A171397 -id:A171397 - cp's OEIS frontend

A375523 a(n) = numerator of Sum_{i=1..n} 1/A171397(i).

1, 3, 11, 25, 137, 49, 363, 761, 7129, 80939, 83249, 1109957, 1135697, 1159721, 2364487, 40916999, 13865893, 267536047, 271415923, 274943083, 6401288429, 6475652719, 32735212187, 33078431987, 300680459483, 43364113769, 1269032646901, 1280123549581, 40016557117411, 3666283538201

Offset: 1

N. J. A. Sloane, Aug 30 2024

Suggested by A375805.

			The first few sums are 1, 3/2, 11/6, 25/12, 137/60, 49/20, 363/140, 761/280, 7129/2520, 80939/27720, 83249/27720, 1109957/360360, 1135697/360360, 1159721/360360, 2364487/720720,  ...

Alois P. Heinz, Table of n, a(n) for n = 1..2082

Cf. A171397, A375805, A375524, A001008/A002805.

b:= n-> (l-> add(l[i]*11^(i-1), i=1..nops(l)))(convert(n,base,10)):
g:= proc(n) option remember; `if`(n<1, 0, g(n-1)+1/b(n)) end:
a:= n-> numer(g(n)):
seq(a(n), n=1..30);  # Alois P. Heinz, Aug 30 2024

A375524 a(n) = denominator of Sum_{i=1..n} 1/A171397(i).

Original entry on oeis.org

1, 2, 6, 12, 60, 20, 140, 280, 2520, 27720, 27720, 360360, 360360, 360360, 720720, 12252240, 4084080, 77597520, 77597520, 77597520, 1784742960, 1784742960, 8923714800, 8923714800, 80313433200, 11473347600, 332727080400, 332727080400, 10314539492400, 937685408400, 937685408400

Offset: 1

N. J. A. Sloane, Aug 30 2024

			The first few sums are 1, 3/2, 11/6, 25/12, 137/60, 49/20, 363/140, 761/280, 7129/2520, 80939/27720, 83249/27720, 1109957/360360, 1135697/360360, 1159721/360360, 2364487/720720,  ...

Alois P. Heinz, Table of n, a(n) for n = 1..2082

Cf. A171397, A375805, A375523, A001008/A002805.

b:= n-> (l-> add(l[i]*11^(i-1), i=1..nops(l)))(convert(n,base,10)):
g:= proc(n) option remember; `if`(n<1, 0, g(n-1)+1/b(n)) end:
a:= n-> denom(g(n)):
seq(a(n), n=1..31);  # Alois P. Heinz, Aug 30 2024

A375805 Decimal expansion of Sum_{n >= 1} 1/A171397(n).

Original entry on oeis.org

2, 6, 2, 8, 3, 3, 2, 8, 2, 0, 4, 8, 8, 1, 4, 2, 0, 7, 6, 9, 9, 4, 0, 1, 5, 1, 6, 8, 7, 4, 4, 4, 2, 2, 2, 9, 2, 4, 1, 8, 8, 7, 9, 8, 0, 9, 2, 5

Offset: 2

Robert C. Lyons, Aug 29 2024

A variation on the harmonic series, in which the denominators are treated as base 11 numbers. Equivalently: sum of reciprocals of positive integers whose base-11 representation contains no digit A (no "10" digit).
Values were calculated using Mathematica code from Baillie & Schmelzer (see link). Note that the code in the Wolfram Library Archive, as it stands, does not support digits > 9 in bases > 10 (and doing the "obvious" thing will be interpreted as asking a different question with a different answer); the code was modified to support this.
Kempner (1914) showed that this series converges. - N. J. A. Sloane, Aug 31 2024
There is a slight ambiguity when we get to 1/10. This is to be regarded as 1/(1*11 + 0*1) = (1/11)-in-base-10 and not as 1/A =  1/(10*1) = (1/10)-in-base-10. - N. J. A. Sloane, Aug 30 2024

			26.2833282048814207699401516874442229241887980925...

Burnol, Jean-François. "Moments in the exact summation of the curious series of Kempner type." arXiv preprint arXiv:2402.08525 (2024).
A. J. Kempner, A Curious Convergent Series, American Mathematical Monthly, 21 (February, 1914), pp. 48-50. (https://dx.doi.org/10.2307/2972074)
Schmelzer, Thomas, and Robert Baillie. "Summing a curious, slowly convergent series." The American Mathematical Monthly 115.6 (2008): 525-540.

Robert Baillie, Sums of reciprocals of integers missing a given digit, Amer. Math. Monthly, 86 (1979), 372-374.
Robert Baillie, Summing the curious series of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015.
Robert Baillie & Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Wolfram Library Archive
G. W. Brewster, An Old Result in a New Dress, The Mathematical Gazette, Vol. 37, No. 322 (Dec., 1953), pp. 269-270.
John D. Cook, A strange take on the harmonic series, Blog, 29 August 2024.
N. J. A. Sloane, A Nasty Surprise in a Sequence and Other OEIS Stories, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; Slides [Mentions this sequence]
Eric Weisstein's World of Mathematics, Kempner Series.
Wikipedia, Kempner series. [From _M. F. Hasler_, Jan 13 2020]

Cf. A171397, A375523/A375524.

Corrected data provided by Gareth McCaughan, Sep 02 2024

A011539 "9ish numbers": decimal representation contains at least one nine.

Original entry on oeis.org

9, 19, 29, 39, 49, 59, 69, 79, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 109, 119, 129, 139, 149, 159, 169, 179, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 209, 219, 229, 239, 249, 259, 269, 279, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298

Offset: 1

N. J. A. Sloane

The 9ish numbers are closed under lunar multiplication. The lunar primes (A087097) are a subset.
Almost all numbers are 9ish, in the sense that the asymptotic density of this set is 1: Among the 9*10^(n-1) n-digit numbers, only a fraction of 0.8*0.9^(n-1) doesn't have a digit 9, and this fraction tends to zero (< 1/10^k for n > 22k-3). This explains the formula a(n) ~ n. - M. F. Hasler, Nov 19 2018
A 9ish number is a number whose largest decimal digit is 9. - Stefano Spezia, Nov 16 2023

			E.g. 9, 19, 69, 90, 96, 99 and 1234567890 are all 9ish.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
D. Applegate, C program for lunar arithmetic and number theory [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
D. Applegate, M. LeBrun, and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
Index entries for sequences related to dismal (or lunar) arithmetic
Index entries for 10-automatic sequences.

Cf. A088924 (number of n-digit terms).
Cf. A087062 (lunar product), A087097 (lunar primes).
A102683 (number of digits 9 in n); fixed points > 8 of A068505.
Cf. Numbers with at least one digit b-1 in base b : A074940 (b=3), A337250 (b=4), A337572 (b=5), A333656 (b=6), A337141 (b=7), A337239 (b=8), A338090 (b=9), this sequence (b=10), A095778 (b=11).
Cf. Numbers with no digit b-1 in base b: A005836 (b=3), A023717 (b=4), A020654 (b=5), A037465 (b=6), A020657 (b=7), A037474 (b=8), A037477 (b=9), A007095 (b=10), A171397 (b=11).
Supersequence of A043525.

Filtered([1..300],n->9 in ListOfDigits(n)); # Muniru A Asiru, Feb 25 2019

a011539 n = a011539_list !! (n-1)
a011539_list = filter ((> 0) . a102683) [1..]  -- Reinhard Zumkeller, Dec 29 2011

seq(`if`(numboccur(9, convert(n, base, 10))>0, n, NULL), n=0..100); # François Marques, Oct 12 2020

Select[ Range[ 0, 100 ], (Count[ IntegerDigits[ #, 10 ], 9 ]>0)& ] (* François Marques, Oct 12 2020 *)
Select[Range[300],DigitCount[#,10,9]>0&] (* Harvey P. Dale, Mar 04 2023 *)

is(n)=n=vecsort(digits(n));n[#n]==9 \\ Charles R Greathouse IV, May 15 2013

select( is_A011539(n)=vecmax(digits(n))==9, [1..300]) \\ M. F. Hasler, Nov 16 2018

def ok(n): return '9' in str(n)
print(list(filter(ok, range(299)))) # Michael S. Branicky, Sep 19 2021

def A011539(n):
    def f(x):
        l = (s:=str(x)).find('9')
        if l >= 0: s = s[:l]+'8'*(len(s)-l)
        return n+int(s,9)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Dec 04 2024

Complement of A007095. A102683(a(n)) > 0 (defines this sequence). A068505(a(n)) = a(n): fixed points of A068505 are the terms of this sequence and the numbers < 9. - Reinhard Zumkeller, Dec 29 2011, edited by M. F. Hasler, Nov 16 2018

a(n) ~ n. - Charles R Greathouse IV, May 15 2013

A037465 a(n) = Sum_{i=0..m} d(i)6^i, where Sum_{i=0..m} d(i)5^i is the base 5 representation of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93

Offset: 0

Clark Kimberling

Numbers without digit 5 in base 6. Complement of A333656. - François Marques, Oct 13 2020

			a(34)=46 because 34 is 114_5 in base 5 and 114_6=46. - _François Marques_, Oct 13 2020

François Marques, Table of n, a(n) for n = 0..10000 (first 1000 terms from Clark Kimberling)

Cf. Numbers with at least one digit b-1 in base b : A074940 (b=3), A337250 (b=4), A337572 (b=5), A333656 (b=6), A337141 (b=7), A337239 (b=8), A338090 (b=9), A011539 (b=10), A095778 (b=11).

Cf. Numbers with no digit b-1 in base b: A005836 (b=3), A023717 (b=4), A020654 (b=5), this sequence (b=6), A020657 (b=7), A037474 (b=8), A037477 (b=9), A007095 (b=10), A171397 (b=11).

Table[FromDigits[RealDigits[n, 5], 6], {n, 0, 100}] (* Clark Kimberling, Aug 14 2012 *)

a(n) = fromdigits(digits(n, 5), 6); \\ François Marques, Oct 13 2020

from gmpy2 import digits
def A037465(n): return int(digits(n,5),6) # Chai Wah Wu, May 06 2025

Offset changed to 0 by Clark Kimberling, Aug 14 2012

A338090 Numbers having at least one 8 in their representation in base 9.

Original entry on oeis.org

8, 17, 26, 35, 44, 53, 62, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 89, 98, 107, 116, 125, 134, 143, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 170, 179, 188, 197, 206, 215, 224, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 251, 260, 269, 278, 287, 296, 305, 314, 315

Offset: 1

François Marques, Oct 09 2020

Blocks of consecutive terms have lengths in A002452. - Devansh Singh, Oct 21 2020

			70 is not in the sequence since it is 77_9 in base 9, but 76 is in the sequence since it is 84_9 in base 9.

François Marques, Table of n, a(n) for n = 1..10000

Cf. A007095 (base 9).
Complement of A037477.
Cf. A043485 (numbers with exactly one 8 in base 9).
Cf. Numbers with at least one digit b-1 in base b: A074940 (b=3), A337250 (b=4), A337572 (b=5), A333656 (b=6), A337141 (b=7), A337239 (b=8), this sequence (b=9), A011539 (b=10), A095778 (b=11).
Cf. Numbers with no digit b-1 in base b: A005836 (b=3), A023717 (b=4), A020654 (b=5), A037465 (b=6), A020657 (b=7), A037474 (b=8), A037477 (b=9), A007095 (b=10), A171397 (b=11).

seq(`if`(numboccur(8, convert(n, base, 9))>0, n, NULL), n=0..100);

Select[ Range[ 0, 100 ], (Count[ IntegerDigits[ #, 9 ], 8 ]>0)& ]

isok(m) = #select(x->(x==8), digits(m, 9)) >= 1;

from gmpy2 import digits
def A338090(n):
    def f(x):
        l = (s:=digits(x,9)).find('8')
        if l >= 0: s = s[:l]+'7'*(len(s)-l)
        return n+int(s,8)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Dec 04 2024

A037474 a(n) = Sum{d(i)8^i: i=0,1,...,m}, where Sum{d(i)7^i: i=0,1,...,m} is the base 7 representation of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85

Offset: 0

Clark Kimberling

Numbers without digit 7 in base 8. Complement of A337239. - François Marques, Oct 13 2020

			a(48)=54 because 48 is 66_7 in base 7 and 66_8=54. - _François Marques_, Oct 13 2020

François Marques, Table of n, a(n) for n = 0..10000 (first 1000 terms from Clark Kimberling)

Cf. Numbers with at least one digit b-1 in base b : A074940 (b=3), A337250 (b=4), A337572 (b=5), A333656 (b=6), A337141 (b=7), A337239 (b=8), A338090 (b=9), A011539 (b=10), A095778 (b=11).

Cf. Numbers with no digit b-1 in base b: A005836 (b=3), A023717 (b=4), A020654 (b=5), A037465 (b=6), A020657 (b=7), this sequence (b=8), A037477 (b=9), A007095 (b=10), A171397 (b=11).

Table[FromDigits[RealDigits[n, 7], 8], {n, 0, 100}] (* Clark Kimberling, Aug 14 2012 *)

a(n) = fromdigits(digits(n, 7), 8); \\ François Marques, Oct 13 2020

from gmpy2 import digits
def A037474(n): return int(digits(n,7),8) # Chai Wah Wu, Dec 04 2024

Offset changed to 0 by Clark Kimberling, Aug 14 2012

A037477 a(n) = Sum{d(i)9^i: i=0,1,...,m}, where Sum{d(i)8^i: i=0,1,...,m} is the base 8 representation of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 81, 82, 83, 84, 85

Offset: 0

Clark Kimberling

Numbers that do not contain the digit 8 in their base 9 expansion. - M. F. Hasler, Oct 05 2014

			a(63) = 7*9+7 = 70 since 63 = 77[8], i.e., "77" when written in base 8;
a(64) = 1*9^2 = 81 since 64 = 100[8]. - _M. F. Hasler_, Oct 05 2014

François Marques, Table of n, a(n) for n = 0..10000 (first 1000 terms from Clark Kimberling)

Cf. A248375.
Cf. Numbers with at least one digit b-1 in base b : A074940 (b=3), A337250 (b=4), A337572 (b=5), A333656 (b=6), A337141 (b=7), A337239 (b=8), A338090 (b=9), A011539 (b=10), A095778 (b=11).
Cf. Numbers with no digit b-1 in base b: A005836 (b=3), A023717 (b=4), A020654 (b=5), A037465 (b=6), A020657 (b=7), A037474 (b=8), this sequence (b=9), A007095 (b=10), A171397 (b=11).

Table[FromDigits[RealDigits[n, 8], 9], {n, 0, 100}]
Select[Range[0,100],DigitCount[#,9,8]==0&] (* Harvey P. Dale, Aug 06 2024 *)

a(n) = vector(#n=digits(n,8),i,9^(#n-i))*n~ \\ M. F. Hasler, Oct 05 2014

a(n) = fromdigits(digits(n, 8), 9); \\ François Marques, Oct 15 2020

def A037477(n): return int(oct(n)[2:],9) # Chai Wah Wu, Jan 27 2025

For n<64, a(n) = floor(9n/8) = A248375(n). - M. F. Hasler, Oct 05 2014

Offset changed to 0 by Clark Kimberling, Aug 14 2012

A095778 Values of n for which A095777(n) is 9 (those terms which are expressible in decimal digits for bases 2 through 10, but not for base 11).

Original entry on oeis.org

10, 21, 32, 43, 54, 65, 76, 87, 98, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 131, 142, 153, 164, 175, 186, 197, 208, 219, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 252, 263, 274, 285, 296, 307, 318, 329, 340, 351, 352, 353

Offset: 1

Chuck Seggelin (seqfan(AT)plastereddragon.com), Jun 05 2004

Numbers with at least one digit A (=10) in their representation in base 11. Complementary sequence to A171397. - François Marques, Oct 11 2020

			a(5)=54 because 54 when expressed in successive bases starting at 2 will produce its first non-decimal digit at base 11. Like so: 110110, 2000, 312, 204, 130, 105, 66, 60, 54. In base 11, 54 is 4A.

François Marques, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)

Cf. A095777.
Cf. Numbers with at least one digit b-1 in base b : A074940 (b=3), A337250 (b=4), A337572 (b=5), A333656 (b=6), A337141 (b=7), A337239 (b=8), A338090 (b=9), A011539 (b=10), this sequence (b=11).
Cf. Numbers with no digit b-1 in base b: A005836 (b=3), A023717 (b=4), A020654 (b=5), A037465 (b=6), A020657 (b=7), A037474 (b=8), A037477 (b=9), A007095 (b=10), A171397 (b=11).

S := []; for n from 1 to 1000 do; if 1>0 then; ct := 0; ok := true; b := 2; if (n>9) then; while ok=true do; L := convert(n, base, b); for e in L while ok=true do; if (e > 9) then ok:=false; fi; od; if ok=true then; ct := ct + 1; b := b + 1; fi; od; fi; if ct=9 then S := [op(S), n]; fi; fi; od; S;
# or
seq(`if`(numboccur(10, convert(n, base, 11))>0, n, NULL), n=0..1000); # François Marques, Oct 11 2020

Select[Range[400],Max[IntegerDigits[#,11]]>9&] (* Harvey P. Dale, Sep 30 2018 *)

isok(m) = #select(x->(x==10), digits(m, 11)) >= 1; \\ François Marques, Oct 11 2020

from gmpy2 import digits
def A095778(n):
    def f(x):
        l = (s:=digits(x,11)).find('a')
        if l >= 0: s = s[:l]+'9'*(len(s)-l)
        return n+int(s)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Dec 04 2024

A337250 Numbers having at least one 3 in their representation in base 4.

Original entry on oeis.org

3, 7, 11, 12, 13, 14, 15, 19, 23, 27, 28, 29, 30, 31, 35, 39, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 71, 75, 76, 77, 78, 79, 83, 87, 91, 92, 93, 94, 95, 99, 103, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119

Offset: 1

François Marques, Sep 19 2020

Complementary sequence of A023717.

			18 is not in the sequence since it is 102_4 in base 4, but 19 is in the sequence since it is 103_4 in base 4.

François Marques, Table of n, a(n) for n = 1..10000

Cf. A196032 (at least one 0 in base 4).
Cf. Numbers with at least one digit b-1 in base b : A074940 (b=3), this sequence, A337572 (b=5), A333656 (b=6), A337141 (b=7), A337239 (b=8), A338090 (b=9), A011539 (b=10), A095778 (b=11).
Cf. Numbers with no digit b-1 in base b: A005836 (b=3), A023717 (b=4), A020654 (b=5), A037465 (b=6), A020657 (b=7), A037474 (b=8), A037477 (b=9), A007095 (b=10), A171397 (b=11).

seq(`if`(numboccur(3, convert(n, base, 4))>0, n, NULL), n=0..100);

Select[ Range[ 0, 100 ], (Count[ IntegerDigits[ #, 4 ], 3 ]>0)& ]

isok(m) = #select(x->(x==3), digits(m, 4)) >= 1; \\ Michel Marcus, Sep 20 2020

from gmpy2 import digits
def A337250(n):
    def f(x):
        l = (s:=digits(x,4)).find('3')
        if l >= 0: s = s[:l]+'2'*(len(s)-l)
        return n+int(s,3)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Dec 04 2024

cp's OEIS Frontend

A375523 a(n) = numerator of Sum_{i=1..n} 1/A171397(i).

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A375524 a(n) = denominator of Sum_{i=1..n} 1/A171397(i).

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A375805 Decimal expansion of Sum_{n >= 1} 1/A171397(n).

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A011539 "9ish numbers": decimal representation contains at least one nine.

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A037465 a(n) = Sum_{i=0..m} d(i)*6^i, where Sum_{i=0..m} d(i)*5^i is the base 5 representation of n.

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A338090 Numbers having at least one 8 in their representation in base 9.

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A037474 a(n) = Sum{d(i)*8^i: i=0,1,...,m}, where Sum{d(i)*7^i: i=0,1,...,m} is the base 7 representation of n.

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A037465 a(n) = Sum_{i=0..m} d(i)6^i, where Sum_{i=0..m} d(i)5^i is the base 5 representation of n.

A037474 a(n) = Sum{d(i)8^i: i=0,1,...,m}, where Sum{d(i)7^i: i=0,1,...,m} is the base 7 representation of n.

A037477 a(n) = Sum{d(i)9^i: i=0,1,...,m}, where Sum{d(i)8^i: i=0,1,...,m} is the base 8 representation of n.