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 1-6 of 6 results.

A253635 Rectangular array read by upwards antidiagonals: a(n,k) = index of largest term <= 10^k in row n of A253572, n >= 1, k >= 0.

Original entry on oeis.org

1, 1, 4, 1, 7, 7, 1, 9, 20, 10, 1, 10, 34, 40, 14, 1, 10, 46, 86, 67, 17, 1, 10, 55, 141, 175, 101, 20, 1, 10, 62, 192, 338, 313, 142, 24, 1, 10, 67, 242, 522, 694, 507, 190, 27, 1, 10, 72, 287, 733, 1197, 1273, 768, 244, 30
Offset: 1

Views

Author

L. Edson Jeffery, Jan 07 2015

Keywords

Comments

Or a(n,k) = the number of positive integers less than or equal to 10^k that are divisible by no prime exceeding prime(n).

Examples

			Array begins:
{1,  4,  7,  10,   14,   17,    20,    24,    27,     30, ...}
{1,  7, 20,  40,   67,  101,   142,   190,   244,    306, ...}
{1,  9, 34,  86,  175,  313,   507,   768,  1105,   1530, ...}
{1, 10, 46, 141,  338,  694,  1273,  2155,  3427,   5194, ...}
{1, 10, 55, 192,  522, 1197,  2432,  4520,  7838,  12867, ...}
{1, 10, 62, 242,  733, 1848,  4106,  8289, 15519,  27365, ...}
{1, 10, 67, 287,  945, 2579,  6179, 13389, 26809,  50351, ...}
{1, 10, 72, 331, 1169, 3419,  8751, 20198, 42950,  85411, ...}
{1, 10, 76, 369, 1385, 4298, 11654, 28434, 63768, 133440, ...}
{1, 10, 79, 402, 1581, 5158, 14697, 37627, 88415, 193571, ...}

Crossrefs

Cf. A123384, A100752, A106598, A106600, A107352, A106629, A108275, A108276, A108277 (rows 1-9).
Cf. A011557, A066343, A253572, A253573.

Programs

  • Mathematica
    r = 10; y[1] = t = Table[2^j, {j, 0, 39}]; max = 10^13; len = 10^10; prev = 0; For[n = 2, n <= r, n++, next = 0; For[k = 1, k <= 43, k++, If[Prime[n]^k < max, t = Union[t, Prime[n]*t]; s = FirstPosition[t, v_ /; v > len, 0]; t = Take[t, s[[1]] - 1]; If[t[[-1]] > len, t = Delete[t, -1]]; next = Length[t]; If[next == prev, Break, prev = next], Break]]; y[n] = t]; b[i_, j_] := FirstPosition[y[i], v_ /; v > 10^j][[1]]; a253635[n_, j_] := If[IntegerQ[b[n, j]], b[n, j] - 1, 0]; Flatten[Table[a253635[n - j, j], {n, r}, {j, 0, n - 1}]] (* array antidiagonals flattened *)

A071604 a(n) is the number of 7-smooth numbers <= n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 11, 11, 12, 13, 14, 14, 15, 15, 16, 17, 17, 17, 18, 19, 19, 20, 21, 21, 22, 22, 23, 23, 23, 24, 25, 25, 25, 25, 26, 26, 27, 27, 27, 28, 28, 28, 29, 30, 31, 31, 31, 31, 32, 32, 33, 33, 33, 33, 34, 34, 34, 35, 36, 36, 36, 36, 36, 36, 37, 37, 38
Offset: 1

Views

Author

Benoit Cloitre, Jun 02 2002

Keywords

Comments

A 7-smooth number is a number of the form 2^x*3^y*5^z*7^u, (x,y,z,u) >= 0.
In other words, a 7-smooth number is a number with no prime factor greater than 7. - Peter Munn, Nov 20 2021

Examples

			a(11) = 10 as there are 10 7-smooth numbers <= 11. Namely 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. - _David A. Corneth_, Apr 19 2021

Links

Crossrefs

Partial sums of A086299.
Column 7 of A080786.
Cf. A002473, A106600, A343598.
Equivalent sequences with other limits on greatest prime factor: A070939 (2), A071521 (3), A071520 (5), A071523 (11), A080684 (13), A080685 (17), A080686 (19), A096300 (log n).

Programs

  • PARI
    for(n=1,100,print1(sum(k=1,n,if(sum(i=5,n,if(k%prime(i),0,1)),0,1)),","))
  • Python
    from sympy import integer_log
def A071604(n):
    c = 0
    for i in range(integer_log(n,7)[0]+1):
        i7 = 7**i
        m = n//i7
        for j in range(integer_log(m,5)[0]+1):
            j5 = 5**j
            r = m//j5
            for k in range(integer_log(r,3)[0]+1):
                c += (r//3**k).bit_length()
    return c # Chai Wah Wu, Sep 16 2024

Formula

a(n) = Card{ k | A002473 (k) <= n }.

Extensions

Name corrected by David A. Corneth, Apr 19 2021

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

A085630 Number of n-digit 7-smooth numbers (A002473).

Original entry on oeis.org

0, 9, 36, 95, 197, 356, 579, 882, 1272, 1767, 2381, 3113, 3984, 5002, 6187, 7545, 9081, 10815, 12759, 14927, 17323, 19960, 22853, 26015, 29458, 33188, 37222, 41568, 46245, 51254, 56618, 62338, 68437, 74917, 81793, 89083, 96786, 104926, 113511
Offset: 0

Views

Author

Jason Earls and Amarnath Murthy, Jul 10 2003

Keywords

Links

Crossrefs

Cf. A002473, A071604, A106600.

Programs

  • PARI
    \\ Here b(n) is A071604.
b(m)={sum(i=0, logint(m,7), my(p=m\7^i); sum(j=0, logint(p,5), my(q=p\5^j); sum(k=0, logint(q,3), logint(q\3^k,2)+1 )))}
a(n)={if(n>0, b(10^n-1))-if(n>1, b(10^(n-1)-1))} \\ Andrew Howroyd, Sep 20 2024

Formula

From Andrew Howroyd, Sep 20 2024: (Start)
a(n) = A106600(n) - A106600(n-1) for n > 0.
a(n) = A071604(10^n-1) - A071604(10^(n-1)-1) for n > 1. (End)

Extensions

More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Nov 18 2004
Name changed by Andrew Howroyd, Sep 20 2024

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 1-6 of 6 results.

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