A002392 -id:A002392 - cp's OEIS frontend

A229124 Scan decimal expansion of log(10) until all n-digit strings have been seen; a(n) is number of digits that must be scanned.

Original entry on oeis.org

22, 701, 7486, 88092, 1189434, 13426407

Offset: 1

Eric W. Weisstein, Sep 14 2013

Eric Weisstein's World of Mathematics, Constant Digit Scanning
Eric Weisstein's World of Mathematics, Natural Logarithm of 10 Digits

Cf. A002392 (decimal expansion of log(10)).

Cf. A229126 (last n-digit string seen when scanning the decimal digits of log(10)).

A229126 Last n-digit number seen when scanning the decimal digits of log(10).

Original entry on oeis.org

7, 38, 351, 8493, 33058, 362945

Offset: 1

Eric W. Weisstein, Sep 14 2013

Eric Weisstein's World of Mathematics, Constant Digit Scanning
Eric Weisstein's World of Mathematics, Natural Logarithm of 10 Digits

Cf. A002392 (decimal expansion of log(10)).

Cf. A229124 (number of digits that must be scanned in the decimal expansion of log(10) to see all n-digit strings).

A229197 Beginning position of n in the decimal expansion of log(10).

Original entry on oeis.org

3, 21, 1, 2, 13, 5, 17, 22, 6, 9, 41, 40, 322, 368, 25, 266, 154, 21, 398, 103, 35, 236, 112, 1, 79, 4, 70, 60, 48, 10, 2, 190, 59, 57, 101, 77, 32, 178, 700, 197, 13, 102, 34, 31, 253, 15, 28, 251, 44, 277, 7

Offset: 0

Eric W. Weisstein, Sep 15 2013

Eric W. Weisstein, Table of n, a(n) for n = 0..999
Eric Weisstein's World of Mathematics, Constant Digit Scanning
Eric Weisstein's World of Mathematics, Natural Logarithm of 10 Digits

Cf. A002392 (decimal expansion of log(10)).

Module[{d=RealDigits[Log[10],10,1000][[1]]}, Table[SequencePosition[ d, IntegerDigits[ n],1][[All,1]],{n,0,50}]]//Flatten (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 28 2019 *)

A334388 Decimal expansion of Sum_{k>=1} A007953(k) / (k*(k+1)) where A007953(k) is the sum of digits of the integer k.

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Sep 08 2020

This series is convergent.
Jeffrey Shallit generalizes this result to any base b (see Amer. Math. Month. link): Sum_{k>=1} digsum(k)_b / (k*(k+1)) = (b/(b-1)) * log(b) where digsum(k)_b is the sum of the digits of k when expressed in base b.
Sum_{n <= x} s(floor(x/n)) = kx + O(x^(2/3 + o(1))) where s(n) is the digital sum A007953 and k is this constant. See Bordellès, Dai, Heyman, Pan, & Shparlinski, Example 3.4. - Charles R Greathouse IV, Mar 22 2022

			2.5584278811044952044644349496492935640012238762541921955925865566

Jean-Paul Allouche, Somme de séries de nombres réels, Image des Mathématiques, CNRS, 2010 (in French).
Olivier Bordellès, Lixia Dai, Randell Heyman, Hao Pan, and Igor E. Shparlinski, On a sum involving the Euler function, arXiv:1808.00188 [math.NT]
J. O. Shallit, Solutions of Advanced Problems, 6450, The American Mathematical Monthly, Vol. 92, No. 7, Aug.-Sep., 1985, pp. 513-514; DOI: 10.2307/2322523.
Index entries for transcendental numbers

Cf. A002392 (log(10)), A007953 (digsum), A016627 (for base 2).

Cf. A308314.

Maple
```
evalf(10*log(10)/9,90);
```

RealDigits[10*Log[10]/9, 10, 100][[1]] (* Amiram Eldar, Sep 08 2020 *)

10*log(10)/9 \\ Charles R Greathouse IV, Mar 22 2022

Equals 1/(1*2) + 2/(2*3) + 3/(3*4) + 4/(4*5) + ... + 1/(10*11) + 2/(11*12) + ...

Equals (10/9) * log(10).

a(90) corrected by Georg Fischer, Jul 12 2021

A372597 Decimal expansion of (10/99)*log(10).

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, May 06 2024

			0.2325843528276813822240395408772085058182930796594720...

Paolo Xausa, Table of n, a(n) for n = 0..10000
J. M. Borwein and P. B. Borwein, Strange Series and High Precision Fraud, The American Mathematical Monthly, Vol. 99, No. 7 (1992), pp. 622-640.
Index entries for transcendental numbers

Cf. A002392.

First[RealDigits[10*Log[10]/99, 10, 100]]

log(10)*10/99 \\ Charles R Greathouse IV, Nov 21 2024

Equals Sum_{k >= 1} e(k)/(k*(k + 1)), where e(k) is the reflection of k through the decimal point (e.g., e(120) = 0.021). See Sum 6 in Borwein and Borwein (1992), p. 623.

A111513 Decimal expansion of (abs(log_10(sine of 1 degree))).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Nov 16 2005, Dec 11 2008

Log denotes the common (base 10) logarithm in the definition, so the constant is A111511/A002392. [From R. J. Mathar, Jan 26 2009]

Cf. A019810.

RealDigits[Abs[Log[10,Sin[1 Degree]]],10,120][[1]] (* Harvey P. Dale, Jul 03 2024 *)

A111514 Decimal expansion of (abs(log_10(sine of 1 degree)))^(-1).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Nov 16 2005

Cf. A019810.

evalf(1/log[10](sin(Pi/180))); # R. J. Mathar, Apr 23 2009

RealDigits[1/Abs[Log10[Sin[1 Degree]]],10,120][[1]] (* Harvey P. Dale, Jul 30 2023 *)

Equals A002392/A111511. - R. J. Mathar, Apr 23 2009

A220260 Decimal expansion of e / log_10(e), where e = A001113.

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Dec 08 2012

Minimum value of the function x / log_10(x) for x > 1, obtained at x = e.

			6.2590752167663952110184773116777272805464599542594694...

Cf. A001113, A002285, A002392.

RealDigits[E/Log[10,E],10,120][[1]] (* Harvey P. Dale, Dec 23 2012 *)

Equals the ratio A001113 / A002285 and also the product A001113*A002392.

A220261 Decimal expansion of (1/e) * log_10(e), where 1/e = A068985, log_10(e) = A002285.

Original entry on oeis.org

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

Offset: 0

Jaroslav Krizek, Dec 08 2012

Maximal value of function x * log_10(x) for x < 0 in x = -1/e.
-0,159768… = minimal value of function x * log_10(x) for x > 0 in x = 1/e.
Decimal expansion of 1 / (e * log(10)), where e = A001113, log(10) = A002392.

			0.15976801130640935267214...

Cf. A001113, A068985, A002285.

RealDigits[1/E Log10[E],10,120][[1]] (* Harvey P. Dale, Aug 11 2019 *)

A228243 First occurrence of n consecutive n's in the decimal expansion of log(10).

Original entry on oeis.org

20, 111, 56, 9041, 4767, 674596, 24611354, 64653957, 131278082

Offset: 1

Eric W. Weisstein, Aug 17 2013

Earls sequence for log(10).

			log(10) = 2.302585092994045684017...4228.., so
a(1) = 20 (the digit string 1 first occurs at position 20 after the decimal point)
a(2) = 111 (the digit string 22 first occurs starting at position 111 after the decimal point)

Eric Weisstein's World of Mathematics, Natural Logarithm of 10 Digits
Eric Weisstein's World of Mathematics, Earls Sequence

Cf. A002392 (decimal expansion of log(10)).

cp's OEIS Frontend

A229124 Scan decimal expansion of log(10) until all n-digit strings have been seen; a(n) is number of digits that must be scanned.

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A229126 Last n-digit number seen when scanning the decimal digits of log(10).

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A229197 Beginning position of n in the decimal expansion of log(10).

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A334388 Decimal expansion of Sum_{k>=1} A007953(k) / (k*(k+1)) where A007953(k) is the sum of digits of the integer k.

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A372597 Decimal expansion of (10/99)*log(10).

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A111513 Decimal expansion of (abs(log_10(sine of 1 degree))).

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A111514 Decimal expansion of (abs(log_10(sine of 1 degree)))^(-1).

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A220260 Decimal expansion of e / log_10(e), where e = A001113.

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A220261 Decimal expansion of (1/e) * log_10(e), where 1/e = A068985, log_10(e) = A002285.

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