A003599 -id:A003599

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

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

A125581

Numbers n such that n does not divide the denominator of the n-th harmonic number nor the denominator of the n-th alternating harmonic number.

+10
12

77, 847, 9317, 102487, 596778, 1127357, 1193556, 6161805, 12323610, 12400927

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

OFFSET

1,1

COMMENTS

Note that a(1) = 7*11, a(2) = 7*11^2, and a(3) = 7*11^3.

Harmonic numbers are defined as H(n) = Sum_{k=1..n} 1/k = A001008(n)/A002805(n).

Alternating harmonic numbers are defined as H'(n) = Sum_{k=1..n} (-1)^(k+1)*1/k = A058313(n)/A058312(n).

Numbers n such that n does not divide the denominator of the n-th harmonic number are listed in A074791. Numbers n such that n does not divide the denominator of the n-th alternating harmonic number are listed in A121594.

This sequence is the intersection of A074791 and A121594.

Comments from Max Alekseyev, Mar 07 2007: (Start)

While A125581 indeed contains the geometric progression 7*11^n as a subsequence, it also contains other geometric progressions such as: 546*1093^n, 1092*1093^n, 1755*3511^n, 3510*3511^n and 4896*5557^n (see A126196 and A126197). It may also contain some "isolated" terms (i.e. not participating in the geometric progressions) but such terms are harder to find and at the moment I have no proof that they exist.

This is a sketch of my proof that geometric progression 7*11^n and the others mentioned above belong to A125581.

Lemma 1. H'(n) = H(n) - H([n/2]).

Lemma 2. For prime p and integer n >= p, valuation(H(n),p) >= valuation(H([n/p]),p) - 1.

Theorem. For an integer b > 1 and a prime number p such that p divides the numerators of both H(b) and H([b/2]), the geometric progression b*p^n belongs to A125581.

Proof. It is enough to show that valuation(H(b*p^n),p) > -n and valuation(H'(b*p^n), p) > -n. By Lemma 2 we have valuation(H(b*p^n), p) >= valuation(H(b), p) - n >= 1 - n > -n.

From this inequality and Lemma 1, we have valuation(H'(b*p^n), p) >= min{ valuation(H(b*p^n), p), valuation(H([b*p^n/2]), p) } >= min{ 1 - n, valuation(H([b*p^n/2]), p) }. It remains to show that valuation(H([b*p^n/2]), p) >= 1 - n.

Again by Lemma 2, we have valuation(H([b*p^n/2]), p) >= valuation(H([b/2]), p) - n >= 1 - n, which completes the proof.

It is easy to check that this Theorem holds for the aforementioned geometric progressions. (End)

LINKS

Table of n, a(n) for n=1..10.

Tanya Khovanova, Non Recursions.

Eric Weisstein's World of Mathematics, Harmonic Number.

MATHEMATICA

f=0; g=0; Do[g=g+1/n; f=f+(-1)^(n+1)/n; If[ !IntegerQ[Denominator[g]/n]&&!IntegerQ[Denominator[f]/n], Print[n]], {n, 1, 10000}]

CROSSREFS

Cf. A001008, A002805, A003599, A058312, A058313, A074791, A119955, A121594.

KEYWORD

hard,more,nonn

AUTHOR

Alexander Adamchuk, Jan 03 2007

EXTENSIONS

More terms from Max Alekseyev, Mar 11 2007

a(8)-a(10) from Max Alekseyev, Mar 19 2007

STATUS

approved