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.

Previous Showing 41-49 of 49 results.

A132030 a(n) = Product_{k=0..floor(log_6(n))} floor(n/6^k), n>=1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 24, 26, 28, 30, 32, 34, 54, 57, 60, 63, 66, 69, 96, 100, 104, 108, 112, 116, 150, 155, 160, 165, 170, 175, 216, 222, 228, 234, 240, 246, 294, 301, 308, 315, 322, 329, 384, 392, 400, 408, 416, 424, 486, 495, 504, 513, 522, 531, 600
Offset: 1

Views

Author

Hieronymus Fischer, Aug 20 2007

Keywords

Comments

If n is written in base 6 as n = d(m)d(m-1)d(m-2)...d(2)d(1)d(0) (where d(k) is the digit at position k) then a(n) is also the product d(m)d(m-1)d(m-2)...d(2)d(1)d(0)*d(m)d(m-1)d(m-2)...d(2)d(1)*d(m)d(m-1)d(m-2)...d(2)*...*d(m)d(m-1)d(m-2)*d(m)d(m-1)*d(m).

Examples

			a(52) = floor(52/6^0)*floor(52/6^1)*floor(52/6^2) = 52*8*1 = 416;
a(58) = 522 since 58 = 134_6 and so a(58) = 134_6 * 13_6 * 1_6 = 58*9*1 = 522.

Links

Crossrefs

Cf. A048651, A132022, A132034, A000217.
For formulas regarding a general parameter p (i.e., terms floor(n/p^k)) see A132264.
For the product of terms floor(n/p^k) for p=2 to p=12 see A098844(p=2), A132027(p=3)-A132033(p=9), A067080(p=10), A132263(p=11), A132264(p=12).
For the products of terms 1+floor(n/p^k) see A132269-A132272, A132327, A132328.

Programs

  • Maple
    f:= proc(n) option remember; n*procname(floor(n/6)) end proc:
f(0):= 1:
seq(f(i),i=1..100); # Robert Israel, Dec 20 2015
  • Mathematica
    Table[Product[Floor[n/6^k], {k, 0, Floor[Log[6, n]]}], {n, 1, 100}] (* G. C. Greubel, Dec 20 2015 *)

Formula

Recurrence: a(n)=n*a(floor(n/6)); a(n*6^m)=n^m*6^(m(m+1)/2)*a(n).
a(k*6^m) = k^(m+1)*6^(m(m+1)/2), for 0
Asymptotic behavior: a(n) = O(n^((1+log_6(n))/2)); this follows from the inequalities below.
a(n) <= b(n), where b(n) = n^(1+floor(log_6(n)))/6^((1+floor(log_6(n)))*floor(log_6(n))/2); equality holds for n=k*6^m, 0=0. b(n) can also be written n^(1+floor(log_6(n)))/6^A000217(floor(log_6(n))).
Also: a(n) <= 2^((1-log_6(2))/2)*n^((1+log_6(n))/2) = 1.236766885...*6^A000217(log_6(n)), equality holds for n=2*6^m and for n=3*6^m, m>=0 (consider 2^((1-log_6(2))/2)=3^((1-log_6(3))/2) since 6=2*3).
a(n) > c*b(n), where c = 0.45071262522603913... (see constant A132022).
Also: a(n) > c*(sqrt(2)/2^log_6(sqrt(2)))*n^((1+log_6(n))/2) = 0.557426449...*6^A000217(log_6(n)).
lim inf a(n)/b(n) = 0.45071262522603913..., for n-->oo.
lim sup a(n)/b(n) = 1, for n-->oo.
lim inf a(n)/n^((1+log_6(n))/2) = 0.45071262522603913...*sqrt(2)/2^log_6(sqrt(2)), for n-->oo.
lim sup a(n)/n^((1+log_6(n))/2) = sqrt(3)/3^log_6(sqrt(3))=1.236766885..., for n-->oo.
lim inf a(n)/a(n+1) = 0.45071262522603913... for n-->oo (see constant A132022).
G.f. g(x) satisfies g(x) = (x+2x^2+3x^3+4x^4+5x^5)*(1 + g(x^6)) + 6*(x^6+x^7+x^8+x^9+x^10+x^11)*g'(x^6). - Robert Israel, Dec 20 2015

A132032 Product{0<=k<=floor(log_8(n)), floor(n/8^k)}, n>=1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 32, 34, 36, 38, 40, 42, 44, 46, 72, 75, 78, 81, 84, 87, 90, 93, 128, 132, 136, 140, 144, 148, 152, 156, 200, 205, 210, 215, 220, 225, 230, 235, 288, 294, 300, 306, 312, 318, 324, 330, 392, 399, 406, 413, 420, 427, 434
Offset: 1

Views

Author

Hieronymus Fischer, Aug 20 2007

Keywords

Comments

If n is written in base-8 as n=d(m)d(m-1)d(m-2)...d(2)d(1)d(0) (where d(k) is the digit at position k) then a(n) is also the product d(m)d(m-1)d(m-2)...d(2)d(1)d(0)*d(m)d(m-1)d(m-2)...d(2)d(1)*d(m)d(m-1)d(m-2)...d(2)*...*d(m)d(m-1)d(m-2)*d(m)d(m-1)*d(m).

Examples

			a(70)=floor(70/8^0)*floor(70/8^1)*floor(70/8^2)=70*8*1=560;
For n=75, 75=113(base-8) and so a(75)=113*11*1(base-8)=75*9*1=675.

Crossrefs

Cf. A048651, A132024, A132036, A000217.
For formulas regarding a general parameter p (i.e. terms floor(n/p^k)) see A132264.
For the product of terms floor(n/p^k) for p=2 to p=12 see A098844(p=2), A132027(p=3)-A132033(p=9), A067080(p=10), A132263(p=11), A132264(p=12).
For the products of terms 1+floor(n/p^k) see A132269-A132272, A132327, A132328.

Formula

Recurrence: a(n)=n*a(floor(n/8)); a(n*8^m)=n^m*8^(m(m+1)/2)*a(n).
a(k*8^m)=k^(m+1)*8^(m(m+1)/2), for 0
Asymptotic behavior: a(n)=O(n^((1+log_8(n))/2)); this follows from the inequalities below.
a(n)<=b(n), where b(n)=n^(1+floor(log_8(n)))/8^((1+floor(log_8(n)))*floor(log_8(n))/2); equality holds for n=k*8^m, 0=0. b(n) can also be written n^(1+floor(log_8(n)))/8^A000217(floor(log_8(n))).
Also: a(n)<=3^((1-log_8(3))/2)*n^((1+log_8(n))/2) = 1.295758534...*8^A000217(log_8(n)), equality holds for n=3*8^m, m>=0.
a(n)>c*b(n), where c = 0.46456888368647639098... (see constant A132024).
Also: a(n)>c*2^(1/3)*n^((1+log_8(n))/2)=0.4645688836...*1.25992105...*8^A000217(log_8(n)).
lim inf a(n)/b(n)=0.46456888368647639098..., for n-->oo.
lim sup a(n)/b(n)=1, for n-->oo.
lim inf a(n)/n^((1+log_8(n))/2)=0.46456888368647639098...*2^(1/3), for n-->oo.
lim sup a(n)/n^((1+log_8(n))/2)=sqrt(3)/3^log_8(sqrt(3))=1.295758534..., for n-->oo.
lim inf a(n)/a(n+1)=0.46456888368647639098... for n-->oo (see constant A132024).

A054900 a(n) = Sum_{j >= 1} floor(n/16^j).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6
Offset: 0

Views

Author

Henry Bottomley, May 23 2000

Keywords

Links

Crossrefs

Cf. A011371 and A054861 for analogs involving powers of 2 and 3.
Cf. A053836, A054897, A054899, A067080, A098844, A132032.

Programs

  • Magma
    m:=16;
function a(n) // a = A054900, m = 16
  if n eq 0 then return 0;
  else return a(Floor(n/m)) + Floor(n/m);
  end if; end function;
[a(n): n in [0..127]]; // G. C. Greubel, Apr 28 2023
  • Mathematica
    a[n_, m_]:= If[n==0, 0, a[Floor[n/m], m] +Floor[n/m]];
Table[a[n, 16], {n,0,127}] (* G. C. Greubel, Apr 28 2023 *)
  • SageMath
    m=16 # a = A054900
def a(n): return 0 if (n==0) else a(n//m) + (n//m)
[a(n) for n in range(128)] # G. C. Greubel, Apr 28 2023

Formula

a(n) = (n - A053836(n))/15.
From Hieronymus Fischer, Aug 14 2007: (Start)
Recurrence:
a(n) = a(floor(n/16)) + floor(n/16).
a(16*n) = a(n) + n.
a(n*16^m) = a(n) + n*(16^m - 1)/15.
a(k*16^m) = k*(16^m - 1)/15, for 0 <= k < 16, m>=0.
Asymptotic behavior:
a(n) = n/15 + O(log(n)).
a(n+1) - a(n) = O(log(n)) (this follows from the inequalities below).
a(n) <= (n-1)/15; equality holds for powers of 16.
a(n) >= (n-15)/15 - floor(log_16(n)); equality holds for n = 16^m - 1, m > 0.
Limits:
lim inf (n/15 - a(n)) = 1/15, for n --> oo.
lim sup (n/15 - log_16(n) - a(n)) = 0, for n --> oo.
lim sup (a(n+1) - a(n) - log_16(n)) = 0, for n --> oo.
Series:
G.f.: (1/(1-x))*Sum_{k > 0} x^(16^k)/(1-x^(16^k)). (End)

A132023 Decimal expansion of Product_{k>=0} 1-1/(2*7^k).

Original entry on oeis.org

4, 5, 8, 7, 6, 6, 7, 2, 6, 6, 9, 9, 7, 6, 8, 9, 8, 5, 0, 2, 0, 0, 0, 5, 1, 5, 3, 3, 6, 9, 7, 4, 3, 7, 2, 1, 7, 8, 2, 5, 4, 6, 6, 8, 8, 7, 1, 4, 7, 3, 1, 8, 7, 0, 0, 7, 8, 2, 4, 4, 0, 1, 3, 8, 5, 0, 6, 9, 9, 7, 4, 4, 0, 3, 2, 6, 5, 9, 3, 0, 3, 6, 5, 2, 3, 7, 8, 1, 7, 1, 0, 9, 0, 4, 0, 5, 8, 4, 7, 5, 9, 8, 2
Offset: 0

Views

Author

Hieronymus Fischer, Aug 14 2007

Keywords

Examples

			0.4587667266997689850200...

Crossrefs

Cf. A048651, A098844, A067080, A132019, A132026, A132031, A132035, A000217, A109808.

Programs

  • Mathematica
    digits = 103; NProduct[1-1/(2*7^k), {k, 0, Infinity}, NProductFactors -> 200, WorkingPrecision -> digits+5] // N[#, digits+5]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
RealDigits[QPochhammer[1/2, 1/7], 10, 120][[1]] (* Amiram Eldar, May 08 2023 *)

Formula

Equals lim inf_{n->oo} Product_{k=0..floor(log_7(n))} floor(n/7^k)*7^k/n.
Equals lim inf_{n->oo} A132031(n)/n^(1+floor(log_7(n)))*7^(1/2*(1+floor(log_7(n)))*floor(log_7(n))).
Equals lim inf_{n->oo} A132031(n)/n^(1+floor(log_7(n)))*7^A000217(floor(log_7(n))).
Equals 1/2*exp(-Sum_{n>0} 7^(-n)*Sum_{k|n} 1/(k*2^k)).
Equals lim inf_{n->oo} A132031(n)/A132031(n+1).
Equals Product_{n>=1} (1 - 1/A109808(n)). - Amiram Eldar, May 08 2023

A132025 Decimal expansion of Product_{k>=0} 1-1/(2*9^k).

Original entry on oeis.org

4, 6, 8, 9, 4, 5, 1, 7, 8, 3, 6, 7, 0, 2, 3, 6, 9, 3, 2, 8, 3, 2, 8, 0, 0, 3, 5, 4, 1, 8, 6, 5, 6, 3, 9, 4, 0, 6, 8, 0, 4, 5, 7, 5, 8, 6, 9, 8, 9, 8, 5, 6, 0, 1, 6, 7, 1, 9, 7, 9, 9, 2, 3, 2, 7, 4, 7, 5, 7, 3, 2, 8, 3, 4, 6, 7, 0, 4, 3, 8, 1, 7, 5, 4, 9, 5, 0, 9, 4, 2, 7, 5, 7, 0, 0, 0, 1, 5, 9, 1, 7, 1, 1
Offset: 0

Views

Author

Hieronymus Fischer, Aug 14 2007

Keywords

Examples

			0.4689451783670236932832800...

Crossrefs

Cf. A048651, A098844, A067080, A132019, A132026, A132033, A132037, A000217, A270369.

Programs

  • Mathematica
    digits = 103; NProduct[1-1/(2*9^k), {k, 0, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+5] // N[#, digits+5]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
RealDigits[QPochhammer[1/2, 1/9], 10, 120][[1]] (* Amiram Eldar, May 08 2023 *)

Formula

Equals lim inf_{n->oo} Product_{k=0..floor(log_9(n))} floor(n/9^k)*9^k/n.
Equals lim inf_{n->oo} A132033(n)/n^(1+floor(log_9(n)))*9^(1/2*(1+floor(log_9(n)))*floor(log_9(n))).
Equals lim inf_{n->oo} A132033(n)/n^(1+floor(log_9(n)))*9^A000217(floor(log_9(n))).
Equals (1/2)*exp(-Sum_{n>0} 9^(-n)*Sum_{k|n} 1/(k*2^k)).
Equals lim inf_{n->oo} A132033(n)/A132033(n+1).
Equals Product_{n>=1} (1 - 1/A270369(n)). - Amiram Eldar, May 08 2023

A132021 Decimal expansion of Product_{k>=0} 1-1/(2*5^k).

Original entry on oeis.org

4, 3, 8, 7, 9, 6, 8, 3, 7, 2, 0, 3, 6, 3, 8, 5, 3, 1, 2, 6, 6, 7, 2, 9, 9, 9, 7, 1, 7, 7, 2, 5, 8, 3, 5, 9, 6, 0, 4, 5, 7, 4, 6, 3, 1, 2, 3, 9, 3, 5, 1, 1, 6, 5, 4, 1, 7, 7, 3, 6, 7, 5, 6, 4, 3, 6, 7, 9, 1, 0, 6, 6, 5, 6, 9, 8, 6, 6, 5, 0, 0, 6, 9, 2, 8, 9, 6, 6, 7, 2, 3, 8, 9, 8, 5, 4, 4, 0, 0, 6, 0, 2, 8
Offset: 0

Views

Author

Hieronymus Fischer, Aug 14 2007

Keywords

Examples

			0.438796837203638531...

Crossrefs

Cf. A048651, A098844, A067080, A132019, A132026, A132029, A100222, A000217, A020729.

Programs

  • Mathematica
    digits = 103; NProduct[1-1/(2*5^k), {k, 0, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+5] // N[#, digits+5]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
RealDigits[QPochhammer[1/2, 1/5], 10, 120][[1]] (* Amiram Eldar, May 08 2023 *)

Formula

Equals lim inf_{n->oo} Product_{k=0..floor(log_5(n))} floor(n/5^k)*5^k/n.
Equals lim inf_{n->oo} A132029(n)/n^(1+floor(log_5(n)))*5^(1/2*(1+floor(log_5(n)))*floor(log_5(n))).
Equals lim inf_{n->oo} A132029(n)/n^(1+floor(log_5(n)))*5^A000217(floor(log_5(n))).
Equals (1/2)*exp(-Sum_{n>0} 5^(-n)*Sum_{k|n} 1/(k*2^k)).
Equals lim inf_{n->oo} A132029(n)/A132029(n+1).
Equals Product_{n>=0} (1 - 1/A020729(n)). - Amiram Eldar, May 08 2023

A132031 Product{0<=k<=floor(log_7(n)), floor(n/7^k)}, n>=1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 28, 30, 32, 34, 36, 38, 40, 63, 66, 69, 72, 75, 78, 81, 112, 116, 120, 124, 128, 132, 136, 175, 180, 185, 190, 195, 200, 205, 252, 258, 264, 270, 276, 282, 288, 343, 350, 357, 364, 371, 378, 385, 448, 456, 464, 472, 480, 488
Offset: 1

Views

Author

Hieronymus Fischer, Aug 20 2007

Keywords

Comments

If n is written in base-7 as n=d(m)d(m-1)d(m-2)...d(2)d(1)d(0) (where d(k) is the digit at position k) then a(n) is also the product d(m)d(m-1)d(m-2)...d(2)d(1)d(0)*d(m)d(m-1)d(m-2)...d(2)d(1)*d(m)d(m-1)d(m-2)...d(2)*...*d(m)d(m-1)d(m-2)*d(m)d(m-1)*d(m).

Examples

			a(52)=floor(52/7^0)*floor(52/7^1)*floor(52/7^2)=52*7*1=364.
a(58)=464 since 58=112(base-7) and so a(58)=112*11*1(base-7)=58*8*1=464.

Crossrefs

Cf. A048651, A132023, A132035, A000217.
For formulas regarding a general parameter p (i.e. terms floor(n/p^k)) see A132264.
For the product of terms floor(n/p^k) for p=2 to p=12 see A098844(p=2), A132027(p=3)-A132033(p=9), A067080(p=10), A132263(p=11), A132264(p=12).
For the products of terms 1+floor(n/p^k) see A132269-A132272, A132327, A132328.

Programs

  • Mathematica
    Table[Times@@Floor[n/7^Range[0,Floor[Log[7,n]]]],{n,70}] (* Harvey P. Dale, Oct 11 2017 *)

Formula

Recurrence: a(n)=n*a(floor(n/7)); a(n*7^m)=n^m*7^(m(m+1)/2)*a(n).
a(k*7^m)=k^(m+1)*7^(m(m+1)/2), for 0
Asymptotic behavior: a(n)=O(n^((1+log_7(n))/2)this follows from the inequalities below.
a(n)<=b(n), where b(n)=n^(1+floor(log_7(n)))/7^((1+floor(log_7(n)))*floor(log_7(n))/2); equality holds for n=k*7^m, 0=0. b(n) can also be written n^(1+floor(log_7(n)))/7^A000217(floor(log_7(n))).
Also: a(n)<=3^((1-log_7(3))/2)*n^((1+log_7(n))/2)=1.270209197...*7^A000217(log_7(n)), equality holds for n=3*7^m, m>=0.
a(n)>c*b(n), where c=0.4587667266997689850200... (see constant A132023).
Also: a(n)>c*(sqrt(2)/2^log_7(sqrt(2)))*n^((1+log_7(n))/2)=0.4587667266...*1.249972544...*7^A000217(log_7(n)).
lim inf a(n)/b(n)=0.4587667266997689850200..., for n-->oo.
lim sup a(n)/b(n)=1, for n-->oo.
lim inf a(n)/n^((1+log_7(n))/2)=0.4587667266997689850200...*sqrt(2)/2^log_7(sqrt(2)), for n-->oo.
lim sup a(n)/n^((1+log_7(n))/2)=sqrt(3)/3^log_7(sqrt(3))=1.270209197..., for n-->oo.
lim inf a(n)/a(n+1)=0.4587667266997689850200... for n-->oo (see constant A132023).

A067082 If n = abc...def in decimal notation then the right digit sum function = abc...def + bc...def + c...def + ... + def + ef + f.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 70, 72
Offset: 0

Views

Author

Amarnath Murthy, Jan 05 2002

Keywords

Examples

			a(256) = 256 + 56 + 6 = 318.

Crossrefs

Cf. A067079, A067080, A065039.

Programs

  • Mathematica
    Table[d = IntegerDigits[n]; rd = 0; While[ Length[d] > 0, rd = rd + FromDigits[d]; d = Drop[d, 1]]; rd, {n, 0, 75} ]

Formula

a(abcd) = abcd+(abcd-1000a)+(abcd-1000a-100b)+(abcd-1000a-100b-10c).
n*length(n)-Sum_{k=1..length(n)} 10^k*floor(n/10^k). - Vladeta Jovovic, Jan 08 2002

Extensions

More terms from Robert G. Wilson v, Jan 07 2002

A071714 Numbers n such that Rd(n) + Ld(n) +/-1 is prime, where Rd and Ld are the right- and left-digital factorial functions.

Original entry on oeis.org

2, 3, 6, 9, 33, 90, 102, 195, 210, 276, 379, 380, 402, 414, 575, 588, 616, 618, 916, 939, 980, 984, 1110, 1112, 1210, 1314, 1614, 2132, 2136, 2166, 2190, 2280, 2372, 2394, 2438, 2468, 2730, 2780, 3360, 3436, 3510, 3816, 3842, 3940, 3950, 4222, 4236, 4344
Offset: 1

Views

Author

Jason Earls, Jun 03 2002

Keywords

Examples

			195 is a term because (195*95*5)+(195*19*1)+1 = 96331 and (195*95*5)+(195*19*1)-1 = 96329 are both prime.

Crossrefs

Cf. A067079, A067080.
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