A170955 -id:A170955 - cp's OEIS frontend

A253213 a(n) = 10^n + 9.

10, 19, 109, 1009, 10009, 100009, 1000009, 10000009, 100000009, 1000000009, 10000000009, 100000000009, 1000000000009, 10000000000009, 100000000000009, 1000000000000009, 10000000000000009, 100000000000000009, 1000000000000000009, 10000000000000000009, 100000000000000000009

Offset: 0

Vincenzo Librandi, Dec 30 2014

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

Cf. similar sequences listed in A253208.

Cf. A170955.

Magma
```
[10^n+9: n in [0..30]];
```

Table[10^n + 9, {n, 0, 40}]
LinearRecurrence[{11,-10},{10,19},40] (* Harvey P. Dale, Jun 29 2018 *)

a(n) = 11*a(n-1) - 10*a(n-2) for n > 1.
G.f.: (10 - 91*x)/((1 - x)*(1 - 10*x)).
E.g.f.: exp(x)*(exp(9*x) + 9). - Elmo R. Oliveira, Sep 15 2024

a(20) from Elmo R. Oliveira, Sep 15 2024

a(20) corrected by Sean A. Irvine, Sep 22 2024

A105259 Number of distinct prime divisors of 99..91 (with n 9's).

Original entry on oeis.org

0, 2, 1, 2, 1, 3, 1, 4, 2, 3, 3, 3, 3, 4, 3, 4, 3, 6, 2, 4, 4, 3, 3, 5, 4, 7, 4, 6, 2, 6, 3, 6, 1, 2, 3, 5, 3, 10, 4, 7, 5, 4, 6, 7, 1, 7, 2, 6, 3, 5, 5, 6, 4, 6, 2, 8, 4, 7, 3, 5, 4, 11, 2, 7, 5, 8, 6, 5, 5, 7, 2, 8, 4, 7, 5, 6, 4, 6, 5, 9, 3, 9, 4, 7, 2, 9, 4

Offset: 0

Parthasarathy Nambi, Apr 14 2005

			If n=1, then the number of distinct prime divisors of 91 is 2.
If n=2, then the number of distinct prime divisors of 991 is 1 (a prime).
If n=3, then the number of distinct prime divisors of 9991 is 2.

Amiram Eldar, Table of n, a(n) for n = 0..229

Cf. A001221, A093177, A170955.

A105259 := proc(n) local x ;x := [1,seq(9,k=1..n)] ; add(op(i,x)*10^(i-1),i=1..nops(x)) ; numtheory[factorset](%) ; nops(%) ; end proc: # R. J. Mathar, Aug 24 2011

Table[PrimeNu[10^(n + 1) - 9], {n, 0, 50}] (* G. C. Greubel, May 10 2017 *)

a(n) = A001221(A170955(n+1)). - R. J. Mathar, Aug 24 2011

More terms from Amiram Eldar, Jan 24 2020

A193871 Square array T(n,k) = k^n - k + 1 read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 25, 13, 1, 1, 31, 79, 61, 21, 1, 1, 63, 241, 253, 121, 31, 1, 1, 127, 727, 1021, 621, 211, 43, 1, 1, 255, 2185, 4093, 3121, 1291, 337, 57, 1, 1, 511, 6559, 16381, 15621, 7771, 2395, 505, 73, 1, 1, 1023, 19681, 65533, 78121, 46651, 16801, 4089, 721, 91, 1

Offset: 1

Omar E. Pol, Aug 21 2011

The columns give 1^n-0, 2^n-1, 3^n-2, 4^n-3, 5^n-4, etc.

The main diagonal gives A006091, which is a sequence related to the famous "coconuts" problem.

			Array begins:
  1,   1,    1,     1,     1,    1,    1,   1,   1,   1
  1,   3,    7,    13,    21,   31,   43,  57,  73
  1,   7,   25,    61,   121,  211,  337, 505
  1,  15,   79,   253,   621, 1291, 2395
  1,  31,  241,  1021,  3121, 7771
  1,  63,  727,  4093, 15621
  1, 127, 2185, 16381
  1, 255, 6559
  1, 511
  1

M. B. Richardson, A Needlessly Complicated and Unhelpful Solution to Ben Ames Williams' Coconuts Problem, The Winnower, 3 (2016), e147175.52128. doi: 10.15200/winn.147175.52128

Row 1: A000012. Rows 2,3: A002061, A061600 but both without repetitions.
Columns 1..10: A000012, positives A000225, A058481, A141725, A164785, A164784, A164783, A164786, A177095, A170955.
Cf. A276135.

Table[k^# - k + 1 &[n - k + 1], {n, 11}, {k, n}] // Flatten (* Michael De Vlieger, Nov 16 2016 *)

A177114 4*(10^n-9).

Original entry on oeis.org

4, 364, 3964, 39964, 399964, 3999964, 39999964, 399999964, 3999999964, 39999999964, 399999999964, 3999999999964, 39999999999964, 399999999999964, 3999999999999964, 39999999999999964, 399999999999999964

Offset: 1

Vincenzo Librandi

Harvey P. Dale, Table of n, a(n) for n = 1..999
Index entries for linear recurrences with constant coefficients, signature (11,-10).

LinearRecurrence[{11, -10}, {4, 364}, 30] (* or *) Join[{4},Table[ 100*FromDigits[PadRight[{3},n,9]]+64,{n,30}]] (* Harvey P. Dale, Jul 31 2017 *)

G.f. 4*x*(1+80*x) / ( (10*x-1)*(x-1) ). - R. J. Mathar, Aug 24 2011

a(n) = 4*A170955(n). - R. J. Mathar, Aug 24 2011

A254648 Numbers n whose square representation in base 10 can be split into three parts whose sum is n.

Original entry on oeis.org

36, 82, 91, 235, 379, 414, 675, 756, 792, 909, 918, 964, 991, 1296, 1702, 1782, 3366, 3646, 3682, 4132, 4906, 5149, 6832, 7543, 8416, 8767, 8856, 9208, 9325, 9586, 9621, 9765, 9901, 9945, 9955, 9991, 12222, 12727, 17271, 22231

Offset: 1

Michel Lagneau, Feb 04 2015

Extension of the Kaprekar numbers (A006886) where the number of parts of n^2 is two. It is probably possible to generalize this property with the division of n^2 into m parts.
By convention, the second and third parts may start with the digit 0, but must be positive. For example, 991 is in the sequence because 991^2 = 982081, which can be split into 982, 08 and 1, and 982 + 08 + 1 = 991. But 100 is not; although 100^2 = 10000 and 100 + 0 + 0 = 100, the second and the third part here are not positive. The number 99 is not in the sequence although 99^2 = 9801 and 98 + 0 + 1 = 99.
Property of the sequence:
The sequence is infinite because the numbers of the form 10^n-9 = 91, 991, 991, ... (A170955) are in the sequence: if m = 99...91 with k digits "9", then m^2 = 99...98200...081 with k-1 digits "9" and k-1 digits "0", and 99...982 + 00...8 + 1 = 99...91 = m.
The prime of the sequence are {379, 9901, ...} union {A093177}.
Calculation method: For each class of squares having k-digit numbers, the number of partitions into 3 parts is n(n+1)/2 (A000217). For instance, if the numbers are of the form (abcde) with k = 5, the 6 partitions into 3 subsets are {a,b,{c,d,e}}, {a,{b,c},{d,e}}, {a,{b,c,d},e}, {{a,b},c,{d,e}}, {{a,b},{c,d},e}, {{a,b,c},d,e} and then we compute the corresponding numbers.
Example: 235^2 = 55225 (abcde) = 55225 => {a,b,{c,d,e}} = {5,5,{2,2,5}} => {5,5,225} and 5+5+225 = 235.

			36^2 = 1296 and 1 + 29 + 6 = 36;
235^2 = 55225 and 5 + 5 + 225 = 235;
1782^2 = 3175524 and 3 + 1755 + 24 = 1782;
12727^2 = 161976529 and 1 + 6197 + 6529 = 12727.

Chai Wah Wu, Table of n, a(n) for n = 1..400

Cf. A000217, A006886, A093177, A170955.

from itertools import combinations
A254648_list, n, n2 = [], 10, 100
while n < 10**4:
    m = str(n2)
    for a in combinations(range(1,len(m)),2):
        x, y, z = int(m[:a[0]]), int(m[a[0]:a[1]]), int(m[a[1]:])
        if y != 0 and z != 0 and x+y+z == n:
            A254648_list.append(n)
            break
    n += 1
    n2 += 2*n-1 # Chai Wah Wu, Aug 27 2017

Removed terms 4879 and 5292 by Chai Wah Wu, Aug 27 2017

cp's OEIS Frontend

A253213 a(n) = 10^n + 9.

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A105259 Number of distinct prime divisors of 99..91 (with n 9's).

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Maple

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A193871 Square array T(n,k) = k^n - k + 1 read by antidiagonals.

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Mathematica

A177114 4*(10^n-9).

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Mathematica

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A254648 Numbers n whose square representation in base 10 can be split into three parts whose sum is n.

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Python

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