A003596 -id:A003596

A036302

Composite numbers k such that the digits of the prime factors of k are either 1 or 2.

+10
25

4, 8, 16, 22, 32, 44, 64, 88, 121, 128, 176, 242, 256, 352, 422, 484, 512, 704, 844, 968, 1024, 1331, 1408, 1688, 1936, 2048, 2321, 2662, 2816, 3376, 3872, 4096, 4222, 4442, 4642, 5324, 5632, 6752, 7744, 8192, 8444, 8884, 9284, 10648, 11264, 13504, 14641, 15488, 16384

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

OFFSET

1,1

COMMENTS

All terms are a product of at least two terms of A020450. - Michel Marcus, Oct 02 2020

LINKS

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Alois P. Heinz)

Index entries sequences related to prime factors

FORMULA

Sum_{n>=1} 1/a(n) = Product_{p in A020450} p/(p-1) - Sum_{p in A020450} 1/p - 1 = 0.616325... - Amiram Eldar, Oct 14 2020

EXAMPLE

422 = 2 * 211 is in the sequence as the digits of its prime factors 2 and 211 are either 1 or 2. - David A. Corneth, Sep 26 2020

MATHEMATICA

Select[Range[2, 14650], !PrimeQ[#] && Complement[Flatten[IntegerDigits[First/@FactorInteger[#]]], {1, 2}]=={} &] (* Jayanta Basu, May 25 2013 *)

PROG

(Magma) [k:k in [2..15000]| not IsPrime(k) and forall{a: a in PrimeDivisors(k)|Intseq(a) subset {1, 2}}]; // Marius A. Burtea, Oct 08 2019

CROSSREFS

Cf. A003596 (a subsequence), A020450, A036303-A036325.

KEYWORD

nonn,base

AUTHOR

Patrick De Geest, Dec 15 1998

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

A108090

Numbers of the form (11^i)*(13^j).

+10
20

1, 11, 13, 121, 143, 169, 1331, 1573, 1859, 2197, 14641, 17303, 20449, 24167, 28561, 161051, 190333, 224939, 265837, 314171, 371293, 1771561, 2093663, 2474329, 2924207, 3455881, 4084223, 4826809, 19487171, 23030293, 27217619

(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) = (11*13)/((11-1)*(13-1)) = 143/120. - Amiram Eldar, Sep 23 2020

a(n) ~ exp(sqrt(2*log(11)*log(13)*n)) / sqrt(143). - Vaclav Kotesovec, Sep 23 2020

MATHEMATICA

mx = 3*10^7; Sort@ Flatten@ Table[ 11^i*13^j, {i, 0, Log[11, mx]}, {j, 0, Log[13, mx/11^i]}] (* Robert G. Wilson v, Aug 17 2012 *)

fQ[n_]:=PowerMod[143, n, n] == 0; Select[Range[2 10^7], fQ] (* Vincenzo Librandi, Jun 27 2016 *)

PROG

(Haskell)

import Data.Set (singleton, deleteFindMin, insert)

a108090 n = a108090_list !! (n-1)

a108090_list = f $ singleton (1, 0, 0) where

f s = y : f (insert (11 * y, i + 1, j) $ insert (13 * y, i, j + 1) s')

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

-- Reinhard Zumkeller, May 15 2015

(Magma) [n: n in [1..10^7] | PrimeDivisors(n) subset [11, 13]]; // Vincenzo Librandi, Jun 27 2016

(PARI) list(lim)=my(v=List(), t); for(j=0, logint(lim\=1, 13), t=13^j; while(t<=lim, listput(v, t); t*=11)); Set(v) \\ Charles R Greathouse IV, Aug 29 2016

CROSSREFS

Cf. A003586, A003592, A003593, A003591, A003594, A003595, A003596, A003597, A003598, A003599, A107326, A107364, A107466, A108056.

KEYWORD

nonn,easy

AUTHOR

Douglas Winston (douglas.winston(AT)srupc.com), Jun 03 2005

STATUS

approved

A107326

Numbers of the form (2^i)*(13^j).

+10
13

1, 2, 4, 8, 13, 16, 26, 32, 52, 64, 104, 128, 169, 208, 256, 338, 416, 512, 676, 832, 1024, 1352, 1664, 2048, 2197, 2704, 3328, 4096, 4394, 5408, 6656, 8192, 8788, 10816, 13312, 16384, 17576, 21632, 26624, 28561, 32768, 35152, 43264, 53248, 57122

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

OFFSET

1,2

COMMENTS

A204455(13*a(n)) = 13, and only for these numbers. - Wolfdieter Lang, Feb 04 2012

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..100 from Vincenzo Librandi)

FORMULA

Sum_{n>=1} 1/a(n) = (2*13)/((2-1)*(13-1)) = 13/6. - Amiram Eldar, Sep 23 2020

a(n) ~ exp(sqrt(2*log(2)*log(13)*n)) / sqrt(26). - Vaclav Kotesovec, Sep 23 2020

MATHEMATICA

fQ[n_] := PowerMod[26, n, n]==0; Select[Range[60000], fQ] (* Vincenzo Librandi, Feb 04 2012 *)

mx = 60000; Sort@ Flatten@ Table[2^i*13^j, {i, 0, Log[2, mx]}, {j, 0, Log[13, mx/2^i]}] (* Robert G. Wilson v, Aug 17 2012 *)

PROG

(PARI) list(lim)=my(v=List(), N); for(n=0, log(lim)\log(13), N=13^n; while(N<=lim, listput(v, N); N<<=1)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

CROSSREFS

Cf. A003586, A003592, A003593, A003591, A003594, A003595, A003596, A003597, A003598, A003599.

KEYWORD

nonn,easy

AUTHOR

Douglas Winston (douglas.winston(AT)srupc.com), May 21 2005

STATUS

approved

A107988

Numbers of the form (4^i)*(11^j), with i, j >= 0.

+10
13

1, 4, 11, 16, 44, 64, 121, 176, 256, 484, 704, 1024, 1331, 1936, 2816, 4096, 5324, 7744, 11264, 14641, 16384, 21296, 30976, 45056, 58564, 65536, 85184, 123904, 161051, 180224, 234256, 262144, 340736, 495616, 644204, 720896, 937024, 1048576

(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) = (4*11)/((4-1)*(11-1)) = 22/15. - Amiram Eldar, Sep 24 2020

a(n) ~ exp(sqrt(2*log(4)*log(11)*n)) / sqrt(44). - Vaclav Kotesovec, Sep 24 2020

MATHEMATICA

n = 10^6; Flatten[Table[4^i*11^j, {i, 0, Log[4, n]}, {j, 0, Log[11, n/4^i]}]] // Sort (* Amiram Eldar, Sep 24 2020 *)

PROG

(Haskell)

import Data.Set (singleton, deleteFindMin, insert)

a107988 n = a107988_list !! (n-1)

a107988_list = f $ singleton (1, 0, 0) where

f s = y : f (insert (4 * y, i + 1, j) $ insert (11 * y, i, j + 1) s')

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

-- Reinhard Zumkeller, May 15 2015

CROSSREFS

Subsequence of A003596.

Cf. A025617, A025618, A025619, A025620, A025621, A107462, A003597, A003598, A003599, A108090.

KEYWORD

nonn

AUTHOR

Douglas Winston (douglas.winston(AT)srupc.com), Jun 12 2005

STATUS

approved

A107364

Numbers of the form (3^i)*(13^j).

+10
12

1, 3, 9, 13, 27, 39, 81, 117, 169, 243, 351, 507, 729, 1053, 1521, 2187, 2197, 3159, 4563, 6561, 6591, 9477, 13689, 19683, 19773, 28431, 28561, 41067, 59049, 59319, 85293, 85683, 123201, 177147, 177957, 255879, 257049, 369603, 371293, 531441

(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) = (3*13)/((3-1)*(13-1)) = 13/8. - Amiram Eldar, Sep 23 2020

a(n) ~ exp(sqrt(2*log(3)*log(13)*n)) / sqrt(39). - Vaclav Kotesovec, Sep 23 2020

MATHEMATICA

mx = 540000; Sort@ Flatten@ Table[3^i*13^j, {i, 0, Log[3, mx]}, {j, 0, Log[13, mx/3^i]}] (* Robert G. Wilson v, Aug 17 2012 *)

fQ[n_]:=PowerMod[39, n, n] == 0; Select[Range[2 10^7], fQ] (* Vincenzo Librandi, Jun 27 2016 *)

PROG

(PARI) list(lim)=my(v=List(), N); for(n=0, log(lim)\log(13), N=13^n; while(N<=lim, listput(v, N); N*=3)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

(Magma) [n: n in [1..10^7] | PrimeDivisors(n) subset [3, 13]]; // Vincenzo Librandi, Jun 27 2016

CROSSREFS

Cf. A003586, A003592, A003593, A003591, A003594, A003595, A003596, A003597, A003598, A003599, A107326.

KEYWORD

nonn,easy

AUTHOR

Douglas Winston (douglas.winston(AT)srupc.com), May 23 2005

STATUS

approved

A107788

Numbers of the form (8^i)*(11^j), with i, j >= 0.

+10
12

1, 8, 11, 64, 88, 121, 512, 704, 968, 1331, 4096, 5632, 7744, 10648, 14641, 32768, 45056, 61952, 85184, 117128, 161051, 262144, 360448, 495616, 681472, 937024, 1288408, 1771561, 2097152, 2883584, 3964928, 5451776, 7496192, 10307264

(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) = (8*11)/((8-1)*(11-1)) = 44/35. - Amiram Eldar, Oct 07 2020

a(n) ~ exp(sqrt(2*log(8)*log(11)*n)) / sqrt(88). - Vaclav Kotesovec, Oct 07 2020

MATHEMATICA

Take[Union[8^First[#]*11^Last[#]&/@Tuples[Range[0, 20], 2]], 40] (* Harvey P. Dale, Jan 17 2015 *)

n = 10^6; Flatten[Table[8^i*11^j, {i, 0, Log[8, n]}, {j, 0, Log[11, n/8^i]}]] // Sort (* Amiram Eldar, Oct 07 2020 *)

PROG

(Haskell)

import Data.Set (singleton, deleteFindMin, insert)

a107788 n = a107788_list !! (n-1)

a107788_list = f $ singleton (1, 0, 0) where

f s = y : f (insert (8 * 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. A025633, A025634, A107764, A003596, A003597, A107988, A003598, A003599, A108090.

KEYWORD

nonn,easy

AUTHOR

Douglas Winston (douglas.winston(AT)srupc.com), Jun 14 2005

STATUS

approved

A108687

Numbers of the form (9^i)*(11^j), with i, j >= 0.

+10
12

1, 9, 11, 81, 99, 121, 729, 891, 1089, 1331, 6561, 8019, 9801, 11979, 14641, 59049, 72171, 88209, 107811, 131769, 161051, 531441, 649539, 793881, 970299, 1185921, 1449459, 1771561, 4782969, 5845851, 7144929, 8732691, 10673289, 13045131

(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) = (9*11)/((9-1)*(11-1)) = 99/80. - Amiram Eldar, Sep 24 2020

a(n) ~ exp(sqrt(2*log(9)*log(11)*n)) / sqrt(99). - Vaclav Kotesovec, Sep 24 2020

MATHEMATICA

f[upto_]:=With[{max9=Floor[Log[9, upto]], max11=Floor[Log[11, upto]]}, Select[Union[Times@@{9^First[#], 11^Last[#]}&/@Tuples[{Range[0, max9], Range[0, max11]}]], #<=upto&]]; f[14000000] (* Harvey P. Dale, Mar 11 2011 *)

PROG

(Haskell)

import Data.Set (singleton, deleteFindMin, insert)

a108687 n = a108687_list !! (n-1)

a108687_list = f $ singleton (1, 0, 0) where

f s = y : f (insert (9 * y, i + 1, j) $ insert (11 * y, i, j + 1) s')

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

-- Reinhard Zumkeller, May 15 2015

CROSSREFS

Subsequence of A003597.

Cf. A025633, A025634, A107788, A107764, A003596, A107988, A003598, A003599, A108090.

KEYWORD

nonn,easy

AUTHOR

Douglas Winston (douglas.winston(AT)srupc.com), Jun 17 2005

STATUS

approved