A061827 -id:A061827 - cp's OEIS frontend

A066960 Duplicate of A061827.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 5, 4, 4, 3, 3, 3, 3, 1, 11, 1, 4, 7, 3, 5, 2, 4, 2, 1, 11, 6, 1, 3

Offset: 1

dead

A119999 Number of partitions of n into parts that occur in decimal representation as substrings of n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 8, 6, 5, 5, 4, 4, 4, 4, 2, 12, 2, 5, 8, 4, 6, 3, 5, 3, 2, 12, 7, 2, 4, 4, 8, 3, 3, 6, 2, 12, 12, 5, 2, 4, 5, 3, 8, 3, 2, 12, 7, 5, 4, 2, 3, 3, 3, 3, 2, 12, 12, 12, 7, 4, 2, 3, 4, 5, 2, 12, 7, 5, 4, 4, 3, 2, 3, 3, 2, 12, 12, 5, 12, 4, 5, 3, 2, 3, 2, 12, 7, 12, 4, 4, 7, 3

Offset: 0

Reinhard Zumkeller, Jun 13 2006

A120002 = first differences; A120003 = partial sums;

see A120000 and A120001 for records and where they occur: A120000(n)=a(A120001(n)).

			a(98) = #{98, 10*9+8, 2*9+10*8} = 3;
a(99) = #{99, 11*9} = 2;
a(100) = #{100, 10*10, 9*10+10*1, 8*10+20*1, 7*10+30*1, 6*10+40*1, 5*10+50*1, 4*10+60*1, 3*10+70*1, 2*10+80*1, 10+90*1, 100*1} = 12;
a(101) = #{101, 10*10+1, 9*10+11*1, 8*10+21*1, 7*10+31*1, 6*10+41*1, 5*10+51*1, 4*10+61*1, 3*10+71*1, 2*10+81*1, 10+91*1, 101*1} = 12;
a(102) = #{102, 10*10+2, 10*10+2*1, 9*10+6*2, ...} = 298.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A000041, A061827, A120004.

import Data.List (isInfixOf)
a119999 n = p (filter ((`isInfixOf` show n) . show) [1..n]) n where
   p _  0 = 1
   p [] _ = 0
   p ks'@(k:ks) m | m < k     = 0
                  | otherwise = p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Aug 14 2011

A125289 Numbers with unique nonzero digit in decimal representation.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 22, 30, 33, 40, 44, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 101, 110, 111, 200, 202, 220, 222, 300, 303, 330, 333, 400, 404, 440, 444, 500, 505, 550, 555, 600, 606, 660, 666, 700, 707, 770, 777, 800, 808, 880, 888, 900, 909

Offset: 1

Reinhard Zumkeller, Nov 26 2006

A043537(a(n)) <= 2.
A043537(A004719(a(n))) = 1: A004719(a(n)) is a repdigit number, see A010785;
also numbers having exactly one partition into digit values of their decimal representations: A061827(a(n))=1.

Rémy Sigrist, Table of n, a(n) for n = 1..9207
Eric Weisstein's World of Mathematics, Repeating Decimal
Eric Weisstein's World of Mathematics, Repdigit

Cf. A125292.

is(n, base=10) = #Set(select(sign, digits(n, base)))==1 \\ Rémy Sigrist, Mar 28 2020

a(n,base=10) = { for (w=0, oo, if (n<=(base-1)*2^w, my (d=1+(n-1)\2^w, k=2^w+(n-1)%(2^w)); return (d*fromdigits(binary(k), base)), n -= (base-1)*2^w)) } \\ Rémy Sigrist, Mar 28 2020

A125289_list = [n for n in range(10**4) if len(set(str(n))-{'0'})==1]
# Chai Wah Wu, Jan 04 2015

from itertools import count, product, islice
def A125289_gen(): # generator of terms
    yield from (int(d+''.join(m)) for l in count(0) for d in '123456789' for m in product('0'+d,repeat=l))
A125289_list = list(islice(A125289_gen(),20)) # Chai Wah Wu, Mar 14 2025

A125291 Number of partitions of n into positive digit values of its ternary representation.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 4, 1, 1, 1, 6, 1, 1, 8, 8, 9, 9, 1, 10, 1, 11, 12, 12, 1, 13, 1, 1, 1, 15, 1, 1, 17, 17, 18, 18, 1, 1, 20, 1, 1, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 1, 28, 1, 29, 30, 30, 1, 31, 1, 32, 33, 33, 34, 34, 35, 35, 36, 36, 1, 37, 1, 38, 39, 39, 1, 40, 1, 1, 1, 42

Offset: 1

Reinhard Zumkeller, Nov 26 2006

a(A125292(n))=1; a(A125293(n))=floor((n+2)/2)=A008619(n).

R. Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Ternary

Cf. A007089, A061827.

a(n) = 1 + floor(n/2) * (1 - 0^(A062756(n)*A081603(n))).

A125290 Numbers with at least two distinct digits in decimal representation, none of which is 0.

Original entry on oeis.org

12, 13, 14, 15, 16, 17, 18, 19, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 48, 49, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 78, 79, 81, 82, 83, 84, 85, 86, 87, 89, 91, 92, 93, 94, 95, 96, 97, 98, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 123

Offset: 1

Reinhard Zumkeller, Nov 26 2006

Also numbers having at least two partitions into digit values of their decimal representations: A061827(a(n)) > 1.

First differs from A101594 at a(83) = 123 != 131 = A101594(83). - Michael S. Branicky, Dec 13 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Subsequence of A052382. Supersequence of A101594.

Cf. A125293, A004719, A043537, A061827, A168046.

a125290 n = a125290_list !! (n-1)
a125290_list = filter ((> 1) . a043537) a052382_list
-- Reinhard Zumkeller, Jun 18 2013

Select[Range[200], FreeQ[#, 0] && Length[Union[#]] > 1 & [IntegerDigits[#]] &] (* Paolo Xausa, May 06 2024 *)

def ok(n): s = set(str(n)); return len(s) >= 2 and "0" not in s
print([k for k in range(124) if ok(k)]) # Michael S. Branicky, Dec 13 2021

A043537(A004719(a(n))) > 1.
A168046(a(n)) * A043537(A004719(a(n))) > 1. - Reinhard Zumkeller, Jun 18 2013
a(n) ~ n. - Charles R Greathouse IV, Feb 13 2017

Name clarified by Michael S. Branicky, Dec 13 2021

A061828 Number of partitions of n into parts using the digits of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 8, 6, 5, 5, 4, 4, 4, 4, 2, 17, 2, 5, 8, 4, 6, 3, 5, 3, 2, 21, 9, 2, 4, 4, 8, 3, 3, 6, 2, 23, 17, 6, 2, 4, 5, 3, 8, 3, 2, 27, 11, 7, 5, 2, 3, 3, 3, 3, 2, 28, 21, 17, 9, 4, 2, 3, 4, 5, 2, 30, 12, 7, 5, 4, 3, 2, 3, 3, 2, 32, 23, 7, 17, 4, 6, 3, 2, 3, 2, 33, 13, 21, 6, 5, 9

Offset: 1

Amarnath Murthy, May 28 2001

			a(20) = 2, the only partitions permitted are 20 and 2+2+2...ten times. a(21) = 16, 5 more than A061827 as the partition 12 + 2+2+2+2+1 etc. ( 5 in number) are also considered.
a(21) = 17. The allowed parts are 1, 2, 12 and 21. The 17 sums are 21, 12+2+2+2+2+1, 12+2+2+2+1+1+1, 12+2+2+1+1+1+1+1, 12+2+1+1+1+1+1+1+1, 12+1+1+1+1+1+1+1+1+1+1 and 11 more involving only 1 and 2.

Cf. A061827.

More terms from David Wasserman, Sep 01 2002

A136460 Number of partitions of n into parts which are not digits of n in decimal representation.

Original entry on oeis.org

0, 1, 2, 4, 6, 10, 14, 21, 29, 11, 13, 8, 10, 18, 23, 33, 40, 55, 64, 170, 35, 244, 119, 174, 234, 303, 384, 475, 577, 1215, 180, 399, 1844, 904, 1194, 1538, 1901, 2326, 2776, 5681, 719, 1517, 2391, 9042, 4426, 5594, 6877, 8286, 9819, 20481, 2184, 4599, 7187, 10050, 33591

Offset: 1

Reinhard Zumkeller, Dec 30 2007

			a(12) = #{9+3,8+4,7+5,6+6,6+3+3,5+4+3,4+4+4,3+3+3+3} = 8;
a(13) = #{9+4,9+2+2,8+5,7+6,7+4+2,7+2+2+2,6+5+2,5+4+4,5+4+2+2,5+2+2+2+2} = 10.

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A061827.

a(n)={my(S=Set(digits(n))); polcoef(1/prod(k=1, 9, 1 - if(!setsearch(S,k), x^k) + O(x*x^n)), n)} \\ Andrew Howroyd, Feb 02 2020

Terms a(38) and beyond from Andrew Howroyd, Feb 02 2020

A193513 Number of partitions of n into parts having at least one common digit in decimal representation.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 3, 8, 5, 9, 9, 13, 9, 16, 12, 18, 16, 23, 20, 31, 30, 38, 38, 51, 49, 64, 62, 79, 77, 95, 101, 118, 118, 143, 145, 179, 181, 216, 223, 267, 286, 325, 341, 399, 416, 485, 500, 575, 600, 686, 735, 823, 864, 981, 1032, 1180

Offset: 0

Reinhard Zumkeller, Jul 30 2011

			a(7) = #{7, 7x1} = 2;
a(8) = #{8, 4+4, 2+2+2+2, 8x1} = 4;
a(9) = #{9, 3+3+3, 9x1} = 3;
a(10) = #{10, 5+5, 2+2+2+2+2, 10x1} = 4;
a(11) = #{11, 10+1, 11x1} = 3;
a(12) = #{12, 11+1, 10+1+1, 6+6, 4+4+4, 3+3+3+3, 6x2, 10x1} = 8;
a(13) = #{13, 12+1, 11+1+1, 10+1+1+1, 13x1} = 5;
a(14) = #{14, 13+1, 12+2, 12+1+1, 11+1+1+1, 10+4x1, 7+7, 7x2, 14x1} = 9.

Cf. A061827, A061828, A109950, A119999.

import Data.List (intersect)
a193513 n = p "0123456789" n 1 where
   p ""        = 0
   p   0       = 1
   p cds m k
     | m < k     = 0
     | otherwise = p (cds `intersect` show k) (m - k) k + p cds m (k + 1)

Thanks to Douglas McNeil, who noticed a program error; data corrected and program fixed by Reinhard Zumkeller, Aug 01 2011

cp's OEIS Frontend

A066960 Duplicate of A061827.

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A119999 Number of partitions of n into parts that occur in decimal representation as substrings of n.

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A125289 Numbers with unique nonzero digit in decimal representation.

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A125291 Number of partitions of n into positive digit values of its ternary representation.

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A125290 Numbers with at least two distinct digits in decimal representation, none of which is 0.

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A061828 Number of partitions of n into parts using the digits of n.

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A136460 Number of partitions of n into parts which are not digits of n in decimal representation.

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A193513 Number of partitions of n into parts having at least one common digit in decimal representation.

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