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 1-5 of 5 results.

A071520 Number of 5-smooth numbers (A051037) <= n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, 10, 10, 11, 12, 12, 13, 13, 14, 14, 14, 14, 15, 16, 16, 17, 17, 17, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 23, 23, 24, 24, 24, 24, 25, 25, 25, 25, 25, 25, 26, 26, 26, 26, 27, 27, 27, 27, 27, 27, 27, 27, 28, 28
Offset: 1

Views

Author

Benoit Cloitre, Jun 02 2002

Keywords

Comments

A 5-smooth number is a number of the form 2^x*3^y*5^z (x,y,z) >= 0.

Links

Crossrefs

Cf. A008683, A051037, A106598, A112751.
Number of p-smooth numbers <= n: A070939 (p=2), A071521 (p=3), this sequence (p=5), A071604 (p=7), A071523 (p=11), A080684 (p=13), A080685 (p=17), A080686 (p=19).

Programs

  • Mathematica
    Accumulate[Table[If[Max[FactorInteger[n][[;;,1]]]<6,1,0],{n,80}]] (* Harvey P. Dale, Aug 04 2024 *)
  • PARI
    for(n=1,100,print1(sum(k=1,n,if(sum(i=4,n,if(k%prime(i),0,1)),0,1)),","))
  • PARI
    a(n)=-sum(k=1,n,moebius(2*3*5*k)*floor(n/k)) \\ Benoit Cloitre, Jun 14 2007
  • Python
    from sympy import integer_log
def A071520(n):
    c = 0
    for i in range(integer_log(n,5)[0]+1):
        for j in range(integer_log(m:=n//5**i,3)[0]+1):
            c += (m//3**j).bit_length()
    return c # Chai Wah Wu, Sep 16 2024

Formula

a(n) = Card{ k | A051037(k) <= n }.
Asymptotically : let a = 1/(6*log(2)*log(3)*log(5)) and b = sqrt(30) then a(n) = a*log(b*n)^3 + O(log(n)).
a(n) = -Sum_{k=1,n} mu(30*k)*floor(n/k). - Benoit Cloitre, Jun 14 2007
a(n) = Sum_{i=0..floor(log_5(n))} Sum_{j=0..floor(log_3(n/5^i))} floor(log_2(2*n/(5^i*3^j))). - Ridouane Oudra, Jul 17 2020

Extensions

Title corrected by Rainer Rosenthal, Aug 30 2020

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 *)

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

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 1-5 of 5 results.

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