A000042 -id:A000042 - cp's OEIS frontend

A046417 Repunit of length a(n) has exactly 6 prime factors (counted with multiplicity).

15, 16, 26, 33, 34, 39, 46, 57, 69, 76, 79, 93, 94, 106, 118, 121, 133, 169, 181, 278, 281, 293

Offset: 1

Patrick De Geest, Jul 15 1998

P. De Geest, Repunits prime factors
Makoto Kamada, Factorizations of 11...11 (Repunit)
Eric Weisstein's World of Mathematics, Repunit

Cf. A000042, A001222, A002275, A004022, A004023, A046053.

More terms from Robert Gerbicz, Nov 22 2010
a(20) from Bo Gyu Jeong, Jun 30 2012
a(21)-a(22) from Ray Chandler, Apr 23 2017

A046418 Repunit of length a(n) has exactly 7 prime factors (counted with multiplicity).

Original entry on oeis.org

12, 20, 21, 22, 27, 35, 61, 65, 74, 82, 85, 141, 146, 177, 187, 194, 226, 299, 323, 329, 337

Offset: 1

Patrick De Geest, Jul 15 1998

P. De Geest, Repunits prime factors
Makoto Kamada, Factorizations of 11...11 (Repunit)
Eric Weisstein's World of Mathematics, Repunit

Cf. A000042, A001222, A002275, A004022, A004023, A046053.

Select[Range[300],PrimeOmega[FromDigits[PadRight[{},#,1]]]==7&] (* Harvey P. Dale, Feb 04 2019 *)

More terms from Robert Gerbicz, Nov 22 2010
Changed offset to 1, a(18) added by Ray Chandler, Apr 23 2017
a(19)-a(21) from Max Alekseyev, May 14 2022

A046419 Repunit of length a(n) has exactly 8 prime factors (counted with multiplicity).

Original entry on oeis.org

28, 51, 55, 58, 77, 86, 95, 98, 107, 115, 119, 124, 155, 161, 193, 209, 217, 218, 221, 233, 253, 265, 295, 298, 303, 314, 346

Offset: 1

Patrick De Geest, Jul 15 1998

P. De Geest, Repunits prime factors
Makoto Kamada, Factorizations of 11...11 (Repunit)
Eric Weisstein's World of Mathematics, Repunit

Cf. A000042, A001222, A002275, A004022, A004023, A046053.

Select[Range[320],PrimeOmega[FromDigits[Table[1,#]]]==8&] (* Harvey P. Dale, Sep 04 2018 *)

More terms from Robert Gerbicz, Nov 22 2010
Offset changed to 1, a(23)-a(26) added by Ray Chandler, Apr 23 2017
a(27) from Max Alekseyev, May 14 2022

A046420 Prime factors of repunit of length a(n) are all of different lengths.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 11, 14, 17, 19, 23, 31, 34, 37, 38, 41, 43, 47, 49, 51, 53, 57, 59, 62, 67, 69, 71, 73, 74, 79, 83, 85, 86, 89, 93, 94, 97, 101, 103, 106, 109, 113, 115, 118, 119, 129, 134, 137, 139, 141, 142, 146, 149, 151, 157, 159, 163

Offset: 1

Patrick De Geest, Jul 15 1998

			a(n)=14 -> 11*239*4649*909091 -> (2)(3)(4)(6) all of different lengths.

Max Alekseyev, Table of n, a(n) for n = 1..117 (first 107 terms from Ray Chandler)
P. De Geest, Repunits prime factors
Eric Weisstein's World of Mathematics, Repunit

Cf. A000042, A004022, A046053.

Select[Range[60],Length[fi=FactorInteger[(10^#-1)/9]]== Plus@@Last/@fi==Length[Union[IntegerLength/@First/@fi]]&] (* Ray Chandler, Apr 24 2017 *)

Offset changed to 1 and more terms added by Ray Chandler, Apr 24 2017

A068743 Digitized partition numbers: numbers with (weakly) decreasing digits ordered by sum of their digits then by the numbers themselves.

Original entry on oeis.org

0, 1, 2, 11, 3, 21, 111, 4, 22, 31, 211, 1111, 5, 32, 41, 221, 311, 2111, 11111, 6, 33, 42, 51, 222, 321, 411, 2211, 3111, 21111, 111111, 7, 43, 52, 61, 322, 331, 421, 511, 2221, 3211, 4111, 22111, 31111, 211111, 1111111, 8, 44, 53, 62, 71, 332, 422, 431, 521

Offset: 0

Henry Bottomley, Feb 27 2002

a(97) cannot be written in decimal since it requires ten to be written as a single digit.

			The partitions of 6 are 6, 5+1, 4+2, 4+1+1, 3+3, 3+2+1, 3+1+1+1, 2+2+2, 2+2+1+1, 2+1+1+1+1, 1+1+1+1+1+1; removing the + signs gives 6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111; ordering these by size gives 6, 33, 42, 51, 222, 321, 411, 2211, 3111, 21111, 111111 as part of the sequence.

Harvey P. Dale, Table of n, a(n) for n = 0..2500 [a(2088) corrected by Sean A. Irvine]
Index entries for sequences related to partitions

Cf. A060002 which writes the partitions with smallest digit first, number of values of a(n) with a digit sum of k is A000041, number of values of a(n) with a digit sum of k and m digits is A008284, a(A000070(n))=n+1 written as a single digit, a(A026905(n))=A000042(n).

Table[Sort[FromDigits[Flatten[IntegerDigits/@#]]&/@IntegerPartitions[n]],{n,0,20}]//Flatten (* Harvey P. Dale, May 18 2024 *)

A082207 Palindromes whose product of digits is a positive palindrome.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 111, 121, 131, 141, 151, 161, 171, 181, 191, 212, 222, 313, 676, 777, 1111, 1221, 1331, 2112, 3113, 11111, 11211, 11311, 11411, 11511, 11611, 11711, 11811, 11911, 12121, 12221, 13131, 16761, 17771, 21112, 21212

Offset: 1

Amarnath Murthy, Apr 10 2003

The unary sequence A000042 is a trivial subsequence.
Conjecture: There are infinitely many terms in the sequence (of the type 777) for which the product of digits > 10.
Subset of A117055, containing terms for which product of digits is greater than 0. - Jayanta Basu, May 15 2013

			777 is a term as 7^3 = 343 is a palindrome.

Harvey P. Dale, Table of n, a(n) for n = 1..89 (all terms up to 10 million)

Cf. A082208.

id[n_]:=IntegerDigits[n]; palQ[n_]:=Reverse[x=id[n]]==x; t={}; Do[If[palQ[n] && (y=Times@@id[n]) > 0 && palQ[y], AppendTo[t,n]], {n,21220}]; t (* Jayanta Basu, May 15 2013 *)
Select[Range[22000],FreeQ[IntegerDigits[#],0]&&AllTrue[{#,Times @@ IntegerDigits[ #]},PalindromeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 22 2019 *)

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 19 2003

A083818 Numbers k such that 2k-1 is the digit reversal of k.

Original entry on oeis.org

1, 37, 397, 3997, 39997, 399997, 3999997, 39999997, 399999997, 3999999997, 39999999997, 399999999997, 3999999999997, 39999999999997, 399999999999997, 3999999999999997, 39999999999999997, 399999999999999997, 3999999999999999997, 39999999999999999997, 399999999999999999997

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 08 2003

a(n) = 1 + 36 + 360 + 3600 + 36000 + ..., for a total of n terms. a(n) = 1 + sum of first n-1 terms of the geometric progression with first term 36 and common ratio 10. a(n) = 1 + 36*A000042(n-1) (the unary sequence).

			2*37 - 1 = 73.

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

Cf. A000042, A083811, A083812, A083813.

Digit reversals of A169830.

my(x='x+O('x^22)); Vec(x*(1+26*x)/((1-x)*(1-10*x))) \\ Elmo R. Oliveira, Jun 12 2025

a(n) = 4*10^(n-1) - 3.
From Elmo R. Oliveira, Jun 12 2025: (Start)
G.f.: x*(26*x+1)/((x-1)*(10*x-1)).
E.g.f.: (13 - 15*exp(x) + 2*exp(10*x))/5.
a(n) = 11*a(n-1) - 10*a(n-2) for n >= 3. (End)

a(1)=1 inserted by David Radcliffe, Jul 25 2015

A110380 a(n) = min{p + q + r + ...} where p,q,r,... are distinct unary numbers - containing only ones, i.e., of the form (10^k - 1)/9 - formed by using a total of n ones.

Original entry on oeis.org

1, 11, 12, 112, 122, 123, 1123, 1223, 1233, 1234, 11234, 12234, 12334, 12344, 12345, 112345, 122345, 123345, 123445, 123455, 123456, 1123456, 1223456, 1233456, 1234456, 1234556, 1234566, 1234567, 11234567, 12234567, 12334567, 12344567, 12345567, 12345667, 12345677

Offset: 1

Amarnath Murthy, Jul 25 2005

The n-th term is the sum of m = A003056(n) repunits A000042, with the last n - m(m+1)/2 terms having one digit more than their index in that sum: see formula. The initial terms can also be described as numbers made of all of the digits 1 through m (m = 1, ..., 9) in increasing order, and at most one of these digits occurring twice in a row. - M. F. Hasler, Aug 08 2020

			Using n ones and only the + sign we get the following sums:
  a(1) = 1;
  a(2) = 11;
  a(3) = 12 = 1 + 11;
  a(4) = 112 = 1 + 111;
  a(5) = 122 = 11 + 111;
  a(6) = 123 = 1 + 11 + 111;
  a(7) = 1123 = 1 + 11 + 1111;
  a(8) = 1223 = 1 + 111 + 1111;
  a(9) = 1233 = 11 + 111 + 1111.

Fred Schneider, Table of n, a(n) for n = 1..5000

Cf. A000042, A003056.

a110380 = drop 1 fn
          where fn    = 0 : 1 : concat (fn' 2)
                fn' n = (map (+ones) (drop nv $ take (n + nv) fn)) : (fn' (n+1))
                        where ones = div (10^n -1) 9
                              nv   = div ((n-1)*(n-2)) 2
-- Fred Schneider, Sep 04 2016

apply( {A110380(n,m=A003056(n))=sum(k=1,m,10^((n+k>(m+3)*m\2)+k)\9)}, [1..40]) \\ with {A003056(n)=(sqrtint(n*8+1)-1)\2}. M. F. Hasler, Aug 08 2020

a(n) = Sum_{k=1..m} A000042(k + [m(m+3)/2 < n+k]), with m = A003056(n). - M. F. Hasler, Aug 08 2020

More terms from Joshua Zucker, May 08 2006

A178865 Triangle read by rows; generalization of A101950.

Original entry on oeis.org

1, 10, 1, 90, 20, 1, 800, 280, 30, 1, 7101, 3400, 570, 40, 1, 63010, 38301, 8800, 960, 50, 1, 559090, 412020, 120601, 18000, 1450, 60, 1, 4960800, 4296280, 1530030, 291001, 32000, 2040, 70, 1

Offset: 1

Mark Dols, Jun 20 2010

Rows sum up to A000042

Erroneous version of A206819. - Philippe Deléham, Feb 23 2013

			Triangle begins:
1,
10, 1,
90, 20, 1,
800, 280, 30, 1

Cf. A000042, A101950, A004189, A206819.

A206819 Riordan array (1/(1-10x-10x^2), x/(1-10x-10x^2)).

Original entry on oeis.org

1, 10, 1, 90, 20, 1, 800, 280, 30, 1, 7100, 3400, 570, 40, 1, 63000, 38300, 8800, 960, 50, 1, 559000, 412000, 120600, 18000, 1450, 60, 1, 4960000, 4296000, 1530000, 291000, 32000, 2040, 70, 1

Offset: 0

Philippe Deléham, Feb 12 2012

Row sums are A000042(n+1).

Subtriangle of triangle given by (0, 10, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

			Triangle begins :
1
10, 1
90, 20, 1
800, 280, 30, 1
7100, 3400, 570, 40, 1
63000, 38300, 8800, 960, 50, 1
559000, 412000, 120600, 18000, 1450, 60, 1
4960000, 4296000, 1530000, 291000, 32000, 2040, 70, 1
Triangle (0, 10, -1, 1, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) begins :
1
0, 1
0, 10, 1
0, 90, 20, 1
0, 800, 280, 30, 1
0, 7100, 3400, 570, 40, 1 ...

Cf. A101950, A178865,

T(n,k) = 10*T(n-1,k) - 10*T(n-2,k) + T(n-1,k-1).
G.f.: 1/(1-10*x+10*x^2-y*x).
Sum_{k, 0<=k} T(n,k)*x^k = A178869(n+1), A057086(n), A000042(n+1) for x = -1, 0, 1 respectively.

cp's OEIS Frontend

A046417 Repunit of length a(n) has exactly 6 prime factors (counted with multiplicity).

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A046418 Repunit of length a(n) has exactly 7 prime factors (counted with multiplicity).

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A046419 Repunit of length a(n) has exactly 8 prime factors (counted with multiplicity).

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A046420 Prime factors of repunit of length a(n) are all of different lengths.

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A068743 Digitized partition numbers: numbers with (weakly) decreasing digits ordered by sum of their digits then by the numbers themselves.

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Mathematica

A082207 Palindromes whose product of digits is a positive palindrome.

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Mathematica

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A083818 Numbers k such that 2k-1 is the digit reversal of k.

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A110380 a(n) = min{p + q + r + ...} where p,q,r,... are distinct unary numbers - containing only ones, i.e., of the form (10^k - 1)/9 - formed by using a total of n ones.

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A206819 Riordan array (1/(1-10x-10x^2), x/(1-10x-10x^2)).