A025616 -id:A025616

A003596

Numbers of the form 2^i * 11^j.

+10
22

1, 2, 4, 8, 11, 16, 22, 32, 44, 64, 88, 121, 128, 176, 242, 256, 352, 484, 512, 704, 968, 1024, 1331, 1408, 1936, 2048, 2662, 2816, 3872, 4096, 5324, 5632, 7744, 8192, 10648, 11264, 14641, 15488, 16384, 21296, 22528, 29282, 30976, 32768

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

A204455(11*a(n)) = 11, and only for these numbers. - Wolfdieter Lang, Feb 04 2012

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 100 terms from Vincenzo Librandi)

FORMULA

The characteristic function of this sequence is given by Sum_{n >= 1} x^a(n) = Sum_{n >= 1} mu(22*n)*x^n/(1 - x^n), where mu(n) is the Möbius function A008683. Cf. with the formula of Hanna in A051037. - Peter Bala, Mar 18 2019

Sum_{n>=1} 1/a(n) = (2*11)/((2-1)*(11-1)) = 11/5. - Amiram Eldar, Sep 23 2020

a(n) ~ exp(sqrt(2*log(2)*log(11)*n)) / sqrt(22). - Vaclav Kotesovec, Sep 23 2020

MATHEMATICA

fQ[n_] := PowerMod[22, n, n]==0; Select[Range[40000], fQ] (* Vincenzo Librandi, Feb 04 2012 *)

PROG

(PARI) list(lim)=my(v=List(), N); for(n=0, log(lim)\log(11), N=11^n; while(N<=lim, listput(v, N); N<<=1)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

(Haskell)

import Data.Set (singleton, deleteFindMin, insert)

a003596 n = a003596_list !! (n-1)

a003596_list = f $ singleton (1, 0, 0) where

f s = y : f (insert (2 * y, i + 1, j) $ insert (11 * y, i, j + 1) s')

where ((y, i, j), s') = deleteFindMin s

-- Reinhard Zumkeller, May 15 2015

(Magma) [n: n in [1..2*10^5] | PrimeDivisors(n) subset [2, 11]]; // Vincenzo Librandi, Jun 27 2016

(GAP) Filtered([1..33000], n->PowerMod(22, n, n)=0); # Muniru A Asiru, Mar 19 2019

CROSSREFS

Cf. A025612, A025616, A025621, A025625, A025629, A025632, A025634, A025635, A108761, A003597, A107988, A003598, A108698, A003599, A107788, A108687, A108779, A108090.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A003598

Numbers of the form 5^i * 11^j.

+10
22

1, 5, 11, 25, 55, 121, 125, 275, 605, 625, 1331, 1375, 3025, 3125, 6655, 6875, 14641, 15125, 15625, 33275, 34375, 73205, 75625, 78125, 161051, 166375, 171875, 366025, 378125, 390625, 805255, 831875, 859375, 1771561, 1830125, 1890625

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

FORMULA

An asymptotic formula for a(n) is roughly 1/sqrt(55)*exp(sqrt(2*log(5)*log(11)*n)). - Benoit Cloitre, Mar 08 2002

The characteristic function of this sequence is given by Sum_{n >= 1} x^a(n) = Sum_{n >= 1} mu(55*n)*x^n/(1 - x^n), where mu(n) is the Möbius function A008683. Cf. with the formula of Hanna in A051037. - Peter Bala, Mar 18 2019

Sum_{n>=1} 1/a(n) = (5*11)/((5-1)*(11-1)) = 11/8. - Amiram Eldar, Sep 23 2020

a(n) ~ exp(sqrt(2*log(5)*log(11)*n)) / sqrt(55). - Vaclav Kotesovec, Sep 23 2020

MATHEMATICA

Take[Union[(5^#[[1]] 11^#[[2]])&/@Tuples[Range[0, 20], {2}]], 50] (* Harvey P. Dale, Dec 26 2010 *)

fQ[n_]:=PowerMod[55, n, n] == 0; Select[Range[2*10^6], fQ] (* Vincenzo Librandi, Jun 27 2016 *)

PROG

(PARI) list(lim)=my(v=List(), N); for(n=0, log(lim)\log(11), N=11^n; while(N<=lim, listput(v, N); N*=5)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

(Haskell)

import Data.Set (singleton, deleteFindMin, insert)

a003598 n = a003598_list !! (n-1)

a003598_list = f $ singleton (1, 0, 0) where

f s = y : f (insert (5 * y, i + 1, j) $ insert (11 * y, i, j + 1) s')

where ((y, i, j), s') = deleteFindMin s

-- Reinhard Zumkeller, May 15 2015

(Magma) [n: n in [1..2*10^6] | PrimeDivisors(n) subset [5, 11]]; // Vincenzo Librandi, Jun 27 2016

(GAP) Filtered([1..2*10^6], n->PowerMod(55, n, n)=0); # Muniru A Asiru, Mar 19 2019

(Sage)

[n for n in (1..2*10^6) if n%55 in {0, 1, 5, 11, 15, 20, 25, 45} and all(x in {5, 11} for x in prime_factors(n))] # F. Chapoton, Mar 16 2020

CROSSREFS

Cf. A025612, A025616, A025621, A025625, A025629, A025632, A025634, A025635, A108761, A003596, A003597, A107988, A108698, A003599, A107788, A108687, A108779, A108090.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A003597

Numbers of the form 3^i*11^j.

+10
21

1, 3, 9, 11, 27, 33, 81, 99, 121, 243, 297, 363, 729, 891, 1089, 1331, 2187, 2673, 3267, 3993, 6561, 8019, 9801, 11979, 14641, 19683, 24057, 29403, 35937, 43923, 59049, 72171, 88209, 107811, 131769, 161051, 177147, 216513, 264627, 323433

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

FORMULA

The characteristic function of this sequence is given by Sum_{n >= 1} x^a(n) = Sum_{n >= 1} mu(33*n)*x^n/(1 - x^n), where mu(n) is the Möbius function A008683. Cf. with the formula of Hanna in A051037. - Peter Bala, Mar 18 2019

Sum_{n>=1} 1/a(n) = (3*11)/((3-1)*(11-1)) = 33/20. - Amiram Eldar, Sep 23 2020

a(n) ~ exp(sqrt(2*log(3)*log(11)*n)) / sqrt(33). - Vaclav Kotesovec, Sep 23 2020

MATHEMATICA

fQ[n_]:=PowerMod[33, n, n] == 0; Select[Range[4*10^5], fQ] (* Vincenzo Librandi, Jun 27 2016 *)

PROG

(PARI) list(lim)=my(v=List(), N); for(n=0, log(lim)\log(11), N=11^n; while(N<=lim, listput(v, N); N*=3)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

(Haskell)

import Data.Set (singleton, deleteFindMin, insert)

a003597 n = a003597_list !! (n-1)

a003597_list = f $ singleton (1, 0, 0) where

f s = y : f (insert (3 * y, i + 1, j) $ insert (11 * y, i, j + 1) s')

where ((y, i, j), s') = deleteFindMin s

-- Reinhard Zumkeller, May 15 2015

(Magma) [n: n in [1..4*10^5] | PrimeDivisors(n) subset [3, 11]]; // Vincenzo Librandi, Jun 27 2016

(GAP) Filtered([1..324000], n->PowerMod(33, n, n)=0); # Muniru A Asiru, Mar 19 2019

CROSSREFS

Cf. A025612, A025616, A025621, A025625, A025629, A025632, A025634, A025635, A108761, A003596, A107988, A003598, A108698, A003599, A107788, A108687, A108779, A108090.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A003599

Numbers of the form 7^i*11^j.

+10
21

1, 7, 11, 49, 77, 121, 343, 539, 847, 1331, 2401, 3773, 5929, 9317, 14641, 16807, 26411, 41503, 65219, 102487, 117649, 161051, 184877, 290521, 456533, 717409, 823543, 1127357, 1294139, 1771561, 2033647, 3195731, 5021863, 5764801

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

FORMULA

The characteristic function of this sequence is given by Sum_{n >= 1} x^a(n) = Sum_{n >= 1} mu(77*n)*x^n/(1 - x^n), where mu(n) is the Möbius function A008683. Cf. with the formula of Hanna in A051037. - Peter Bala, Mar 18 2019

Sum_{n>=1} 1/a(n) = (7*11)/((7-1)*(11-1)) = 77/60. - Amiram Eldar, Sep 23 2020

a(n) ~ exp(sqrt(2*log(7)*log(11)*n)) / sqrt(77). - Vaclav Kotesovec, Sep 23 2020

MATHEMATICA

Take[Union[7^#[[1]] 11^#[[2]]&/@Tuples[Range[0, 9], 2]], 40] (* Harvey P. Dale, Mar 11 2015 *)

fQ[n_]:=PowerMod[77, n, n] == 0; Select[Range[6 10^6], fQ] (* Vincenzo Librandi, Jun 27 2016 *)

PROG

(PARI) list(lim)=my(v=List(), N); for(n=0, log(lim)\log(11), N=11^n; while(N<=lim, listput(v, N); N*=7)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

(Haskell)

import Data.Set (singleton, deleteFindMin, insert)

a003599 n = a003599_list !! (n-1)

a003599_list = f $ singleton (1, 0, 0) where

f s = y : f (insert (7 * y, i + 1, j) $ insert (11 * y, i, j + 1) s')

where ((y, i, j), s') = deleteFindMin s

-- Reinhard Zumkeller, May 15 2015

(Magma) [n: n in [1..6*10^6] | PrimeDivisors(n) subset [7, 11]]; // Vincenzo Librandi, Jun 27 2016

CROSSREFS

Cf. A025612, A025616, A025621, A025625, A025629, A025632, A025634, A025635, A108761, A003596, A003597, A107988, A003598, A108698, A107788, A108687, A108779, A108090.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A025629

Numbers of form 6^i*10^j with i, j >= 0.

+10
12

1, 6, 10, 36, 60, 100, 216, 360, 600, 1000, 1296, 2160, 3600, 6000, 7776, 10000, 12960, 21600, 36000, 46656, 60000, 77760, 100000, 129600, 216000, 279936, 360000, 466560, 600000, 777600, 1000000, 1296000, 1679616, 2160000, 2799360, 3600000, 4665600

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

FORMULA

Sum_{n>=1} 1/a(n) = (6*10)/((6-1)*(10-1)) = 4/3. - Amiram Eldar, Sep 26 2020

a(n) ~ exp(sqrt(2*log(6)*log(10)*n)) / sqrt(60). - Vaclav Kotesovec, Sep 26 2020

MATHEMATICA

n = 10^6; Flatten[Table[6^i*10^j, {i, 0, Log[6, n]}, {j, 0, Log10[n/6^i]}]] // Sort (* Amiram Eldar, Sep 26 2020 *)

PROG

(PARI) list(lim)=my(v=List(), N); for(n=0, logint(lim\=1, 10), N=10^n; while(N<=lim, listput(v, N); N*=6)); Set(v) \\ Charles R Greathouse IV, Jan 10 2018

CROSSREFS

Cf. A025610, A025618, A025622, A025626, A025627, A025628.

Cf. A025612, A025616, A025621, A025625, A025632, A025634.

KEYWORD

easy,nonn

AUTHOR

David W. Wilson

STATUS

approved

A025632

Numbers of form 7^i*10^j, with i, j >= 0.

+10
11

1, 7, 10, 49, 70, 100, 343, 490, 700, 1000, 2401, 3430, 4900, 7000, 10000, 16807, 24010, 34300, 49000, 70000, 100000, 117649, 168070, 240100, 343000, 490000, 700000, 823543, 1000000, 1176490, 1680700, 2401000, 3430000, 4900000, 5764801, 7000000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

FORMULA

Sum_{n>=1} 1/a(n) = (7*10)/((7-1)*(10-1)) = 35/27. - Amiram Eldar, Sep 25 2020

a(n) ~ exp(sqrt(2*log(7)*log(10)*n)) / sqrt(70). - Vaclav Kotesovec, Sep 25 2020

MATHEMATICA

n = 10^6; Flatten[Table[7^i*10^j, {i, 0, Log[7, n]}, {j, 0, Log10[n/7^i]}]] // Sort (* Amiram Eldar, Sep 25 2020 *)

PROG

(Haskell)

import Data.Set (singleton, deleteFindMin, insert)

a025632 n = a025632_list !! (n-1)

a025632_list = f $ singleton (1, 0, 0) where

f s = y : f (insert (7 * y, i + 1, j) $ insert (10 * y, i, j + 1) s')

where ((y, i, j), s') = deleteFindMin s

-- Reinhard Zumkeller, May 15 2015

(PARI) list(lim)=my(v=List(), N); for(n=0, logint(lim\=1, 10), N=10^n; while(N<=lim, listput(v, N); N*=7)); Set(v) \\ Charles R Greathouse IV, Jan 10 2018

CROSSREFS

Cf. A025612, A025616, A025621, A025625, A025629, A025634, A025635, A108761, A003596, A003597, A107988, A003598, A108698, A003599, A107788, A108687, A108779, A108090.

KEYWORD

easy,nonn

AUTHOR

David W. Wilson

STATUS

approved

A025635

Numbers of form 9^i*10^j, with i, j >= 0.

+10
9

1, 9, 10, 81, 90, 100, 729, 810, 900, 1000, 6561, 7290, 8100, 9000, 10000, 59049, 65610, 72900, 81000, 90000, 100000, 531441, 590490, 656100, 729000, 810000, 900000, 1000000, 4782969, 5314410, 5904900, 6561000, 7290000, 8100000, 9000000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

PROG

(Haskell)

import Data.Set (singleton, deleteFindMin, insert)

a025635 n = a025635_list !! (n-1)

a025635_list = f $ singleton (1, 0, 0) where

f s = y : f (insert (9 * y, i + 1, j) $ insert (10 * y, i, j + 1) s')

where ((y, i, j), s') = deleteFindMin s

-- Reinhard Zumkeller, May 15 2015

(PARI) list(lim)=my(v=List(), N); for(n=0, logint(lim\=1, 10), N=10^n; while(N<=lim, listput(v, N); N*=9)); Set(v) \\ Charles R Greathouse IV, Jan 10 2018

CROSSREFS

Cf. A025612, A025616, A025621, A025625, A025629, A025632, A025634, A108761, A003596, A003597, A107988, A003598, A108698, A003599, A107788, A108687, A108779, A108090.

KEYWORD

easy,nonn

AUTHOR

David W. Wilson

STATUS

approved

A108779

Numbers of the form (10^i)*(11^j), with i, j >= 0.

+10
9

1, 10, 11, 100, 110, 121, 1000, 1100, 1210, 1331, 10000, 11000, 12100, 13310, 14641, 100000, 110000, 121000, 133100, 146410, 161051, 1000000, 1100000, 1210000, 1331000, 1464100, 1610510, 1771561, 10000000, 11000000, 12100000, 13310000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

FORMULA

Sum_{n>=1} 1/a(n) = (10*11)/((10-1)*(11-1)) = 11/9. - Amiram Eldar, Sep 25 2020

a(n) ~ exp(sqrt(2*log(10)*log(11)*n)) / sqrt(110). - Vaclav Kotesovec, Sep 25 2020

MATHEMATICA

n = 10^7; Flatten[Table[10^i*11^j, {i, 0, Log10[n]}, {j, 0, Log[11, n/10^i]}]] // Sort (* Amiram Eldar, Sep 25 2020 *)

PROG

(Haskell)

import Data.Set (singleton, deleteFindMin, insert)

a108779 n = a108779_list !! (n-1)

a108779_list = f $ singleton (1, 0, 0) where

f s = y : f (insert (10 * y, i + 1, j) $ insert (11 * y, i, j + 1) s')

where ((y, i, j), s') = deleteFindMin s

-- Reinhard Zumkeller, May 15 2015

CROSSREFS

Cf. A025612, A025616, A025621, A025625, A025629, A025632, A025634, A025635, A108761, A003596, A003597, A107988, A003598, A108698, A003599, A107788, A108687, A108090.

KEYWORD

nonn,easy

AUTHOR

Douglas Winston (douglas.winston(AT)srupc.com), Jun 26 2005

STATUS

approved

A317804

Numbers of form 2^i*12^j, with i, j >= 0.

+10
2

1, 2, 4, 8, 12, 16, 24, 32, 48, 64, 96, 128, 144, 192, 256, 288, 384, 512, 576, 768, 1024, 1152, 1536, 1728, 2048, 2304, 3072, 3456, 4096, 4608, 6144, 6912, 8192, 9216, 12288, 13824, 16384, 18432, 20736, 24576, 27648, 32768, 36864, 41472, 49152, 55296, 65536

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Dario Ch, Table of n, a(n) for n = 1..10000

PROG

(Python)

from heapq import heappush, heappop

def sequence():

pq = [1]

seen = set(pq)

while True:

value = heappop(pq)

yield value

seen.remove(value)

for x in 2 * value, 12 * value:

if x not in seen:

heappush(pq, x)

seen.add(x)

seq = sequence()

finalsequence_list = [next(seq) for i in range(100)] # Dario Ch, Sep 01 2018

CROSSREFS

Cf. A025612, A003596, A107326, A003597, A107364, A025616, A108201, A108238.

KEYWORD

nonn,easy

AUTHOR

Dario Ch, Sep 01 2018

STATUS

approved

A025685

Exponent of 10 (value of j) in n-th number of form 3^i*10^j.

+10
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,9

COMMENTS

Different from A055087, since a(143)=0 and a(144)=11.

LINKS

Table of n, a(n) for n=1..108.

FORMULA

a(n) = A007814(A025616(n)). - R. J. Mathar, Feb 26 2024

CROSSREFS

Cf. A007814, A025616, A055087.

KEYWORD

nonn

AUTHOR

David W. Wilson

STATUS

approved