cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A005422 Largest prime factor of 10^n - 1.

Original entry on oeis.org

3, 11, 37, 101, 271, 37, 4649, 137, 333667, 9091, 513239, 9901, 265371653, 909091, 2906161, 5882353, 5363222357, 333667, 1111111111111111111, 27961, 10838689, 513239, 11111111111111111111111, 99990001, 182521213001, 1058313049
Offset: 1

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Author

Keywords

References

  • J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Crossrefs

Same as A003020 except for the additional a(1) = 3.
Cf. similar sequences listed in A274906.

Programs

Formula

For n > 1, a(n) = A003020(n). For 1 < n < 10, a(n) = A075024(n). - M. F. Hasler, Jul 30 2015
a(n) = A006530(A002283(n)). - Vincenzo Librandi, Jul 13 2016
a(A004023(n)) = A002275(A004023(n)). - Bernard Schott, May 24 2022

Extensions

Terms to a(100) in b-file from Yousuke Koide added by T. D. Noe, Dec 06 2006
Edited by M. F. Hasler, Jul 30 2015
a(101)-a(322) in b-file from Ray Chandler, Apr 22 2017
a(323)-a(352) in b-file from Max Alekseyev, Apr 26 2022

A099656 a(n) is the least prime following A002276(n) repdigits.

Original entry on oeis.org

2, 3, 23, 223, 2237, 22229, 222247, 2222239, 22222223, 222222227, 2222222243, 22222222223, 222222222301, 2222222222243, 22222222222229, 222222222222227, 2222222222222281, 22222222222222301, 222222222222222281
Offset: 0

Views

Author

Labos Elemer, Nov 17 2004

Keywords

Examples

			n=2: 22 is followed by 23.
		

Crossrefs

Programs

  • Maple
    seq(nextprime(2*(10^i-1)/9), i=0..20); # Robert Israel, Aug 25 2017
  • Mathematica
    Table[NextPrime[2*(10^n-1)/9], {n, 0, 35}]
    Table[ NextPrime[2*(10^n - 1)/9], {n, 0, 18}] (* Robert G. Wilson v, Nov 20 2004 *)
    Table[NextPrime[FromDigits[PadRight[{},n,2]]],{n,0,20}] (* Harvey P. Dale, Dec 15 2021 *)

A178631 a(n) = 27*((10^n - 1)/9)^2.

Original entry on oeis.org

27, 3267, 332667, 33326667, 3333266667, 333332666667, 33333326666667, 3333333266666667, 333333332666666667, 33333333326666666667, 3333333333266666666667, 333333333332666666666667, 33333333333326666666666667, 3333333333333266666666666667, 333333333333332666666666666667
Offset: 1

Views

Author

Reinhard Zumkeller, May 31 2010

Keywords

Examples

			n=1: ..................... 27 = 9 * 3;
n=2: ................... 3267 = 99 * 33;
n=3: ................. 332667 = 999 * 333;
n=4: ............... 33326667 = 9999 * 3333;
n=5: ............. 3333266667 = 99999 * 33333;
n=6: ........... 333332666667 = 999999 * 333333;
n=7: ......... 33333326666667 = 9999999 * 3333333;
n=8: ....... 3333333266666667 = 99999999 * 33333333;
n=9: ..... 333333332666666667 = 999999999 * 333333333.
		

Crossrefs

Programs

  • Magma
    [27*((10^n-1)/9)^2: n in [1..50]]; // Vincenzo Librandi, Dec 28 2010
    
  • Mathematica
    27*(FromDigits/@Table[PadRight[{},n,1],{n,20}])^2 (* or *) LinearRecurrence[ {111,-1110,1000},{27,3267,332667},20] (* Harvey P. Dale, Oct 11 2012 *)
  • Maxima
    A178631(n):=27*((10^n-1)/9)^2$ makelist(A178631(n),n,1,10); /* Martin Ettl, Nov 12 2012 */
    
  • PARI
    a(n)=27*(10^n\9)^2 \\ Charles R Greathouse IV, Jul 02 2013

Formula

a(n) = 27*A002477(n) = A002283(n)*A002277(n).
a(n) = ((A002277(n-1)*10 + 2)*10^(n-1) + A002280(n-1))*10 + 7.
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n>3, a(1)=27, a(2)=3267, a(3)=332667. - Harvey P. Dale, Oct 11 2012
G.f.: 27*x*(1 + 10*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)). - Ilya Gutkovskiy, Feb 24 2017
E.g.f.: exp(x)*(1 - 2*exp(9*x) + exp(99*x))/3. - Elmo R. Oliveira, Aug 01 2025

A178632 a(n) = 45*((10^n - 1)/9)^2.

Original entry on oeis.org

45, 5445, 554445, 55544445, 5555444445, 555554444445, 55555544444445, 5555555444444445, 555555554444444445, 55555555544444444445, 5555555555444444444445, 555555555554444444444445, 55555555555544444444444445, 5555555555555444444444444445, 555555555555554444444444444445
Offset: 1

Views

Author

Reinhard Zumkeller, May 31 2010

Keywords

Examples

			n=1: ..................... 45 = 9 * 5;
n=2: ................... 5445 = 99 * 55;
n=3: ................. 554445 = 999 * 555;
n=4: ............... 55544445 = 9999 * 5555;
n=5: ............. 5555444445 = 99999 * 55555;
n=6: ........... 555554444445 = 999999 * 555555;
n=7: ......... 55555544444445 = 9999999 * 5555555;
n=8: ....... 5555555444444445 = 99999999 * 55555555;
n=9: ..... 555555554444444445 = 999999999 * 555555555.
		

Crossrefs

Programs

Formula

a(n) = 45*A002477(n) = A002283(n)*A002279(n).
a(n) = (A002279(n-1)*10^n + A002278(n))*10 + 5.
G.f.: 45*x*(1 + 10*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)). - Ilya Gutkovskiy, Feb 24 2017
From Elmo R. Oliveira, Aug 01 2025: (Start)
E.g.f.: 5*exp(x)*(1 - 2*exp(9*x) + exp(99*x))/9.
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 3. (End)

A178633 a(n) = 54*((10^n - 1)/9)^2.

Original entry on oeis.org

54, 6534, 665334, 66653334, 6666533334, 666665333334, 66666653333334, 6666666533333334, 666666665333333334, 66666666653333333334, 6666666666533333333334, 666666666665333333333334, 66666666666653333333333334, 6666666666666533333333333334, 666666666666665333333333333334
Offset: 1

Views

Author

Reinhard Zumkeller, May 31 2010

Keywords

Examples

			n = 1:                   54 = 9 * 6;
n = 2:                 6534 = 99 * 66;
n = 3:               665334 = 999 * 666;
n = 4:             66653334 = 9999 * 6666;
n = 5:           6666533334 = 99999 * 66666;
n = 6:         666665333334 = 999999 * 666666;
n = 7:       66666653333334 = 9999999 * 6666666;
n = 8:     6666666533333334 = 99999999 * 66666666;
n = 9:   666666665333333334 = 999999999 * 666666666.
		

References

  • Walther Lietzmann, Lustiges und Merkwuerdiges von Zahlen und Formen, (F. Hirt, Breslau 1921-43), p. 149.

Crossrefs

Programs

Formula

a(n) = 54*A002477(n) = A002283(n)*A002280(n).
a(n) = ((A002280(n-1)*10 + 5)*10^(n-1) + A002277(n-1))*10 + 4 = (2/3)*(10^n - 1)^2.
From Colin Barker, Dec 07 2015: (Start)
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n>3.
G.f.: 54*x*(1+10*x)/((1-x)*(1-10*x)*(1-100*x)). (End)
E.g.f.: 2*exp(x)*(1 - 2*exp(9*x) + exp(99*x))/3. - Elmo R. Oliveira, Aug 01 2025

A178634 a(n) = 63*((10^n - 1)/9)^2.

Original entry on oeis.org

63, 7623, 776223, 77762223, 7777622223, 777776222223, 77777762222223, 7777777622222223, 777777776222222223, 77777777762222222223, 7777777777622222222223, 777777777776222222222223, 77777777777762222222222223, 7777777777777622222222222223, 777777777777776222222222222223
Offset: 1

Views

Author

Reinhard Zumkeller, May 31 2010

Keywords

Examples

			n=1: ..................... 63 = 9 * 7;
n=2: ................... 7623 = 99 * 77;
n=3: ................. 776223 = 999 * 777;
n=4: ............... 77762223 = 9999 * 7777;
n=5: ............. 7777622223 = 99999 * 77777;
n=6: ........... 777776222223 = 999999 * 777777;
n=7: ......... 77777762222223 = 9999999 * 7777777;
n=8: ....... 7777777622222223 = 99999999 * 77777777;
n=9: ..... 777777776222222223 = 999999999 * 777777777.
		

References

  • Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 33 at p. 62.
  • Walther Lietzmann, Lustiges und Merkwuerdiges von Zahlen und Formen, (F. Hirt, Breslau 1921-43), p. 149.

Crossrefs

Programs

  • GAP
    List([1..20], n -> 63*((10^n - 1)/9)^2); # G. C. Greubel, Jan 28 2019
  • Magma
    [63*((10^n - 1)/9)^2: n in [1..20]]; // Vincenzo Librandi, Dec 28 2010
    
  • Mathematica
    63((10^Range[15]-1)/9)^2 (* or *) Table[FromDigits[Join[PadRight[{},n,7],{6},PadRight[{},n,2],{3}]],{n,0,15}] (* Harvey P. Dale, Apr 23 2012 *)
  • PARI
    a(n)=63*(10^n\9)^2 \\ Charles R Greathouse IV, Jul 02 2013
    
  • Sage
    [63*((10^n - 1)/9)^2 for n in (1..20)] # G. C. Greubel, Jan 28 2019
    

Formula

a(n) = 63*A002477(n) = A002283(n)*A002281(n).
a(n) = ((A002281(n-1)*10 + 6)*10^(n-1) + A002276(n-1))*10 + 3.
G.f.: 63*x*(1 + 10*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)). - Ilya Gutkovskiy, Feb 24 2017
E.g.f.: 7*exp(x)*(1 - 2*exp(9*x) + exp(99*x))/9. - Stefano Spezia, Jul 31 2024
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 3. - Elmo R. Oliveira, Aug 01 2025

A178635 a(n) = 72*((10^n - 1)/9)^2.

Original entry on oeis.org

72, 8712, 887112, 88871112, 8888711112, 888887111112, 88888871111112, 8888888711111112, 888888887111111112, 88888888871111111112, 8888888888711111111112, 888888888887111111111112, 88888888888871111111111112, 8888888888888711111111111112, 888888888888887111111111111112
Offset: 1

Views

Author

Reinhard Zumkeller, May 31 2010

Keywords

Examples

			n=1: ..................... 72 = 9 * 8;
n=2: ................... 8712 = 99 * 88;
n=3: ................. 887112 = 999 * 888;
n=4: ............... 88871112 = 9999 * 8888;
n=5: ............. 8888711112 = 99999 * 88888;
n=6: ........... 888887111112 = 999999 * 888888;
n=7: ......... 88888871111112 = 9999999 * 8888888;
n=8: ....... 8888888711111112 = 99999999 * 88888888;
n=9: ..... 888888887111111112 = 999999999 * 888888888.
		

References

  • Walther Lietzmann, Lustiges und Merkwuerdiges von Zahlen und Formen, (F. Hirt, Breslau 1921-43), p. 149.

Crossrefs

Programs

Formula

a(n) = 72*A002477(n) = A002283(n)*A002282(n).
a(n) = ((A002282(n-1)*10 + 7)*10^(n-1) + A002275(n-1))*10 + 2.
G.f.: 72*x*(1 + 10*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)). - Ilya Gutkovskiy, Feb 24 2017
From Elmo R. Oliveira, Aug 01 2025: (Start)
E.g.f.: 8*exp(x)*(1 - 2*exp(9*x) + exp(99*x))/9.
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 3. (End)

A333402 Numbers m such that the largest digit in the decimal expansion of 1/m is 1.

Original entry on oeis.org

1, 9, 10, 90, 99, 100, 900, 909, 990, 999, 1000, 9000, 9009, 9090, 9900, 9990, 9999, 10000, 90000, 90009, 90090, 90900, 90909, 99000, 99900, 99990, 99999, 100000, 900000, 900009, 900090, 900900, 909000, 909090, 990000, 990099, 999000, 999900, 999990, 999999, 1000000
Offset: 1

Views

Author

Bernard Schott, Mar 19 2020

Keywords

Comments

If m is a term, 10*m is also a term.
If m is a term then m has only digits {1}, {9}, {1,0} or {9,0} in its decimal representation, but this is not sufficient to be a term (see examples).
Some subsequences below (not exhaustive, see crossrefs):
m = 10^k, k >= 0, hence m is in A011557 = {1, 10, 100, 1000, 10000, ...};
m = 9*10^k, k >= 0, hence m is in A052268 = {9, 90, 900, 9000, 90000, ...};
m = 10^k-1, k >= 1, hence m is in A002283 = {9, 99, 999, 9999, 99999, ...};
m = 9*(10^k+1), k >= 1, hence m is in 9*A000533 = {99, 909, 9009, 90009, ...};
m = 9+100*(100^k-1)/11, k >= 0, hence m is in 9*A094028 = {9, 909, 90909, 9090909, ...}.

Examples

			As 1/101 = 0.009900990099..., 101 is not a term.
As 1/909 = 0.001100110011..., 909 is a term.
As 1/9099 = 0.000109902187..., 9099 is not a term.
As 1/9999 = 0.000100010001..., 9999 is also a term.
		

Crossrefs

Cf. A333236, A333237 (similar, with 9).
Subsequences: A002283, A011557, A052268.
Subsequences: 9*A000533, 9*A094028, 9*A135577, 9*A261544, 9*A330135.

Programs

  • Mathematica
    Select[Range[10^4], Max @ RealDigits[1/#][[1]] == 1 &] (* Amiram Eldar, Mar 19 2020 *)
  • Python
    from itertools import count, islice
    def A333402_gen(startvalue=1): # generator of terms >= startvalue
        for m in count(max(startvalue,1)):
            k = 1
            while k <= m:
                k *= 10
            rset = {0}
            while True:
                k, r = divmod(k, m)
                if max(str(k)) > '1':
                    break
                else:
                    if r in rset:
                        yield m
                        break
                rset.add(r)
                k = r
                while k <= m:
                    k *= 10
    A333402_list = list(islice(A333402_gen(),30)) # Chai Wah Wu, Feb 17 2022

Formula

A333236(a(n))= 1.

Extensions

More terms from Jinyuan Wang, Mar 19 2020

A066138 a(n) = 10^(2*n) + 10^n + 1.

Original entry on oeis.org

3, 111, 10101, 1001001, 100010001, 10000100001, 1000001000001, 100000010000001, 10000000100000001, 1000000001000000001, 100000000010000000001, 10000000000100000000001, 1000000000001000000000001, 100000000000010000000000001, 10000000000000100000000000001, 1000000000000001000000000000001
Offset: 0

Views

Author

Henry Bottomley, Dec 07 2001

Keywords

Comments

Palindromes whose digit sum is 3.
Essentially the same as A135577. - R. J. Mathar Apr 29 2008
From Peter Bala, Sep 25 2015: (Start)
For n >= 1, the simple continued fraction expansion of sqrt(a(n)) = [10^n; 1, 1, 2/3*(10^n - 1), 1, 1, 2*10^n, ...] has period 6. As n increases, the expansion has the large partial quotients 2/3*(10^n - 1) and 2*10^n.
For n >= 1, the continued fraction expansion of sqrt(a(2*n))/a(n) = [0; 1, 10^n - 1, 1, 1, 1/3*(10^n - 4), 1, 4, 1, 1/3*(10^n - 4), 1, 1, 10^n - 1, 2, 10^n - 1, ...] has pre-period of length 3 and period 12 beginning 1, 1, 1/3*(10^n - 4), .... As n increases, the expansion has the large partial quotients 10^n - 1 and 1/3*(10^n - 4).
A theorem of Kuzmin in the measure theory of continued fractions says that large partial quotients are the exception in continued fraction expansions.
Empirically, we also see exceptionally large partial quotients in the continued fraction expansions of the m-th root of the numbers a(m*n), for m >= 3. For example, it appears that the continued fraction expansion of a(3*n)^(1/3), for n >= 2, begins [10^(2*n); 3*10^n - 1, 1, 0.5*10^(2*n) - 1, 1.44*10^n - 1, 1, ...]. Cf. A000533, A002283 and A168624. (End)

Examples

			From _Peter Bala_, Sep 25 2015: (Start)
Simple continued fraction expansions showing large partial quotients:
a(9)^(1/3) =[1000000; 2999, 1, 499999, 1439, 1, 2582643, 1, 1, 1, 2, 3, 3, ...].
a(20)^(1/4) = [10000000000; 39999999999, 1, 3999999999, 16949152542, 2, 1, 2, 6, 1, 4872106, 3, 9, 2, 3, ...].
a(25)^(1/5) = [10000000000; 4999999999999999, 1, 3333333332, 2, 1, 217391304347825, 2, 2, 1, 1, 1, 2, 1, 23980814, 1, 1, 1, 1, 1, 7, ...]. (End)
		

Crossrefs

Programs

  • Magma
    [10^(2*n) + 10^n + 1: n in [0..20]]; // Vincenzo Librandi, Sep 27 2015
  • Mathematica
    Table[10^(2 n) + 10^n + 1, {n, 0, 15}] (* Michael De Vlieger, Sep 27 2015 *)
    CoefficientList[Series[(3 - 222 x + 1110 x^2)/((1 - 100 x) (1 - 10 x) (1 - x)), {x, 0, 33}], x] (* Vincenzo Librandi, Sep 27 2015 *)
  • PARI
    a(n) = { 10^(2*n) + 10^n + 1 } \\ Harry J. Smith, Feb 02 2010
    
  • PARI
    Vec(-3*(370*x^2-74*x+1)/((x-1)*(10*x-1)*(100*x-1)) + O(x^20)) \\ Colin Barker, Sep 27 2015
    

Formula

A168624(n) = a(2*n)/a(n). - Peter Bala, Sep 24 2015
G.f.: (3 - 222*x + 1110*x^2)/((1 - 100*x)*(1 - 10*x)*(1 - x)). - Vincenzo Librandi, Sep 27 2015
From Colin Barker, Sep 27 2015: (Start)
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.
G.f.: -3*(370*x^2-74*x+1)/((x-1)*(10*x-1)*(100*x-1)). (End)
From Elmo R. Oliveira, Aug 27 2024: (Start)
E.g.f.: exp(x)*(exp(99*x) + exp(9*x) + 1).
a(n) = 3*A074992(n). (End)

Extensions

Offset changed from 1 to 0 by Harry J. Smith, Feb 02 2010
More terms from Michael De Vlieger, Sep 27 2015

A295503 a(n) = phi(10^n-1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

6, 60, 648, 6000, 64800, 466560, 6637344, 58752000, 648646704, 5890320000, 66663457344, 461894400000, 6458084523072, 60339430569600, 610154104320000, 5529599115264000, 66666634474902192, 441994921381739520, 6666666666666666660, 58301444908800000000
Offset: 1

Views

Author

Seiichi Manyama, Nov 22 2017

Keywords

Crossrefs

phi(k^n-1): A053287 (k=2), A295500 (k=3), A295501 (k=4), A295502 (k=5), A366623 (k=6), A366635 (k=7), A366654 (k=8), A366663 (k=9), this sequence (k=10), A366685 (k=11), A366711 (k=12).

Programs

  • Mathematica
    Array[ EulerPhi[10^# - 1] &, 20] (* Robert G. Wilson v, Nov 22 2017 *)
  • PARI
    {a(n) = eulerphi(10^n-1)}

Formula

a(n) = n*A295497(n).
a(n) = A000010(A002283(n)). - Michel Marcus, Nov 25 2017
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