A206245 -id:A206245 - cp's OEIS frontend

A179051 Number of partitions of n into powers of 10 (cf. A011557).

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

Offset: 0

Reinhard Zumkeller, Jun 27 2010

nonn

A179052 and A008592 give record values and where they occur.

			a(19) = #{10 + 9x1, 19x1} = 2;
a(20) = #{10 + 10, 10 + 10x1, 20x1} = 3;
a(21) = #{10 + 10 + 1, 10 + 11x1, 21x1} = 3.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for related partition-counting sequences

Number of partitions of n into powers of b: A018819 (b=2), A062051 (b=3).
Cf. A206245, A000041, A179051.
Cf. A133880, A132272.
Cf. A008592, A179052.

a179051 = p 1 where
   p _ 0 = 1
   p k m = if m < k then 0 else p k (m - k) + p (k * 10) m
-- Reinhard Zumkeller, Feb 05 2012

terms = 10001;
CoefficientList[Product[1/(1 - x^(10^k)) + O[x]^terms,
     {k, 0, Log[10, terms] // Ceiling}], x]
(* Jean-François Alcover, Dec 12 2021, after Ilya Gutkovskiy *)

a(n) = A133880(n) for n < 90; a(n) = A132272(n) for n < 100.
a(10^n) = A145513(n).
a(10*n) = A179052(n).
A179052(n) = a(A008592(n));
a(n) = p(n,1) where p(n,k) = if k<=n then p(10*[(n-k)/10],k)+p(n,10*k) else 0^n.
G.f.: Product_{k>=0} 1/(1 - x^(10^k)). - Ilya Gutkovskiy, Jul 26 2017

A083278 Repunit powers.

Original entry on oeis.org

1, 11, 111, 121, 1111, 1331, 11111, 12321, 14641, 111111, 161051, 1111111, 1234321, 1367631, 1771561, 11111111, 19487171, 111111111, 123454321, 151807041, 214358881, 1111111111, 1371330631, 2357947691, 11111111111

Offset: 1

Reinhard Zumkeller, Jun 02 2003

			a(13)=1234321=1111^2; a(14)=1367631=111^3; a(15)=1771561=11^6.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Repunit
Wikipedia, Repunit

Cf. A002275, A206245.

import Data.Set (empty, findMin, deleteMin, deleteMin, insert)
import qualified Data.Set as Set (null)
a083278 n = a083278_list !! (n-1)
a083278_list = 1 : f empty (drop 2 a002275_list) where
   f rups rus'@(ru:rus)
     | Set.null rups || m > ru = f (insert (ru,ru) rups) rus
     | otherwise = m : f (insert (m*m',m') (deleteMin rups)) rus'
     where (m,m') = findMin rups
-- Reinhard Zumkeller, Feb 05 2012

With[{intlen=12},Select[Union[Flatten[#^Range[intlen]&/@(FromDigits/@ Table[ PadRight[{},n,1],{n,intlen}])]],IntegerLength[#]<=intlen&]] (* Harvey P. Dale, Apr 25 2016 *)

lista(nn) = {my(list = List(), r); for (n=1, nn, my(r = (10^n-1)/9); listput(list, r); if (r > 1, my(e=2); while(#Str(x=r^e) <= nn, listput(list, x); e++));); Vec(vecsort(list));} \\ Michel Marcus, May 28 2019

A206244 Number of partitions of n into repunits (A002275).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 0

Reinhard Zumkeller, Feb 05 2012

nonn

a(n) = A206245(n) for n <= 120, a(n) < A206245(n) for n > 120.

			a(12)=2 is the first nontrivial term, from the partitions 12 = 1+1+...+1 = 11+1. - _N. J. A. Sloane_, Jul 26 2017

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Repunit
Wikipedia, Repunit
Index entries for related partition-counting sequences

Cf. A000041, A002275, A179051.

a206244 = p $ tail a002275_list where
   p _             0 = 1
   p rus'@(ru:rus) n = if n < ru then 0 else p rus' (n - ru) + p rus n

With[{nn = 50}, Table[Count[IntegerPartitions@ n, k_ /; ContainsAll[Array[Floor[10^#/9] &, IntegerLength[nn + 1]], Union@ k]], {n, 0, nn}]] (* Michael De Vlieger, Jul 26 2017 *)

G.f.: Product_{k>=1} 1/(1 - x^((10^k-1)/9)). - Ilya Gutkovskiy, Jul 26 2017

cp's OEIS Frontend

A179051 Number of partitions of n into powers of 10 (cf. A011557).

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A083278 Repunit powers.

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A206244 Number of partitions of n into repunits (A002275).

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