programs

  • Mathematica
    n = 10^6; Flatten[Table[4^i*13^j, {i, 0, Log[4, n]}, {j, 0, Log[13, n/4^i]}]] // Sort (* Amiram Eldar, Sep 24 2020 *)

Formula

Sum_{n>=1} 1/a(n) = (4*13)/((4-1)*(13-1)) = 13/9. - Amiram Eldar, Sep 24 2020
a(n) ~ exp(sqrt(2*log(4)*log(13)*n)) / sqrt(52). - Vaclav Kotesovec, Sep 24 2020

A107710 Numbers of the form (6^i)*(13^j), with i, j >= 0.

Original entry on oeis.org

1, 6, 13, 36, 78, 169, 216, 468, 1014, 1296, 2197, 2808, 6084, 7776, 13182, 16848, 28561, 36504, 46656, 79092, 101088, 171366, 219024, 279936, 371293, 474552, 606528, 1028196, 1314144, 1679616, 2227758, 2847312, 3639168, 4826809, 6169176
Offset: 1

Views

Author

Douglas Winston (douglas.winston(AT)srupc.com), Jun 10 2005

Keywords

Links

Crossrefs

Cf. A025626, A025627, A025628, A025629, A064476, A107326, A107364, A107462, A107466, A108056, A108090.

Programs

  • Mathematica
    n = 10^6; Flatten[Table[6^i*13^j, {i, 0, Log[6, n]}, {j, 0, Log[13, n/6^i]}]] // Sort (* Amiram Eldar, Sep 25 2020 *)

Formula

Sum_{n>=1} 1/a(n) = (6*13)/((6-1)*(13-1)) = 13/10. - Amiram Eldar, Sep 25 2020
a(n) ~ exp(sqrt(2*log(6)*log(13)*n)) / sqrt(78). - Vaclav Kotesovec, Sep 25 2020

A107764 Numbers of the form (8^i)*(13^j), with i, j >= 0.

Original entry on oeis.org

1, 8, 13, 64, 104, 169, 512, 832, 1352, 2197, 4096, 6656, 10816, 17576, 28561, 32768, 53248, 86528, 140608, 228488, 262144, 371293, 425984, 692224, 1124864, 1827904, 2097152, 2970344, 3407872, 4826809, 5537792, 8998912, 14623232, 16777216
Offset: 1

Views

Author

Douglas Winston (douglas.winston(AT)srupc.com), Jun 11 2005

Keywords

Links

Crossrefs

Cf. A025633, A025634, A107326, A107364, A107462, A107466, A107710, A108056, A108090.

Programs

  • Mathematica
    n = 10^6; Flatten[Table[8^i*13^j, {i, 0, Log[8, n]}, {j, 0, Log[13, n/8^i]}]] // Sort (* Amiram Eldar, Sep 25 2020 *)

Formula

Sum_{n>=1} 1/a(n) = (8*13)/((8-1)*(13-1)) = 26/21. - Amiram Eldar, Sep 25 2020
a(n) ~ exp(sqrt(2*log(8)*log(13)*n)) / sqrt(104). - Vaclav Kotesovec, Sep 25 2020

A108201 Numbers of the form (5^i)*(12^j), with i, j >= 0.

Original entry on oeis.org

1, 5, 12, 25, 60, 125, 144, 300, 625, 720, 1500, 1728, 3125, 3600, 7500, 8640, 15625, 18000, 20736, 37500, 43200, 78125, 90000, 103680, 187500, 216000, 248832, 390625, 450000, 518400, 937500, 1080000, 1244160, 1953125, 2250000, 2592000
Offset: 1

Views

Author

Douglas Winston (douglas.winston(AT)srupc.com), Jun 15 2005

Keywords

Links

Crossrefs

Cf. A025622, A003595, A025623, A025624, A025625, A003598, A107466, A064476.

Programs

  • Mathematica
    Take[Union[5^First[#] 12^Last[#]&/@Tuples[Range[0,20],2]],50] (* Harvey P. Dale, Mar 23 2012 *)

Formula

Sum_{n>=1} 1/a(n) = 15/11. - Amiram Eldar, Mar 29 2025

A108748 Numbers of the form (9^i)*(13^j), with i, j >= 0.

Original entry on oeis.org

1, 9, 13, 81, 117, 169, 729, 1053, 1521, 2197, 6561, 9477, 13689, 19773, 28561, 59049, 85293, 123201, 177957, 257049, 371293, 531441, 767637, 1108809, 1601613, 2313441, 3341637, 4782969, 4826809, 6908733, 9979281, 14414517, 20820969
Offset: 1

Views

Author

Douglas Winston (douglas.winston(AT)srupc.com), Jun 23 2005

Keywords

Links

Crossrefs

Cf. A025635, A108687, A107326, A107364, A107462, A107466, A107710, A108056, A107764, A108090.

Programs

  • Mathematica
    n = 10^6; Flatten[Table[9^i*13^j, {i, 0, Log[9, n]}, {j, 0, Log[13, n/9^i]}]] // Sort (* Amiram Eldar, Sep 25 2020 *)

Formula

Sum_{n>=1} 1/a(n) = (9*13)/((9-1)*(13-1)) = 39/32. - Amiram Eldar, Sep 25 2020
a(n) ~ exp(sqrt(2*log(9)*log(13)*n)) / sqrt(117). - Vaclav Kotesovec, Sep 25 2020

A108771 Numbers of the form (12^i)*(13^j), with i, j >= 0.

Original entry on oeis.org

1, 12, 13, 144, 156, 169, 1728, 1872, 2028, 2197, 20736, 22464, 24336, 26364, 28561, 248832, 269568, 292032, 316368, 342732, 371293, 2985984, 3234816, 3504384, 3796416, 4112784, 4455516, 4826809, 35831808, 38817792, 42052608, 45556992
Offset: 1

Views

Author

Douglas Winston (douglas.winston(AT)srupc.com), Jun 25 2005

Keywords

Links

Crossrefs

Cf. A107326, A107364, A107462, A107466, A107710, A108056, A107764, A108748, A108761, A108090.

Programs

  • Maple
    N:= 10^8: # to get all terms <= N
sort([seq(seq(12^i*13^j, j = 0 .. floor(log[13](N/12^i))), i=0..floor(log[12](N)))]); # Robert Israel, Jun 16 2019
  • Mathematica
    With[{max = 5*10^7}, Flatten[Table[12^i*13^j, {i, 0, Log[12, max]}, {j, 0, Log[13, max/12^i]}]] // Sort] (* Amiram Eldar, Mar 29 2025 *)

Formula

Sum_{n>=1} 1/a(n) = 13/11. - Amiram Eldar, Mar 29 2025
Showing 1-9 of 9 results.

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

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