A010716 -id:A010716 - cp's OEIS frontend

A370361 Minimum greatest prime factor for a length n number with 2 distinct digits, excluding multiples of 10.

2, 3, 3, 7, 11, 29, 19, 19, 23, 67, 29, 139, 107, 71, 101, 137, 127, 307, 173, 347, 383, 439, 271, 853, 521, 587, 883, 571, 823, 941

Offset: 2

Ed Pegg Jr, Mar 05 2024

Multiples of ten are disallowed, because that would give A010716 (all 5) preceded by 2, 3, 3.

Corresponds to 2-distinct-digit numbers in A370849, except at a(21) where 101010110010001010011 with zero digits is more smooth than 222229999999292992929.

			a(7) = 29 as the largest prime factor of the 7-digit number with exactly two distinct digits, 1111222, is 29 and no 7-digit number with exactly two distinct digits has a smaller largest prime factor and no 7-digit number with exactly two distinct digits smaller than 1111222 has a largest prime factor that is equal to 29. - _David A. Corneth_, Mar 05 2024
a(9) = 19, because 799779977 = 17*19^6 has nine digits, two distinct digits and largest prime factor 19. - _Ed Pegg Jr_, Mar 05 2024

Lucas A. Brown, Python program.

Cf. A006530, A370849.

from sympy import primefactors
from sympy.utilities.iterables import multiset_permutations
def A370361(n): return min((max(primefactors(a:=int(''.join(s)))),a) for i in range(10) for j in range(i+1,10) for k in range(1,n) for s in multiset_permutations(str(i)*k+str(j)*(n-k)) if s[0] != '0' and s[-1] != '0')[0] # Chai Wah Wu, Mar 09 2024

# See LINKS. Lucas A. Brown, Mar 30 2024

a(n) = A006530(A370849(n)) unless the smoothest solution is (as for n = 21) a number made of digits {0, 1}, currently excluded in A370849. - M. F. Hasler, Mar 05 2024

a(21)-a(23) from Michael S. Branicky, Mar 05 2024
a(24)-a(25) from David A. Corneth, Mar 05 2024
a(26)-a(30) from Don Reble, Mar 06 2024
a(31) from Lucas A. Brown, Mar 30 2024

A021022 Decimal expansion of 1/18.

Original entry on oeis.org

0, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 0

N. J. A. Sloane

			0.05555555555555555555555555555555555555555555555555...

Hideyuki Ohtsuka, Problem B-1174, Elementary Problems and Solutions, Fibonacci Quart. 54 (2016), no. 3, p. 273.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A010716, A244855.

PadRight[{0}, 100, 5] (* Paolo Xausa, Nov 24 2024 *)

Equals -Sum_{n>=3} (-1)^n/(Fibonacci(n)^4-1). - Michel Marcus, May 14 2021
From Elmo R. Oliveira, Aug 05 2024: (Start)
G.f.: 5*x/(1-x).
E.g.f.: 5*(exp(x) - 1).
a(n) = 5 for n >= 1. (End)

A153981 a(n) = 36Fibonacci(2n+1) - 4.

Original entry on oeis.org

32, 68, 176, 464, 1220, 3200, 8384, 21956, 57488, 150512, 394052, 1031648, 2700896, 7071044, 18512240, 48465680, 126884804, 332188736, 869681408, 2276855492, 5960885072, 15605799728, 40856514116, 106963742624, 280034713760, 733140398660

Offset: 0

Paul Curtz, Jan 04 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (4, -4, 1).

[36*Fibonacci(2*n+1)-4: n in [0..30]]; // Vincenzo Librandi, Aug 07 2011

36*Fibonacci[2*Range[0,30]+1]-4 (* or *) LinearRecurrence[{4,-4,1},{32,68,176},30] (* Harvey P. Dale, Jan 26 2013 *)

a(n) = 4*A153873(n) = 2*A153819(n).
a(n) = 5 (mod 9) = A010716(n) (mod 9).
a(n) = 3*a(n-1) - a(n-2) + 4.
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3).
a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4).
G.f.: 4*(8 - 15*x + 8x^2)/((1-x)*(1 -3*x +x^2)). - R. J. Mathar, Jan 23 2009

Edited and extended by R. J. Mathar and N. J. A. Sloane, Jan 23 2009

A168330 Period 2: repeat [3, -2].

Original entry on oeis.org

3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2

Offset: 1

Klaus Brockhaus, Nov 23 2009

Interleaving of A010701 and -A007395.
Binomial transform of 3 followed by a signed version of A020714.
Inverse binomial transform of 3 followed by A000079.
A084964 without first two terms gives partial sums.

Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A168309 (repeat 4, -3), A010701 (all 3's sequence), A007395 (all 2's sequence), A010716 (all 5's sequence), A020714 (5*2^n), A000079 (powers of 2), A084964 (follow n+2 by n).

Magma
```
&cat[[3,-2]: n in [1..42]];
```

[n eq 1 select 3 else -Self(n-1)+1:n in [1..84]];

[(-5*(-1)^n+1)/2: n in [1..100]]; // Vincenzo Librandi, Jul 19 2016

LinearRecurrence[{0, 1}, {3, -2}, 25] (* G. C. Greubel, Jul 18 2016 *)
PadRight[{},120,{3,-2}] (* Harvey P. Dale, Oct 05 2016 *)

a(n)=3-n%2*5 \\ Charles R Greathouse IV, Jul 13 2016

a(n) = (-5*(-1)^n + 1)/2.
a(n+1) - a(n) = 5*(-1)^n.
a(n) = -a(n-1) + 1 for n > 1; a(1) = 3.
a(n) = a(n-2) for n > 2; a(1) = 3, a(2) = -2.
G.f.: x*(3 - 2*x)/((1-x)*(1+x)).
a(n) = A049071(n). - R. J. Mathar, Nov 25 2009
E.g.f.: (1/2)*(1 - exp(-x))*(5 + exp(x)). - G. C. Greubel, Jul 18 2016

A236203 Interleave A005563(n), A028347(n).

Original entry on oeis.org

0, 0, 3, 5, 8, 12, 15, 21, 24, 32, 35, 45, 48, 60, 63, 77, 80, 96, 99, 117, 120, 140, 143, 165, 168, 192, 195, 221, 224, 252, 255, 285, 288, 320, 323, 357, 360, 396, 399, 437, 440, 480, 483, 525, 528, 572, 575, 621, 624, 672, 675, 725, 728, 780, 783, 837, 840, 896

Offset: 2

Paul Curtz, Jan 20 2014

A175628 gives the numerators of interleaved Lyman and Balmer series, i.e., A005563(n)/A000290(n+1) and A061037(n+2)/A061038(n+2).
Difference table of a(n):
   -1,  -3,  0,  0,  3,   5,   8,  12,  15,  21,  24, ...
   -2,   3,  0,  3,  2,   3,   4,   3,   6,   3,   8, ...
    5,  -3,  3, -1,  1,   1,  -1,   3,  -3,   5,  -5, ...
   -8,   6, -4,  2,  0,  -2,   4,  -6,   8, -10,  12, ...
   14, -10,  6, -2, -2,   6, -10,  14, -18,  22, -26, ...
  -24,  16, -8,  0,  8, -16,  24, -32,  40, -48,  56, ... .
a(n+2) gives the numerators of 0/1, 0/16, 3/4, 5/36, 8/9, 12/64, 15/16, 21/100, 24/25, 32/144, ... . The denominators are A097362(n+1)^2. (Compare A097362 to A029578.)
Note the particular distribution of a(-n). Example:
a(n-9) = 12,15, 5,8, 0,3, -3,0, -4,-1, -3,0, 0,3, 5,8, 12,15, ... .
a(2n) + a(2n+1) = a(-2n-1) + a(-2n-2) = -4,0,8,20,36,56,80,... = 4*A000096(n-1).
a(2n) + a(2n-1) = a(-2n) + a(-2n-1) = -5,-3,3,13,... = A001105(n) - A010716(n).

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A005843, A008590, A016825, A109613, A147658.

List([2..60], n-> (2*n^2 +2*n -19 -(2*n-11)*(-1)^n)/8 ); # G. C. Greubel, Dec 04 2019

[(2*n^2 + 2*n - 19 - (2*n - 11)*(-1)^n)/8: n in [2..60]]; // Vincenzo Librandi, Jul 27 2014

seq( (2*n^2 +2*n -19 -(2*n-11)*(-1)^n)/8, n=2..60); # G. C. Greubel, Dec 04 2019

CoefficientList[Series[x^2(3x^2-2x-3)/((x-1)^3(x+1)^2), {x, 0, 60}], x] (* Vincenzo Librandi, Jul 27 2014 *)
LinearRecurrence[{1,2,-2,-1,1},{0,0,3,5,8},60] (* Harvey P. Dale, Aug 30 2018 *)

concat([0,0], Vec(x^4*(3*x^2-2*x-3)/((x-1)^3*(x+1)^2) + O(x^60))) \\ Colin Barker, Jan 26 2014

[(2*n^2 +2*n -19 -(2*n-11)*(-1)^n)/8 for n in (2..60)] # G. C. Greubel, Dec 04 2019

a(n+2) = (period 8: repeat 1, 16, 1, 1, 1, 4, 1, 1)*A175628(n+1).
a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).
a(n+4) - a(n-4) = 0, 8, 8, ... = A168397.
From Colin Barker, Jan 26 2014: (Start)
a(n) = (n^2 -4)/4 for n even, a(n) = (n^2 +2*n -15)/4 for n odd.
G.f.: x^4*(3 + 2*x - 3*x^2)/ ((1-x)^3*(1+x)^2). (End)
a(n) = (2*n^2 + 2*n - 19 - (2*n - 11)*(-1)^n)/8. - Luce ETIENNE, Jul 26 2014
Sum_{n>=4} (-1)^n/a(n) = 11/48. - Amiram Eldar, Aug 21 2022

More terms from Colin Barker, Jan 26 2014

A141721 A141631(n) mod 10.

Original entry on oeis.org

2, 7, 8, 5, 8, 7, 2, 3, 0, 3, 2, 7, 8, 5, 8, 7, 2, 3, 0, 3, 2, 7, 8, 5, 8, 7, 2, 3, 0, 3, 2, 7, 8, 5, 8, 7, 2, 3, 0, 3, 2, 7, 8, 5, 8, 7, 2, 3, 0, 3, 2, 7, 8, 5, 8, 7, 2, 3, 0, 3, 2, 7, 8, 5, 8, 7, 2, 3, 0, 3, 2, 7, 8, 5, 8, 7, 2, 3, 0, 3, 2, 7, 8, 5, 8, 7

Offset: 1

Paul Curtz, Sep 12 2008

Mentioned in A010716.

The sequence and its first differences (5, 1, -3, 3, -1, -5, 1, -3, 3, -1) are both periodic with period length 10.

Mod[#,10]&/@Table[3n^2-4n+3,{n,200}] (* Harvey P. Dale, Oct 28 2012 *)

a(n)=a(n-10), period 10 (2, 7, 8, 5, 8, 7, 2, 3, 0, 3).

Also: decimal expansion of the constant 309541367/1111111111. - R. J. Mathar, Oct 15 2008

Edited by R. J. Mathar, Oct 15 2008

More terms from Harvey P. Dale, Oct 28 2012

A210032 a(n)=n for n=1,2,3 and 4; a(n)=5 for n >= 5.

Original entry on oeis.org

1, 2, 3, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

A. Timothy Royappa, Mar 16 2012

In atomic spectroscopy, a(n) is the number of D term symbols with spin multiplicity equal to n, i.e., there is one singlet-D term (n=1), and there are two doublet-D terms (n=2), three triple-D terms (n=3), four quartet-D terms (n=4) and five terms for every other D term of multiplicity 5 or higher (n >= 5).

Decimal expansion of 11111/9000. - Arkadiusz Wesolowski, Mar 29 2012

Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A010716, A101272, A158411, A168092, A128999.

Join[Range@4, Table[5, {83}]] (* Arkadiusz Wesolowski, Mar 29 2012 *)
PadRight[{1,2,3,4},120,{5}] (* Harvey P. Dale, Sep 23 2017 *)

a(n) = min(n,5). - Wesley Ivan Hurt, Apr 16 2014
From Elmo R. Oliveira, Jun 26 2024: (Start)
G.f.: x*(1+x+x^2+x^3+x^4)/(1-x) = x*(1-x^5)/(1-x)^2.
a(n) = 1 + A158411(n-1) = A101272(n+1) - 1 = A168093(n-1) - 2. (End)

A257936 Decimal expansion of 11/18.

Original entry on oeis.org

6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Bruno Berselli, May 13 2015

Decimal expansion of Sum_{i>=1} 1/A028552(i).

Also, continued fraction expansion of 5+A001622.

			.6111111111111111111111111111111111111111111111111111111111111111...

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A028552, A030880.

Cf. A010716 (decimal expansion of 5/9 = 10/18), A010722 (decimal expansion of 2/3 = 12/18).

Magma
```
[6,1^^90];
```
Mathematica
```
RealDigits[11/18, 10, 90][[1]]
```

11/18. \\ Charles R Greathouse IV, Apr 18 2016

Equals A020773 + A142464.
From Elmo R. Oliveira, Aug 05 2024: (Start)
G.f.: (6-5*x)/(1-x).
E.g.f.: exp(x) + 5.
a(n) = 1, n >= 1. (End)

cp's OEIS Frontend

A370361 Minimum greatest prime factor for a length n number with 2 distinct digits, excluding multiples of 10.

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A021022 Decimal expansion of 1/18.

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A153981 a(n) = 36*Fibonacci(2*n+1) - 4.

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A168330 Period 2: repeat [3, -2].

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A236203 Interleave A005563(n), A028347(n).

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A141721 A141631(n) mod 10.

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A210032 a(n)=n for n=1,2,3 and 4; a(n)=5 for n >= 5.

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A153981 a(n) = 36Fibonacci(2n+1) - 4.