A132269 -id:A132269 - cp's OEIS frontend

A132028 Product{0<=k<=floor(log_4(n)), floor(n/4^k)}, n>=1.

1, 2, 3, 4, 5, 6, 7, 16, 18, 20, 22, 36, 39, 42, 45, 64, 68, 72, 76, 100, 105, 110, 115, 144, 150, 156, 162, 196, 203, 210, 217, 512, 528, 544, 560, 648, 666, 684, 702, 800, 820, 840, 860, 968, 990, 1012, 1034, 1728, 1764, 1800, 1836, 2028, 2067, 2106, 2145, 2352

Offset: 1

Hieronymus Fischer, Aug 20 2007

nonn

If n is written in base-4 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).

			a(26)=floor(26/4^0)*floor(26/4^1)*floor(26/4^2)=26*6*1=156; a(34)=544 since 34=202(base-4) and so
a(34)=202*20*2(base-4)=34*8*2=544.

Cf. A048651, A132020, A100221, 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.

Recurrence: a(n)=n*a(floor(n/4)); a(n*4^m)=n^m*4^(m(m+1)/2)*a(n).

a(k*4^m)=k^(m+1)*4^(m(m+1)/2), for 0

Asymptotic behavior: a(n)=O(n^((1+log_4(n))/2)); this follows from the inequalities below.

a(n)<=b(n), where b(n)=n^(1+floor(log_4(n)))/4^((1+floor(log_4(n)))*floor(log_4(n))/2); equality holds for n=k*4^m, 0=0. b(n) can also be written n^(1+floor(log_4(n)))/4^A000217(floor(log_4(n))).

Also: a(n)<=2^(1/4)*n^((1+log_4(n))/2)=1.189207...*4^A000217(log_4(n)), equality holds for n=2*4^m, m>=0.

a(n)>c*b(n), where c=0.4194224417951075977... (see constant A132020).

Also: a(n)>c*2^(1/4)*n^((1+log_4(n))/2)=0.498780...*4^A000217(log_4(n)).

lim inf a(n)/b(n)=0.4194224417951075977..., for n-->oo.

lim sup a(n)/b(n)=1, for n-->oo.

lim inf a(n)/n^((1+log_4(n))/2)=0.4194224417951075977...*2^(1/4), for n-->oo.

lim sup a(n)/n^((1+log_4(n))/2)=2^(1/4), for n-->oo.

lim inf a(n)/a(n+1)=0.4194224417951075977... for n-->oo (see constant A132020).

A132029 Product{0<=k<=floor(log_5(n)), floor(n/5^k)}, n>=1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 20, 22, 24, 26, 28, 45, 48, 51, 54, 57, 80, 84, 88, 92, 96, 125, 130, 135, 140, 145, 180, 186, 192, 198, 204, 245, 252, 259, 266, 273, 320, 328, 336, 344, 352, 405, 414, 423, 432, 441, 1000, 1020, 1040, 1060, 1080, 1210, 1232, 1254, 1276

Offset: 1

Hieronymus Fischer, Aug 20 2007

nonn

If n is written in base-5 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).

			a(26)=floor(26/5^0)*floor(26/5^1)*floor(26/5^2)=26*5*1=130; a(34)=204 since 34=114(base-5) and so a(34)=114*11*1(base-5)=34*6*1=204.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A048651, A132021, A100222, 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.

Table[Product[Floor[n/5^k],{k,0,Floor[Log[5,n]]}],{n,60}] (* Harvey P. Dale, Oct 16 2019 *)

Recurrence: a(n)=n*a(floor(n/5)); a(n*5^m)=n^m*5^(m(m+1)/2)*a(n).

a(k*5^m)=k^(m+1)*5^(m(m+1)/2), for 0

Asymptotic behavior: a(n)=O(n^((1+log_5(n))/2)); this follows from the inequalities below.

a(n)<=b(n), where b(n)=n^(1+floor(log_5(n)))/5^((1+floor(log_5(n)))*floor(log_5(n))/2); equality holds for n=k*5^m, 0=0. b(n) can also be written n^(1+floor(log_5(n)))/5^A000217(floor(log_5(n))).

Also: a(n)<=2^((1-log_5(2))/2)*n^((1+log_5(n))/2)=1.2181246...*5^A000217(log_5(n)), equality holds for n=2*5^m, m>=0.

a(n)>c*b(n), where c=0.438796837203638531... (see constant A132021).

Also: a(n)>c*(sqrt(2)/2^log_5(sqrt(2)))*n^((1+log_5(n))/2)=0.534509224...*5^A000217(log_5(n)).

lim inf a(n)/b(n)=0.438796837203638531..., for n-->oo.

lim sup a(n)/b(n)=1, for n-->oo.

lim inf a(n)/n^((1+log_5(n))/2)=0.438796837203638531...*sqrt(2)/2^log_5(sqrt(2)), for n-->oo.

lim sup a(n)/n^((1+log_5(n))/2)=sqrt(2)/2^log_5(sqrt(2))=1.2181246..., for n-->oo.

lim inf a(n)/a(n+1)=0.438796837203638531... for n-->oo (see constant A132021).

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

Hieronymus Fischer, Aug 20 2007

nonn

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).

			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.

G. C. Greubel, Table of n, a(n) for n = 1..10000

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.

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

Table[Product[Floor[n/6^k], {k, 0, Floor[Log[6, n]]}], {n, 1, 100}] (* G. C. Greubel, Dec 20 2015 *)

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

Hieronymus Fischer, Aug 20 2007

nonn

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).

			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.

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.

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).

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

Hieronymus Fischer, Aug 20 2007

nonn

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).

			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.

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.

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

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).

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A132028 Product{0<=k<=floor(log_4(n)), floor(n/4^k)}, n>=1.

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A132029 Product{0<=k<=floor(log_5(n)), floor(n/5^k)}, n>=1.

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

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

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

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