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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
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
(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.
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
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
(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.
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
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
(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.
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
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
(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.
Numbers of form 3^i*10^j, with i, j >= 0.
+10
11
1, 3, 9, 10, 27, 30, 81, 90, 100, 243, 270, 300, 729, 810, 900, 1000, 2187, 2430, 2700, 3000, 6561, 7290, 8100, 9000, 10000, 19683, 21870, 24300, 27000, 30000, 59049, 65610, 72900, 81000, 90000, 100000, 177147, 196830, 218700, 243000, 270000
FORMULA
Sum_{n>=1} 1/a(n) = (3*10)/((3-1)*(10-1)) = 5/3. - Amiram Eldar, Sep 25 2020 a(n) ~ exp(sqrt(2*log(3)*log(10)*n)) / sqrt(30). - Vaclav Kotesovec, Sep 25 2020
MATHEMATICA
n = 10^6; Flatten[Table[3^i*10^j, {i, 0, Log[3, n]}, {j, 0, Log10[n/3^i]}]] // Sort (* Amiram Eldar, Sep 25 2020 *)
PROG
(Haskell)
import Data.Set (singleton, deleteFindMin, insert)
a025616 n = a025616_list !! (n-1)
a025616_list = f $ singleton (1, 0, 0) where
f s = y : f (insert (3 * y, i + 1, j) $ insert (10 * y, i, j + 1) s')
where ((y, i, j), s') = deleteFindMin s
(PARI) list(lim)=my(v=List(), N); for(n=0, logint(lim\=1, 10), N=10^n; while(N<=lim, listput(v, N); N*=3)); Set(v) \\ Charles R Greathouse IV, Jan 10 2018
CROSSREFS
Cf. A025612, A025621, A025625, A025629, A025632, A025634, A025635, A108761, A003596, A003597, A107988, A003598, A108698, A003599, A107788, A108687, A108779, A108090.
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
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
(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.
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
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
(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.
Numbers of the form (11^i)*(12^j), with i, j >= 0.
+10
1
1, 11, 12, 121, 132, 144, 1331, 1452, 1584, 1728, 14641, 15972, 17424, 19008, 20736, 161051, 175692, 191664, 209088, 228096, 248832, 1771561, 1932612, 2108304, 2299968, 2509056, 2737152, 2985984, 19487171, 21258732, 23191344, 25299648
PROG
(Haskell)
import Data.Set (singleton, deleteFindMin, insert)
a108218 n = a108218_list !! (n-1)
a108218_list = f $ singleton (1, 0, 0) where
f s = y : f (insert (11 * y, i + 1, j) $ insert (12 * y, i, j + 1) s')
where ((y, i, j), s') = deleteFindMin s
CROSSREFS
Cf. A003596, A003597, A107988, A003598, A108201, A108698, A064476, A003599, A108238, A107788, A108687, A108779, A108090, A108771.
AUTHOR
Douglas Winston (douglas.winston(AT)srupc.com), Jun 28 2005
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