A002277 -id:A002277 - cp's OEIS frontend

A281860 Curious identities based on the Armstrong number 371 = A005188(12).

371, 336701, 333667001, 333366670001, 333336666700001, 333333666667000001, 333333366666670000001, 333333336666666700000001, 333333333666666667000000001, 333333333366666666670000000001, 333333333336666666666700000000001, 333333333333666666666667000000000001

Offset: 1

Wolfdieter Lang, Feb 08 2017

See a comment in A067275.

			n=1: 371 = 3^3 + 7^3 + 1^3;
n=2: 336701 = 33^3 + 67^3 + (01)^3;
n=3: 333667001 = 333^3 + 667^3 + (001)^3.

Colin Barker, Table of n, a(n) for n = 1..333
Index entries for linear recurrences with constant coefficients, signature (1111,-112110,1111000,-1000000).

Cf. A002277, A067275, A281857, A281858, A281859.

LinearRecurrence[{1111,-112110,1111000,-1000000},{371,336701,333667001,333366670001},20] (* Harvey P. Dale, May 28 2024 *)

Vec(x*(371 - 75480*x + 1185000*x^2 - 2000000*x^3) / ((1 - x)*(1 - 10*x)*(1 - 100*x)*(1 - 1000*x)) + O(x^30)) \\ Colin Barker, Feb 09 2017

a(n) = A002277(n) * 10^(2*n) + A067275(n+1) * 10^n + 0(n-1)1, where 0(n-1)1 stands for n-1 0's followed by a 1, for n >= 1.
a(n) = A002277(n)^3 + A067275(n+1)^3 + (0(n-1)1)^3.
From Colin Barker, Feb 09 2017: (Start)
G.f.: x*(371 - 75480*x + 1185000*x^2 - 2000000*x^3)/((1 - x)*(1 - 10*x)*(1 - 100*x)*(1 - 1000*x)).
a(n) = 1111*a(n-1) - 112110*a(n-2) + 1111000*a(n-3) - 1000000*a(n-4) for n>4.
a(n) = (3 + 10^n + 100^n + 1000^n)/3. (End)

A332131 a(n) = (10^(2n+1)-1)/3 - 2*10^n.

Original entry on oeis.org

1, 313, 33133, 3331333, 333313333, 33333133333, 3333331333333, 333333313333333, 33333333133333333, 3333333331333333333, 333333333313333333333, 33333333333133333333333, 3333333333331333333333333, 333333333333313333333333333, 33333333333333133333333333333, 3333333333333331333333333333333

Offset: 0

M. F. Hasler, Feb 09 2020

See A183174 = {1, 3, 7, 61, 90, 92, 269, ...} for the indices of primes.

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).
Patrick De Geest, Palindromic Wing Primes: (3)1(3), updated: June 25, 2017.
Makoto Kamada, Factorization of 33...33133...33, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077775-1)/2 = A183174: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A002277 (3*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332121 .. A332191 (variants with different repeated digit 2, ..., 9).
Cf. A332130 .. A332139 (variants with different middle digit 0, ..., 9).

A332131 := n -> (10^(2*n+1)-1)/3-2*10^n;

Array[3 (10^(2 # + 1)-1)/9 - 2*10^# &, 15, 0]

apply( {A332131(n)=10^(n*2+1)\3-2*10^n}, [0..15])

def A332131(n): return 10**(n*2+1)//3-2*10**n

a(n) = 3*A138148(n) + 1*10^n = A002277(2n+1) - 2*10^n.
G.f.: (1 + 202*x - 500*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A004178 Omit 3's from n.

Original entry on oeis.org

0, 1, 2, 0, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 14, 15, 16, 17, 18, 19, 20, 21, 22, 2, 24, 25, 26, 27, 28, 29, 0, 1, 2, 0, 4, 5, 6, 7, 8, 9, 40, 41, 42, 4, 44, 45, 46, 47, 48, 49, 50, 51, 52, 5, 54, 55, 56, 57, 58, 59, 60, 61, 62, 6, 64, 65, 66, 67, 68, 69, 70, 71, 72, 7, 74

Offset: 0

N. J. A. Sloane

a(n) = 0 when n = 0 or when n is a 3-repdigit (A002277). a(n) = n when it contains no 3 among its digits. - Alonso del Arte, Oct 16 2012

			a(13) = 1 because after deleting the 3 from its base 10 representation, we're left with 1.
a(14) = 14 because there are no 3s to delete.

Cf. A004722, A052405.

Table[FromDigits[DeleteCases[IntegerDigits[n], 3]], {n, 0, 74}] (* Alonso del Arte, Oct 16 2012 *)

a(n)=subst(Pol(select(k->k-3,digits(n))),'x,10) \\ Charles R Greathouse IV, Oct 16 2012

A181718 a(n) = (1/9)(10^(2n) + 10^n - 2).

Original entry on oeis.org

0, 12, 1122, 111222, 11112222, 1111122222, 111111222222, 11111112222222, 1111111122222222, 111111111222222222, 11111111112222222222, 1111111111122222222222, 111111111111222222222222, 11111111111112222222222222, 1111111111111122222222222222

Offset: 0

Paul Curtz, Nov 17 2010

In decimal, n times 1 followed by n times 2.

a(n) = 3 + 3*3, 33 + 33*33, 333 + 333*333, written with 3,6,9,12,... = A008585(n+1) 3's.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002277, A008585, A074992, A093137.

[(1/9)*(10^(2*n) + 10^n - 2): n in [0..20]]; // Vincenzo Librandi, Aug 04 2011

Table[FromDigits[Join[PadRight[{},n,1],PadRight[{},n,2]]],{n,0,20}] (* or *) LinearRecurrence[{111,-1110,1000},{0,12,1122},20] (* Harvey P. Dale, Jul 31 2013 *)

vector(30, n, n--; (10^(2*n) + 10^n - 2)/9) \\ G. C. Greubel, Nov 02 2018

for n in range(30):
    print((10**(2*n)+10**n-2)//9, end=', ')
# Stefano Spezia, Nov 02 2018

[(100^n +10^n -2)//9 for n in range(31)] # G. C. Greubel, Mar 25 2024

a(n) = A002277(n)*A093137(n).
G.f.: 6*x*(2-35*x) / ( (1-x)*(1-10*x)*(1-100*x) ). - R. J. Mathar, Feb 28 2011
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3), a(0)=0, a(1)=12, a(2)=1122. - Harvey P. Dale, Jul 31 2013
a(n) = (A074992(n) - 1)/3. - Michel Marcus, Sep 14 2013
E.g.f.: (1/9)*(-2*exp(x) + exp(10*x) + exp(100*x)). - G. C. Greubel, Mar 25 2024

A241217 Largest number that when multiplied by 7 produces an n-digit number.

Original entry on oeis.org

1, 14, 142, 1428, 14285, 142857, 1428571, 14285714, 142857142, 1428571428, 14285714285, 142857142857, 1428571428571, 14285714285714, 142857142857142, 1428571428571428, 14285714285714285, 142857142857142857, 1428571428571428571, 14285714285714285714

Offset: 1

J. Lowell, Apr 17 2014

The definition "largest number that when multiplied by 3 produces an n-digit number" gives A002277.

			14*7 = 98 but 15*7 = 105 (too large) so a(2) = 14.

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 34 at p. 62.

Index entries for linear recurrences with constant coefficients, signature (11,-10,-1,11,-10).

Cf. A020806, A002277.

LinearRecurrence[{11,-10,-1,11,-10},{1,14,142,1428,14285},30] (* Harvey P. Dale, Mar 03 2024 *)

a(n) = floor(10^n/7); \\ Michel Marcus, Apr 21 2014

a(n) = floor(10^n/7). - Michel Marcus, Apr 21 2014
G.f.: x*(1+3*x-2*x^2+7*x^3)/((x-1)*(10*x-1)*(x+1)*(x^2-x+1)). - Alois P. Heinz, Apr 30 2014
E.g.f.: (3*cosh(10*x) - 7*cosh(x) + 2*exp(x/2)*(2*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)) - 14*sinh(x) + 3*sinh(10*x))/21. - Stefano Spezia, Jul 31 2024

More terms from Michel Marcus, Apr 21 2014

A322570 Positive integers k such that A270710(k) (= (k+1)(3k-1)) have only 1 or 2 different digits in base 10.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 12, 16, 17, 33, 34, 48, 54, 285, 333, 334, 365, 385, 430, 471, 516, 816, 1049, 3333, 3334, 33333, 33334, 333333, 333334, 483048, 3333333, 3333334, 33333333, 33333334, 333333333, 333333334, 3333333333, 3333333334, 33333333333, 33333333334

Offset: 1

Seiichi Manyama, Aug 29 2019

Giovanni Resta, Table of n, a(n) for n = 1..50

Cf. A002277, A016069, A093137, A213517 (in case of triangular numbers), A270710, A322571.

[k:k in [1..10000000]| #Set(Intseq((k+1)*(3*k-1))) le 2]; // Marius A. Burtea, Aug 29 2019

Select[Range@ 50000, Length@ Union@ IntegerDigits[3 #^2 + 2 # - 1] <= 2 &] (* Giovanni Resta, Sep 04 2019 *)

for(k=1, 1e8, if(#Set(digits(3*k^2+2*k-1))<=2, print1(k", ")))

For k > 0, A002277(k) is a term.

For k >= 0, A002277(k) + 1 (= A093137(k)) is a term.

a(35)-a(36) from Jinyuan Wang, Aug 30 2019

a(37)-a(40) from Giovanni Resta, Sep 04 2019

A323639 a(n) = 3*(10^n - 4)/9.

Original entry on oeis.org

-1, 2, 32, 332, 3332, 33332, 333332, 3333332, 33333332, 333333332, 3333333332, 33333333332, 333333333332, 3333333333332, 33333333333332, 333333333333332, 3333333333333332, 33333333333333332, 333333333333333332, 3333333333333333332, 33333333333333333332

Offset: 0

Seiichi Manyama, Aug 31 2019

			        (0+1) * (3*0-1) = -1.
        (3+1) * (3*3-1) = 32.
      (33+1) * (3*33-1) = 3332.
    (333+1) * (3*333-1) = 333332.
  (3333+1) * (3*3333-1) = 33333332.
(33333+1) * (3*33333-1) = 3333333332.
-------------------------------------
        8 * 4 = 32.
      68 * 49 = 3332.
    668 * 499 = 333332.
  6668 * 4999 = 33333332.
66668 * 49999 = 3333333332.

Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A002277, A073555, A086948, A093137, A198971, A246057.

Table[(10^n-4)/3,{n,0,20}] (* or *) LinearRecurrence[{11,-10},{-1,2},21] (* Harvey P. Dale, Jan 09 2021 *)

PARI
```
{a(n) = 3*(10^n-4)/9}
```

N=40; x='x+O('x^N); Vec((-1+13*x)/((1-x)*(1-10*x)))

G.f.: (-1+13*x)/((1-x)*(1-10*x)).
a(n) = 11*a(n-1) - 10*a(n-2).
a(n) = A002277(n) - 1.
a(n) = 2*A246057(n-1) for n > 0.
a(2*n) = (A002277(n)+1) * (3*A002277(n)-1).
a(2*n) = A073555(n+1) * A198971(n-1) for n > 0.
E.g.f.: exp(x)*(exp(9*x) - 4)/3. - Stefano Spezia, May 02 2025
a(n) = A086948(n)/6 for n >= 1. - Elmo R. Oliveira, May 06 2025

A332132 a(n) = (10^(2n+1)-1)/3 - 10^n.

Original entry on oeis.org

2, 323, 33233, 3332333, 333323333, 33333233333, 3333332333333, 333333323333333, 33333333233333333, 3333333332333333333, 333333333323333333333, 33333333333233333333333, 3333333333332333333333333, 333333333333323333333333333, 33333333333333233333333333333, 3333333333333332333333333333333

Offset: 0

M. F. Hasler, Feb 09 2020

There are no primes > 2 in this list because a(n) = round(10^n/.6)*(2*10^n-1) = 16...67*19...99.

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275 (repunits R_n = (10^n-1)/9), A002277 (3*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332112 .. A332192 (variants with different repeated digit 1, ..., 9).
Cf. A332130 .. A332139 (variants with different middle digit 0, ..., 9).

Maple
```
A332132 := n -> (10^(2*n+1)-1)/3-10^n;
```

Array[ (10^(2 # + 1)-1)/3 - 10^# &, 15, 0]

apply( {A332132(n)=10^(n*2+1)\3-10^n}, [0..15])

def A332132(n): return 10**(n*2+1)//3-10**n

a(n) = 3*A138148(n) + 2*10^n = A002277(2n+1) - 10^n.
G.f.: (2 + 101*x - 400*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A289006 Conversion to octal of the binary expansion given by the first n terms of the period-3 sequence A011655 (repeat 0, 1, 1).

Original entry on oeis.org

0, 1, 3, 6, 15, 33, 66, 155, 333, 666, 1555, 3333, 6666, 15555, 33333, 66666, 155555, 333333, 666666, 1555555, 3333333, 6666666, 15555555, 33333333, 66666666, 155555555, 333333333, 666666666, 1555555555, 3333333333, 6666666666, 15555555555, 33333333333, 66666666666, 155555555555, 333333333333, 666666666666

Offset: 1

Peter Schonefeld, Jun 21 2017

The length of the n-th term is floor((n+1)/3) digits, for all n>1. [Corrected by M. F. Hasler, Jun 23 2017]

Index entries for linear recurrences with constant coefficients, signature (0, 0, 11, 0, 0, -10).

A033129(n-1) written in base 8.

Cf. A011655. Trisections: A099915, A002277, A002280.

{ my(x='x+O('x^33)); concat([0],Vec( x*(1+x)*(1+2*x+4*x^2)/((1-x)*(1+x+x^2)*(1-10*x^3)) )) } \\ Joerg Arndt, Jun 21 2017

A289006(n)=if(n%3==2,10^(n\3+1)\6-10^(n\3)\9,10^(n\3)\3<<(n%3)) \\ M. F. Hasler, Jun 23 2017

a(3n) = floor(10^n/3) (= n times the digit '3'), a(3n+1) = floor(10^n/3)*2 (= n times the digit '6'), a(3n+2) = floor(10^(n+1)/6) - floor(10^n/9) (= digit '1' followed by n digits '5'). - M. F. Hasler, Jun 23 2017

G.f.: x^2*(1+x)*(4*x^2+2*x+1) / ( (x-1)*(1+x+x^2)*(10*x^3-1) ). - R. J. Mathar, Jun 29 2017

A322571 Positive integers A270710(k) (= (k+1)(3k-1)) which have only 1 or 2 different digits in base 10.

Original entry on oeis.org

4, 15, 32, 55, 84, 119, 455, 799, 900, 3332, 3535, 7007, 8855, 244244, 333332, 335335, 400404, 445444, 555559, 666464, 799799, 1999199, 3303300, 33333332, 33353335, 3333333332, 3333533335, 333333333332, 333335333335, 700007077007, 33333333333332, 33333353333335, 3333333333333332, 3333333533333335

Offset: 1

Seiichi Manyama, Aug 29 2019

Giovanni Resta, Table of n, a(n) for n = 1..50

Cf. A002277, A018885 (in case of squares), A213516 (in case of triangular numbers), A270710, A322570, A323639.

[a:k in [1..10000000]| #Set(Intseq(a)) le 2 where a is (k+1)*(3*k-1)]; // Marius A. Burtea, Aug 29 2019

Select[Table[(n+1)(3n-1),{n,3334*10^4}],Count[DigitCount[#],0]>7&] (* Harvey P. Dale, Jun 12 2022 *)

for(k=1, 1e8, if(#Set(digits(j=3*k^2+2*k-1))<=2, print1(j", ")))

a(n) = A270710(A322570(n)).

For k > 0, A002277(2*k) - 1 is a term.

cp's OEIS Frontend

A281860 Curious identities based on the Armstrong number 371 = A005188(12).

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A332131 a(n) = (10^(2n+1)-1)/3 - 2*10^n.

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A004178 Omit 3's from n.

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A181718 a(n) = (1/9)*(10^(2*n) + 10^n - 2).

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A241217 Largest number that when multiplied by 7 produces an n-digit number.

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A322570 Positive integers k such that A270710(k) (= (k+1)*(3*k-1)) have only 1 or 2 different digits in base 10.

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A323639 a(n) = 3*(10^n - 4)/9.

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A181718 a(n) = (1/9)(10^(2n) + 10^n - 2).

A322570 Positive integers k such that A270710(k) (= (k+1)(3k-1)) have only 1 or 2 different digits in base 10.

A322571 Positive integers A270710(k) (= (k+1)(3k-1)) which have only 1 or 2 different digits in base 10.