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.

A123384 Number of bits in binary expansion of 10^n.

Original entry on oeis.org

1, 4, 7, 10, 14, 17, 20, 24, 27, 30, 34, 37, 40, 44, 47, 50, 54, 57, 60, 64, 67, 70, 74, 77, 80, 84, 87, 90, 94, 97, 100, 103, 107, 110, 113, 117, 120, 123, 127, 130, 133, 137, 140, 143, 147, 150, 153, 157, 160, 163, 167, 170, 173, 177, 180, 183, 187, 190, 193, 196
Offset: 0

Views

Author

Andrew Caldwell (spongebobpj(AT)yahoo.com), Nov 09 2006

Keywords

Comments

Number of powers of 2 less than or equal to 10^n. - Peter Munn, Nov 13 2019

Examples

			a(3)=10 because 10^3 = 1111101000_2.
10^1 = 10 = 1010_2 has 4 digits.

Crossrefs

Cf. A000079, A007524, A011557, A066343, A067497.
Row 1 of A253635.

Programs

  • Maple
    A007524 := log[10](2.0) ; for n from 0 to 40 do printf("%d,", 1+floor(n/A007524)) ; od: # R. J. Mathar, Nov 12 2006
a:=n->nops(convert(10^n,base,2)): seq(a(n),n=0..70); # Emeric Deutsch, Mar 26 2007
  • Mathematica
    a[n_]:=1 + Floor[n/Log10[2]]; Array[a,60,0] (* Stefano Spezia, Aug 31 2024 *)

Formula

a(n) = 1 + floor(n/A007524) = 1 + floor(n/log_10(2)). - R. J. Mathar, Nov 12 2006
a(n) = 1 + A066343(n). - R. J. Mathar, Mar 02 2007
a(n) = A067497(n) for n >= 1. - Georg Fischer, Nov 02 2018

Extensions

More terms from Emeric Deutsch, Mar 26 2007

A106600 Number of positive integers <= 10^n that are divisible by no prime exceeding 7.

Original entry on oeis.org

1, 10, 46, 141, 338, 694, 1273, 2155, 3427, 5194, 7575, 10688, 14672, 19674, 25861, 33406, 42487, 53302, 66061, 80988, 98311, 118271, 141124, 167139, 196597, 229785, 267007, 308575, 354820, 406074, 462692, 525030, 593467, 668384, 750177, 839260
Offset: 0

Views

Author

Robert G. Wilson v, May 27 2005

Keywords

Comments

Position of 10^n among the 7-smooth numbers (A002473). Note that all powers of 10 are in A002473. - Zak Seidov, Nov 18 2013

Examples

			A002473(a(2)) = A002473(46)=100.

Links

Crossrefs

Row 4 of A253635.
Cf. A002473, A011557.

Programs

  • Mathematica
    n = 35; t = Select[ Flatten[ Table[ 7^d*Select[ Flatten[ Table[ 5^c*Select[ Flatten[ Table[2^a*3^b, {a, 0, n*Log[2, 10]}, {b, 0, n*Log[3, 10]}]], # <= 10^n &], {c, 0, n*Log[5, 10]}]], # <= 10^n &], {d, 0, n*Log[5, 10]}]], # <= 10^n &]; Table[ Length[ Select[t, # <= 10^n &]], {n, 0, 35}]

A106598 Number of positive integers <= 10^n that are divisible by no prime exceeding 5.

Original entry on oeis.org

1, 9, 34, 86, 175, 313, 507, 768, 1105, 1530, 2053, 2683, 3429, 4301, 5310, 6466, 7780, 9259, 10917, 12761, 14801, 17048, 19511, 22201, 25127, 28300, 31730, 35425, 39397, 43654, 48207, 53066, 58243, 63746, 69584, 75769, 82310, 89216, 96499, 104168
Offset: 0

Views

Author

Robert G. Wilson v, May 27 2005

Keywords

Links

Crossrefs

Row 3 of A253635.
Cf. A011557, A051037, A071520.

Programs

  • Mathematica
    n = 40; t = Select[ Flatten[ Table[ 5^c*Select[ Flatten[ Table[2^a*3^b, {a, 0, n*Log[2, 10]}, {b, 0, n*Log[3, 10]}]], # <= 10^n &], {c, 0, n*Log[5, 10]}]], # <= 10^n &]; Table[ Length[ Select[t, # <= 10^n &]], {n, 0, 40}]

A100752 a(n) is the number of positive integers <= 10^n that are divisible by no prime exceeding 3.

Original entry on oeis.org

1, 7, 20, 40, 67, 101, 142, 190, 244, 306, 376, 452, 534, 624, 720, 824, 935, 1052, 1178, 1309, 1447, 1593, 1745, 1905, 2071, 2244, 2424, 2611, 2806, 3006, 3214, 3429, 3652, 3881, 4117, 4360, 4610, 4866, 5131, 5401, 5679, 5964, 6255, 6553, 6859, 7172, 7491
Offset: 0

Views

Author

Robert G. Wilson v, May 27 2005

Keywords

Comments

A good approximation seems to be ceiling(log(10^n)*log(6*10^n)/(log(3)*log(4))). - Horst H. Manninger, Oct 29 2022

Examples

			a(1) = 7 as there are 7 3-smooth numbers less than 10^1 = 10; they are 1, 2, 3, 4, 6, 8, 9. - _David A. Corneth_, Nov 14 2019

Links

Crossrefs

Cf. A003586, A011557, A071521.
Cf. A066343, A106598, A106600, A107352, A106629.
Row 2 of A253635.

Programs

  • Mathematica
    f[n_] := Sum[ Floor@ Log[2, n/3^i] + 1, {i, 0, Log[3, n]}]; Table[ f[10^n], {n, 0, 46}] (* Robert G. Wilson v, Nov 07 2012 *)
  • Python
    from sympy import integer_log
def A100752(n): return sum((10**n//3**i).bit_length() for i in range(integer_log(10**n,3)[0]+1)) # Chai Wah Wu, Oct 23 2024

Formula

a(n) = A071521(10^n). - Chai Wah Wu, Oct 23 2024

A107352 Number of positive integers <= 10^n that are divisible by no prime exceeding 11.

Original entry on oeis.org

1, 10, 55, 192, 522, 1197, 2432, 4520, 7838, 12867, 20193, 30524, 44696, 63694, 88658, 120895, 161885, 213294, 276997, 355082, 449849, 563834, 699826, 860861, 1050260, 1271598, 1528765, 1825937, 2167611, 2558606, 3004075, 3509523
Offset: 0

Views

Author

N. J. A. Sloane, May 23 2005

Keywords

Comments

Lehmer quotes A. E. Western as computing a(5) = 1197, a(8) = 7838 and a(10) = 20193.
Number of integers of the form 2^a*3^b*5^c*7^d*11^e <= 10^n.

Links

Crossrefs

Row 5 of A253635.
Cf. A011557, A051038.

Programs

  • Mathematica
    fQ[n_] := FactorInteger[n][[ -1, 1]] < 13; c = 1; k = 1; Do[ While[k <= 10^n, If[ fQ[k], c++ ]; k++ ]; Print[c], {n, 0, 9}] (* Or *)
n = 32; t = Select[ Flatten[ Table[11^e*Select[ Flatten[ Table[7^d*Select[ Flatten[ Table[5^c*Select[ Flatten[ Table[2^a*3^b, {a, 0, Log[2, 10^n]}, {b, 0, Log[3, 10^n]}]], # <= 10^n &], {c, 0, Log[5, 10^n]}]], # <= 10^n &], {d, 0, Log[7, 10^n]}]], # <= 10^n &], {e, 0, Log[11, 10^n]}]], # <= 10^n &]; Table[ Length[ Select[t, # <= 10^n &]], {n, 0, 32}] (* Robert G. Wilson v, May 24 2005 *)
  • Python
    from sympy import integer_log, prevprime
def A107352(n):
    def g(x, m): return sum((x//3**i).bit_length() for i in range(integer_log(x, 3)[0]+1)) if m==3 else sum(g(x//(m**i), prevprime(m))for i in range(integer_log(x, m)[0]+1))
    return g(10**n,11) # Chai Wah Wu, Oct 22 2024

Formula

Does a(n)/(a(n-1) - a(n-2)) tend to c*n + d for large n where c ~= 0.20 and d ~= 1.37? - David A. Corneth, Nov 14 2019

Extensions

More terms from Robert G. Wilson v and Don Reble, May 26 2005

A106629 Number of positive integers <= 10^n that are divisible by no prime exceeding 13.

Original entry on oeis.org

1, 10, 62, 242, 733, 1848, 4106, 8289, 15519, 27365, 45914, 73908, 114831, 173077, 254065, 364385, 511985, 706293, 958460, 1281500, 1690506, 2202871, 2838489, 3620013, 4573071, 5726533, 7112760, 8767880, 10732089, 13049906, 15770500, 18948010, 22641849, 26917042, 31844560
Offset: 0

Views

Author

Robert G. Wilson v, May 27 2005

Keywords

Crossrefs

Row 6 of A253635.
Cf. A011557, A080197.

Programs

  • Mathematica
    n = 10; t = Select[ Flatten[ Table[13^f*Select[ Flatten[ Table[11^e*Select[ Flatten[ Table[7^d*Select[ Flatten[ Table[5^c*Select[ Flatten[ Table[2^a*3^b, {a, 0, n*Log[2, 10]}, {b, 0, n*Log[3, 10]}]], # <= 10^n &], {c, 0, n*Log[5, 10]}]], # <= 10^n &], {d, 0, n*Log[7, 10]}]], # <= 10^n &], {e, 0, n*Log[11, 10]}]], # <= 10^n &], {f, 0, n*Log[13, 10]}]], # <= 10^n &]; Table[ Length[ Select[t, # <= 10^n &]], {n, 0, 10}];

Extensions

a(21)-a(34) from Daniel Suteu, Nov 14 2019

A108275 Number of positive integers <= 10^n that are divisible by no prime exceeding 17.

Original entry on oeis.org

1, 10, 67, 287, 945, 2579, 6179, 13389, 26809, 50351, 89679, 152751, 250420, 397227, 612185, 919814, 1351182, 1945205, 2750000, 3824456, 5239923, 7082118, 9453184, 12473898, 16286197, 21055745, 26974877, 34265658, 43183191, 54019174
Offset: 0

Views

Author

Robert G. Wilson v, May 31 2005

Keywords

Crossrefs

Row 7 of A253635.
Cf. A011557, A080681.

Programs

  • Mathematica
    n = 10; t = Select[ Flatten[ Table[17^g* Select[ Flatten[ Table[13^f* Select[ Flatten[ Table[11^e* Select[ Flatten[ Table[7^d* Select[ Flatten[ Table[5^c* Select[ Flatten[ Table[2^a*3^b, {a, 0, n*Log[2, 10]}, {b, 0, n*Log[3, 10]}]], # <= 10^n &], {c, 0, n*Log[5, 10]}]], # <= 10^n &], {d, 0, n*Log[7, 10]}]], # <= 10^n &], {e, 0, n*Log[11, 10]}]], # <= 10^n &], {f, 0, n*Log[13, 10]}]], # <= 10^n &], {g, 0, n*Log[17, 10]}]], # <= 10^n &]; Table[ Length[ Select[t, # <= 10^n &]], {n, 0, 10}]

Extensions

a(11)-a(18) from Donovan Johnson, Sep 16 2009
a(19)-a(29) from Max Alekseyev, Apr 27 2010

A108276 Number of positive integers <= 10^n that are divisible by no prime exceeding 19.

Original entry on oeis.org

1, 10, 72, 331, 1169, 3419, 8751, 20198, 42950, 85411, 160626, 288126, 496303, 825326, 1330766, 2088013, 3197529, 4791093, 7039193, 10159603, 14427309, 20186026, 27861175, 37974797, 51162295, 68191379, 89983125, 117635672
Offset: 0

Views

Author

Robert G. Wilson v, May 31 2005

Keywords

Crossrefs

Row 8 of A253635.
Cf. A123384, A100752, A106598, A106600, A107352, A106629, A108275, A108277, A011557, A080682.

Programs

  • Mathematica
    n = 9; t = Select[ Flatten[ Table[19^h*Select[ Flatten[ Table[17^g*Select[ Flatten[ Table[13^f*Select[ Flatten[ Table[11^e*Select[ Flatten[ Table[7^d*Select[ Flatten[ Table[5^c*Select[ Flatten[ Table[2^a*3^b, {a, 0, n*Log[2, 10]}, {b, 0, n*Log[3, 10]}]], # <= 10^n &], {c, 0, n*Log[5, 10]}]], # <= 10^n &], {d, 0, n*Log[7, 10]}]], # <= 10^n &], {e, 0, n*Log[11, 10]}]], # <= 10^n &], {f, 0, n*Log[13, 10]}]], # <= 10^n &], {g, 0, n*Log[17, 10]}]], # <= 10^n &], {h, 0, n*Log[19, 10]}]], # <= 10^n &]; Table[ Length[ Select[t, # <= 10^n &]], {n, 0, 10}]

Extensions

a(10)-a(18) from Donovan Johnson, Sep 16 2009
a(19)-a(27) from Max Alekseyev, Apr 28 2010

A108277 Number of positive integers <= 10^n that are divisible by no prime exceeding 23.

Original entry on oeis.org

1, 10, 76, 369, 1385, 4298, 11654, 28434, 63768, 133440, 263529, 495412, 892644, 1550012, 2605342, 4254753, 6771752, 10531080, 16038303, 23965659, 35195450, 50872227, 72464493, 101837746, 141340075, 193902062, 263152095, 353549942
Offset: 0

Views

Author

Robert G. Wilson v, May 31 2005

Keywords

Crossrefs

Row 9 of A253635.
Cf. A123384, A100752, A106598, A106600, A107352, A106629, A108275, A108276, A011557, A080683.

Programs

  • Mathematica
    n = 6; t = Select[ Flatten[ Table[23^i*Select[ Flatten[ Table[19^h*Select[ Flatten[ Table[17^g*Select[ Flatten[ Table[13^f*Select[ Flatten[ Table[11^e*Select[ Flatten[ Table[7^d*Select[ Flatten[ Table[5^c*Select[ Flatten[ Table[2^a*3^b, {a, 0, n*Log[2, 10]}, {b, 0, n*Log[3, 10]}]], # <= 10^n &], {c, 0, n*Log[5, 10]}]], # <= 10^n &], {d, 0, n*Log[7, 10]}]], # <= 10^n &], {e, 0, n*Log[11, 10]}]], # <= 10^n &], {f, 0, n*Log[13, 10]}]], # <= 10^n &], {g, 0, n*Log[17, 10]}]], # <= 10^n &], {h, 0, n*Log[19, 10]}]], # <= 10^n &], {i, 0, n*Log[23, 10]}]], # <= 10^n &]; Table[ Length[ Select[t, # <= 10^n &]], {n, 0, 10}]

Extensions

a(7)-a(18) from Donovan Johnson, Sep 16 2009
a(19)-a(27) from Max Alekseyev, Apr 28 2010
Showing 1-9 of 9 results.

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

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