Displaying 1-10 of 11 results found.
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 the form (6^i)*(11^j), with i, j >= 0.
+10
10
1, 6, 11, 36, 66, 121, 216, 396, 726, 1296, 1331, 2376, 4356, 7776, 7986, 14256, 14641, 26136, 46656, 47916, 85536, 87846, 156816, 161051, 279936, 287496, 513216, 527076, 940896, 966306, 1679616, 1724976, 1771561, 3079296, 3162456
FORMULA
Sum_{n>=1} 1/a(n) = (6*11)/((6-1)*(11-1)) = 33/25. - Amiram Eldar, Oct 07 2020 a(n) ~ exp(sqrt(2*log(6)*log(11)*n)) / sqrt(66). - Vaclav Kotesovec, Oct 07 2020
MATHEMATICA
n = 10^6; Flatten[Table[6^i*11^j, {i, 0, Log[6, n]}, {j, 0, Log[11, n/6^i]}]] // Sort (* Amiram Eldar, Oct 07 2020 *)
PROG
(Haskell)
import Data.Set (singleton, deleteFindMin, insert)
a108698 n = a108698_list !! (n-1)
a108698_list = f $ singleton (1, 0, 0) where
f s = y : f (insert (6 * y, i + 1, j) $ insert (11 * y, i, j + 1) s')
where ((y, i, j), s') = deleteFindMin s
CROSSREFS
Cf. A025626, A025627, A025628, A025629, A064476, A107710, A003596, A003597, A107988, A003598, A108687, A003599, A107788, A108090.
AUTHOR
Douglas Winston (douglas.winston(AT)srupc.com), Jun 19 2005
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 (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
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
CROSSREFS
Cf. A025612, A025616, A025621, A025625, A025629, A025632, A025634, A025635, A108761, A003596, A003597, A107988, A003598, A108698, A003599, A107788, A108687, A108090.
AUTHOR
Douglas Winston (douglas.winston(AT)srupc.com), Jun 26 2005
Numbers of the form (9^i)*(13^j), with i, j >= 0.
+10
3
1, 9, 13, 81, 117, 169, 729, 1053, 1521, 2197, 6561, 9477, 13689, 19773, 28561, 59049, 85293, 123201, 177957, 257049, 371293, 531441, 767637, 1108809, 1601613, 2313441, 3341637, 4782969, 4826809, 6908733, 9979281, 14414517, 20820969
FORMULA
Sum_{n>=1} 1/a(n) = (9*13)/((9-1)*(13-1)) = 39/32. - Amiram Eldar, Sep 25 2020 a(n) ~ exp(sqrt(2*log(9)*log(13)*n)) / sqrt(117). - Vaclav Kotesovec, Sep 25 2020
MATHEMATICA
n = 10^6; Flatten[Table[9^i*13^j, {i, 0, Log[9, n]}, {j, 0, Log[13, n/9^i]}]] // Sort (* Amiram Eldar, Sep 25 2020 *)
AUTHOR
Douglas Winston (douglas.winston(AT)srupc.com), Jun 23 2005
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