A007908 -id:A007908 - cp's OEIS frontend

A277635 Number of 7's appearing in the sequence of consecutive natural numbers from 1 to A007908(n), where A007908 = (1, 12, 123, 1234, ...).

0, 1, 22, 343, 4664, 58985, 713307, 8367637, 96022049

Offset: 1

Alexander R. Povolotsky, Oct 24 2016

First 6 terms are the same as in A083449, also see A272525. [See the OEIS wiki page for more details. - M. F. Hasler, Dec 29 2020]
a(n) gives the number of times the digit 7 occurs in all terms of A000027 in the interval [A000027(1), A007908(n)]. - Felix Fröhlich, Oct 28 2016
The sequence was initially defined only up to n = 9 and then extended using A007908 = concat(1..n); see A277837 for the extension using A014824 (a(n) = 10 a(n-1) + n) leading to a smoother growth, in particular at powers of 10. - M. F. Hasler, Nov 01 2016, edited Dec 29 2020

			22 is the third term of the sequence because there are 22 occurrences of the digit '7' contained in numbers within the range of 1 to 123.
96022049 is the 9th term of the sequence because there are 96022049 occurrences of the digit '7' contained in numbers within the range of 1 to 123456789.

Puzzling Stack Exchange, How many sevens?
M. F. Hasler, Digits d in 0 through 123...n, OEIS Wiki, Nov. 2016.

Cf. A083449, A272525, A014824.

Cf. A277830 - A277838 and A277849: analog for digits 0 .. 9, but based on A014824 instead of A083449.

Table[a[n] = Count[Flatten@ Map[IntegerDigits, Range@ FromDigits@ Range@ n], k_ /; k == 8]; Print@ a@ n; an = a[n]; an, {n, 0, 9}] (* Michael De Vlieger, Oct 30 2016 *)

print1(c=0);N=1;for(n=2,8,print1(","c+=sum(k=N+1,N=eval(Str(N,n)),#select(d->d==7,digits(k))))) \\ For illustration; more efficient code below. - M. F. Hasler, Oct 31 2016

A277635(n, m=7)=if(n>m,A277635(n, m+1)+(m+2)*10^(n-m-1),A277830(n)-(m>n)) \\ Valid only for n <= 9. - M. F. Hasler, Nov 02 2016

A050926 Binary representation of A007908(n).

Original entry on oeis.org

1, 1100, 1111011, 10011010010, 11000000111001, 11110001001000000, 100101101011010000111, 101111000110000101001110, 111010110111100110100010101, 1011011111110111000001110000111110

Offset: 1

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 30 1999

Seiichi Manyama, Table of n, a(n) for n = 1..136

Cf. A007908, A097580, A097582, A097583.

Cf. A007088.

a(n) = A007088(A007908(n)). - Seiichi Manyama, Apr 23 2022

A353104 Base-4 representation of A007908(n).

Original entry on oeis.org

1, 30, 1323, 103102, 3000321, 132021000, 10231122013, 233012011032, 13112330310111, 23133313001300332, 101331301332300201003, 130010202012003303220320, 223313011101131330223313231, 1010202010033102302310203002202

Offset: 1

Seiichi Manyama, Apr 23 2022

Seiichi Manyama, Table of n, a(n) for n = 1..236

Cf. A007908, A050926, A097580, A097582, A097583, A353105, A353106, A353107.

Cf. A007090.

def A(k, n)
  (1..n).map{|i| (1..i).to_a.join.to_i.to_s(k).to_i}
end
p A(4, 20)

a(n) = A007090(A007908(n)).

A353105 Base-5 representation of A007908(n).

Original entry on oeis.org

1, 22, 443, 14414, 343340, 12422311, 304001232, 11130030203, 223101104124, 200240443211120, 130211343340003021, 112140204001002213422, 100421133100401442024323, 40324014240311242321340224, 31241112311230113034201201130

Offset: 1

Seiichi Manyama, Apr 23 2022

Seiichi Manyama, Table of n, a(n) for n = 1..269

Cf. A007908, A050926, A097580, A097582, A097583, A353104, A353106, A353107.

Cf. A007091.

def A(k, n)
  (1..n).map{|i| (1..i).to_a.join.to_i.to_s(k).to_i}
end
p A(5, 20)

a(n) = A007091(A007908(n)).

A353106 Base-6 representation of A007908(n).

Original entry on oeis.org

1, 20, 323, 5414, 133053, 2351320, 42243331, 1120335530, 20130035113, 5401014424514, 2343052550252003, 1114323133240053240, 321321022332303252301, 132140014431255340214310, 42015444551405453142112503

Offset: 1

Seiichi Manyama, Apr 23 2022

Seiichi Manyama, Table of n, a(n) for n = 1..295

Cf. A007908, A050926, A097580, A097582, A097583, A353104, A353105, A353107.

Cf. A007092.

def A(k, n)
  (1..n).map{|i| (1..i).to_a.join.to_i.to_s(k).to_i}
end
p A(6, 20)

a(n) = A007092(A007908(n)).

A353107 Base-9 representation of A007908(n).

Original entry on oeis.org

1, 13, 146, 1621, 17836, 207313, 2281451, 25206070, 277266780, 34771513601, 4330564256733, 535110486286816, 65858371163036861, 8202185121837583406, 1113465620754570813253, 134741562223525280514741, 16425841240157671153405780

Offset: 1

Seiichi Manyama, Apr 23 2022

Seiichi Manyama, Table of n, a(n) for n = 1..354

Cf. A007908, A050926, A097580, A097582, A097583, A353104, A353105, A353106.

Cf. A007095.

def A(k, n)
  (1..n).map{|i| (1..i).to_a.join.to_i.to_s(k).to_i}
end
p A(9, 20)

a(n) = A007095(A007908(n)).

A075002 a(n) = smallest k such that A007908(k) is divisible by n.

Original entry on oeis.org

1, 2, 2, 2, 5, 2, 11, 6, 8, 10, 106, 2, 11, 22, 5, 6, 21, 8, 27, 20, 11, 106, 33, 6, 25, 30, 26, 24, 31, 20, 46, 6, 113, 64, 45, 36, 21, 64, 11, 20, 3, 24, 19, 128, 35, 68, 8, 6, 53, 50, 21, 120, 83, 26, 135, 128, 27, 82, 104, 20

Offset: 1

Amarnath Murthy, Sep 01 2002

Conjecture: For every n, a(n) is well-defined.

The maximum of the first 10000 terms is a(6314) = 1002902 and A007908(1002902) is a 5902910-digit number. - Rick L. Shepherd, Oct 11 2015

			a(4) = 2 since A007908(2)=12 is divisible by 4.
a(7) = 11 as 7 divides A007908(11)=1234567891011 and 7 does not divide any smaller concatenations.

Rick L. Shepherd, Table of n, a(n) for n = 1..10000

Cf. A007908.

a(n)=my(m=Mod(1,n),k=1); while(m, m=10^#Str(k++)*m+k); k \\ Charles R Greathouse IV, Oct 11 2015

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

Edited by Max Alekseyev, Sep 30 2015

A165770 Smallest divisor of 123...n (=A007908) larger than the preceding term.

Original entry on oeis.org

1, 2, 3, 617, 823, 1286, 9721, 14593, 32463, 1234567891, 1417414341, 2110805449, 98220894641, 205761315168520219, 8230452606740808761, 3086419727527803285379, 8755800645468945490437, 70708355728013820970883

Offset: 1

M. F. Hasler, Sep 30 2009

Inspired by Layman's comment on a sequence from Angelini, cf. link.

John W. Layman, Re: Sum divides concatenation, SeqFan list, Sep 30 2009.

Cf. A151995.

c=""; s=0; for(i=1,19, fordiv(c=eval(Str(c,i)),d, d>s || next; print1(s=d,", "); break))

Edited by Charles R Greathouse IV, Apr 24 2010

A072725 Integers which are exactly the concatenation of the first m numbers (A007908) divided by their sum (A000217 = m*(m+1)/2).

Original entry on oeis.org

1, 4, 823

Offset: 1

Henry Bottomley, Jul 06 2002

A probability argument suggests that this sequence may be finite.

			a(1) = 1/1 =1; a(2) = 12/(1+2) = 4; a(3) = 12345/(1+2+3+4+5).

Cf. A067116, A072723, A072724.

A165771 Smallest positive integer such that a(1)+...a(n) divides 123...n (=A007908(n)).

Original entry on oeis.org

1, 1, 1, 614, 206, 463, 8435, 4872, 17870, 1234535428, 182846450, 693391108, 96110089192, 205761216947625578, 8024691291572288542, 3078189274921062476618, 5669380917941142205058, 61952555082544875480446

Offset: 1

M. F. Hasler, Sep 30 2009

Without the initial term, these are the first differences of A165770.

John W. Layman, Re: Sum divides concatenation, SeqFan list, Sep 30 2009.

Cf. A151995, A165770, A166064.

c=""; s=0; for(i=1,19, fordiv(c=eval(Str(c,i)),d, d>s || next; print1(d-s,", "); s=d; break))

Edited by Charles R Greathouse IV, Apr 24 2010

cp's OEIS Frontend

A277635 Number of 7's appearing in the sequence of consecutive natural numbers from 1 to A007908(n), where A007908 = (1, 12, 123, 1234, ...).

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A050926 Binary representation of A007908(n).

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A353104 Base-4 representation of A007908(n).

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A353105 Base-5 representation of A007908(n).

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A353106 Base-6 representation of A007908(n).

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A353107 Base-9 representation of A007908(n).

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A075002 a(n) = smallest k such that A007908(k) is divisible by n.

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A165770 Smallest divisor of 123...n (=A007908) larger than the preceding term.

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Comments

Links

Crossrefs

Programs

PARI

Extensions

A072725 Integers which are exactly the concatenation of the first m numbers (A007908) divided by their sum (A000217 = m*(m+1)/2).

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