A242614 -id:A242614 - cp's OEIS frontend

A235229 Numbers whose sum of digits is 20.

299, 389, 398, 479, 488, 497, 569, 578, 587, 596, 659, 668, 677, 686, 695, 749, 758, 767, 776, 785, 794, 839, 848, 857, 866, 875, 884, 893, 929, 938, 947, 956, 965, 974, 983, 992, 1199, 1289, 1298, 1379, 1388, 1397, 1469, 1478, 1487, 1496, 1559, 1568, 1577, 1586

Offset: 1

Vincenzo Librandi, Jan 05 2014

A007953(a(n)) = 20; number of repdigits = #{5555,44444,2222222222,1^20} = A242627(20) = 4. - Reinhard Zumkeller, Jul 17 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A052217 - A052224, A143164, A166311, A166370, A166459.
Cf. A011557 (1), A052216 (2), A052217 (3), A052218 (4), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052223 (9), A052224 (10), A166311 (11), A235151 (12), A143164 (13), A235225(14), A235226 (15), A235227 (16), A166370 (17), A235228 (18), A166459 (19).
Cf. A242614, A242627.

a235229 n = a235229_list !! (n-1)
a235229_list = filter ((== 20) . a007953) [0..]
-- Reinhard Zumkeller, Jul 17 2014

[n: n in [1..2000] | &+Intseq(n) eq 20];

Select[Range[2000], Total[IntegerDigits[#]]==20&]

A235226 Numbers whose sum of digits is 15.

Original entry on oeis.org

69, 78, 87, 96, 159, 168, 177, 186, 195, 249, 258, 267, 276, 285, 294, 339, 348, 357, 366, 375, 384, 393, 429, 438, 447, 456, 465, 474, 483, 492, 519, 528, 537, 546, 555, 564, 573, 582, 591, 609, 618, 627, 636, 645, 654, 663, 672, 681, 690, 708, 717, 726, 735

Offset: 1

Vincenzo Librandi, Jan 05 2014

A007953(a(n)) = 15; number of repdigits = #{555,33333,1^15} = A242627(15) = 3. - Reinhard Zumkeller, Jul 17 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A052217 - A052224, A143164, A166311, A166370, A166459, A235227, A235229.
Cf. A011557 (1), A052216 (2), A052217 (3), A052218 (4), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052223 (9), A052224 (10), A166311 (11), A235151 (12), A143164 (13), A235225(14), A235227 (16), A166370 (17), A235228 (18), A166459 (19), A235229 (20).
Cf. A242614, A242627.

a235226 n = a235226_list !! (n-1)
a235226_list = filter ((== 15) . a007953) [0..]
-- Reinhard Zumkeller, Jul 17 2014

[n: n in [1..1000] | &+Intseq(n) eq 15];

Select[Range[1000], Total[IntegerDigits[#]] == 15 &]

A242627 Number of divisors of n that are less than 10.

Original entry on oeis.org

9, 1, 2, 2, 3, 2, 4, 2, 4, 3, 3, 1, 5, 1, 3, 3, 4, 1, 5, 1, 4, 3, 2, 1, 6, 2, 2, 3, 4, 1, 5, 1, 4, 2, 2, 3, 6, 1, 2, 2, 5, 1, 5, 1, 3, 4, 2, 1, 6, 2, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 6, 1, 2, 4, 4, 2, 4, 1, 3, 2, 4, 1, 7, 1, 2, 3, 3, 2, 4, 1, 5, 3, 2, 1, 6, 2

Offset: 0

Reinhard Zumkeller, Jul 16 2014

Number of numbers <= 9, dividing n;
a(n) <= 9; a(2520*n) = 9;
a(n) = (number of repdigit numbers in row n of triangle A242614) = sum(A202022(A242614(n,k)): k=1..A242622(n)), for n > 0.
Periodic with period 2520. Each period there are 576 1's, 720 2's, 464 3's, 360 4's, 206 5's, 122 6's, 58 7's, 13 8's, and 1 9 (average 2.82...). - Charles R Greathouse IV, Sep 27 2015

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (-2,-4,-7,-11,-15,-20,-24,-27,-28,-27,-23,-17,-9,0,9,17,23,27,28,27,24,20,15,11,7,4,2,1).

Cf. A165412.

a242627 n = length $ filter ((== 0) . mod n) [1..9]

a:= n -> numboccur(0,map2(`modp`,n,[$1..9])):
map(a,[$0..100]); # Robert Israel, Jul 31 2014

a[n_] := If[n == 0, 9, Count[Divisors[n], d_ /; d < 10]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Dec 13 2021 *)

a(n)=1+sum(k=2,9,n%k<1) \\ Zak Seidov, Jul 31 2014

G.f.: Sum_(j=1..9, 1/(1-x^j)). - Robert Israel, Jul 31 2014

A242622 a(n) = number of numbers with digit sum n, not greater than the n-th repunit (cf. A052222).

Original entry on oeis.org

1, 1, 2, 6, 21, 77, 287, 1079, 4082, 15522, 59271, 227131, 873051, 3364827, 12998867, 50321075, 195162350, 758156366, 2949614789, 11490869489, 44819531180, 175009397380, 684059403670, 2676260628574, 10479320689274, 41065736472890, 161043272100440, 631974335773536

Offset: 0

Reinhard Zumkeller, Jul 16 2014

a(n) = length of row n in triangle A242614.

Jens Kruse Andersen, Table of n, a(n) for n = 0..100

Haskell
```
a242622 = length . a242614_row
```

\\ c[x,y] = #numbers < 10^x with digit sum y-1.
m=50; c=matrix(m,m+1); for(x=1, m, for(y=1, m+1, c[x,y]=sum(d=0, min(9,y-1), if(x>1, c[x-1,y-d], d==y-1)))); for(n=0, m, print1(1+sum(i=2, n, c[i-1,i+1])", ")); \\ Jens Kruse Andersen, Jul 17 2014

More terms from Jens Kruse Andersen, Jul 17 2014

A114034 Let f(n) be the number of sequences of 1's and 2's which sum to n. Sequence contains the string of sequences.

Original entry on oeis.org

1, 2, 11, 12, 21, 111, 22, 112, 121, 211, 1111, 122, 212, 221, 1112, 1121, 1211, 2111, 11111, 222, 1122, 1212, 1221, 2112, 2121, 2211, 11112, 11121, 11211, 12111, 21111, 111111, 1222, 2122, 2212, 2221, 11122, 11212, 11221, 12112, 12121, 12211, 21112, 21121, 21211, 22111, 111112, 111121, 111211, 112111, 121111, 211111, 1111111

Offset: 1

Amarnath Murthy, Nov 13 2005

Number of sequences of ones and twos that sum to n are Fibonacci(n+1). The maximum number of terms in a sequence is n. (111111 n times). Following is the triangle of the frequency of sequences of each size:
1
1 1
0 2 1
0 1 3 1
0 0 3 4 1
0 0 1 6 5 1
...
This is a vertical Pascal's triangle and the horizontal sum gives the Fibonacci numbers.
Each row of the irregular triangle provides a list of increasing positive integers of only 1s and 2s that sum up to n (see Example section). - Stefano Spezia, Jan 14 2024

			The irregular triangle begins:
  n
  1:   1;                                          f(1) = 1.
  2:   2,  11;                                     f(2) = 2.
  3:  12,  21, 111;                                f(3) = 3.
  4:  22, 112, 121,  211, 1111;                    f(4) = 5.
  5: 122, 212, 221, 1112, 1121, 1211, 2111, 11111; f(5) = 8.
...

N. Karimilla Bi, Amritanshu Prasad, and P. Giftson Santhosh, Residues modulo powers of two in the Young-Fibonacci lattice, arXiv:1702.06684 [math.CO], 2017. See Figure 1.

Cf. A000045, A242614.

row[n_] := Select[Range[(10^n-1)/9], SubsetQ[{1,2}, DeleteDuplicates[digits = IntegerDigits[#]]] && Total[digits]==n &]; Array[row,7]//Flatten (* Stefano Spezia, Jan 14 2024 *)

More terms from Terryjames Morris (trm5002(AT)psu.edu), Mar 09 2007

Duplicate term removed by Stefano Spezia, Jan 14 2024

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A235229 Numbers whose sum of digits is 20.

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A235226 Numbers whose sum of digits is 15.

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A242627 Number of divisors of n that are less than 10.

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A242622 a(n) = number of numbers with digit sum n, not greater than the n-th repunit (cf. A052222).

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A114034 Let f(n) be the number of sequences of 1's and 2's which sum to n. Sequence contains the string of sequences.

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