A016630 -id:A016630 - cp's OEIS frontend

A154158 Decimal expansion of log_7(10).

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

Offset: 1

N. J. A. Sloane, Oct 30 2009

			1.1832946624549383268179285616468591481654445229423947233563...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers

Cf. decimal expansion of log_7(m): A152713 (m=2), A152945 (m=3), A153103 (m=4), A153203 (m=5), A153463 (m=6), A153755 (m=8), A113211 (m=9), this sequence, A154179 (m=11), A154200 (m=12), A154294 (m=13), A154467 (m=14), A154572 (m=15), A154793 (m=16), A154857 (m=17), A154912 (m=18), A155059 (m=19), A155496 (m=20), A155591 (m=21), A155735 (m=22), A155824 (m=23), A155964 (m=24).

SetDefaultRealField(RealField(100)); Log(10)/Log(7); // G. C. Greubel, Sep 02 2018

RealDigits[Log[7, 10], 10,100][[1]] (* Vincenzo Librandi, Aug 31 2013 *)

default(realprecision, 100); log(10)/log(7) \\ G. C. Greubel, Sep 02 2018

Equals A002392 / A016630 = 1/A153620. - R. J. Mathar, Jul 31 2025

A020860 Decimal expansion of log(7)/log(2).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			2.807354922...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
L. Adleman, Molecular Computation of Solutions to Combinatorial Problems, Science 266 (1994): 1021-1024.
Aran Nayebi, Fast matrix multiplication techniques based on the Adleman-Lipton model, arXiv:0912.0750 [q-bio.QM], 2009-2011.
D. K. Nguyen, I. Lavallée, and M. Bui, A New Direction to Parallelize Winograd's Algorithm on Distributed Memory Computers, Modeling, Simulation and Optimization of Complex Processes Proceedings of the Third International Conference on High Performance Scientific Computing, March 6-10, 2006, Hanoi, Vietnam: 445-457.
V. Pan, How can we speed up matrix multiplication?, SIAM Review, 26 (1984): 393-416.
S. Robinson, Toward an Optimal Algorithm for Matrix Multiplication, SIAM News 38 (2005): 1-3.
V. Strassen, Gaussian elimination is not optimal, Numer. Math. 13 (1969): 354-356. MR 40:2223.
Yahoo Questions, Prove log(base 2)7 is rational?
Index entries for transcendental numbers

Cf. A002162, A016630.

RealDigits[Log[2, 7], 10, 100][[1]] (* Vincenzo Librandi, Aug 29 2013 *)

log(7)/log(2) \\ Charles R Greathouse IV, May 15 2019

A279061 Number of divisors of n of the form 7*k + 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 2, 3, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2

Offset: 0

Ilya Gutkovskiy, Dec 05 2016

Möebius transform is a period-7 sequence {1, 0, 0, 0, 0, 0, 0, ...}.

			a(8) = 2 because 8 has 4 divisors {1,2,4,8} among which 2 divisors {1,8} are of the form 7*k + 1.

Robert Israel, Table of n, a(n) for n = 0..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A001227, A001817, A001826, A001876, A188169.

Cf. A001620, A016630, A354627 (psi(1/7)).

N:= 200: # to get a(0)..a(N)
V:= Vector(N):
for k from 1 to N do
  R:= [seq(i,i=k..N,7*k)];
  V[R]:= map(`+`,V[R],1);
od:
0,seq(V[i],i=1..N); # Robert Israel, Dec 05 2016

nmax = 120; CoefficientList[Series[Sum[x^k/(1 - x^(7 k)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 120; CoefficientList[Series[Sum[x^(7 k + 1)/(1 - x^(7 k + 1)), {k, 0, nmax}], {x, 0, nmax}], x]
Table[Count[Divisors[n],?(IntegerQ[(#-1)/7]&)],{n,0,100}] (* _Harvey P. Dale, Nov 08 2022 *)

concat([0], Vec(sum(k=1, 100, x^k / (1 - x^(7*k))) + O(x^101))) \\ Indranil Ghosh, Mar 29 2017

G.f.: Sum_{k>=1} x^k/(1 - x^(7*k)).
G.f.: Sum_{k>=0} x^(7*k+1)/(1 - x^(7*k+1)).
Sum_{k=1..n} a(k) = n*log(n)/7 + c*n + O(n^(1/3)*log(n)), where c = gamma(1,7) - (1 - gamma)/7 = 0.713612..., gamma(1,7) = -(psi(1/7) + log(7))/7 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A363795 Number of divisors of n of the form 7*k + 2.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 2, 0, 2, 0, 1, 0, 2, 1, 1, 0, 1, 0, 1, 0, 2, 1, 2, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 1, 1, 2, 1, 1, 0, 1, 1, 1, 0, 3, 0, 2, 0, 1, 0, 1, 1, 2, 1, 1, 0, 1, 0, 2, 0, 2, 0, 3, 0, 2, 1, 1, 0, 2, 0, 1, 1

Offset: 1

Seiichi Manyama, Jun 23 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A279061, A363805, A363806, A363807, A363808.
Cf. A001822, A001877.
Cf. A284443.
Cf. A001620, A016630, A354628 (psi(2/7)).

a[n_] := DivisorSum[n, 1 &, Mod[#, 7] == 2 &]; Array[a, 100] (* Amiram Eldar, Jun 23 2023 *)

PARI
```
a(n) = sumdiv(n, d, d%7==2);
```

G.f.: Sum_{k>0} x^(2*k)/(1 - x^(7*k)).
G.f.: Sum_{k>0} x^(7*k-5)/(1 - x^(7*k-5)).
Sum_{k=1..n} a(k) = n*log(n)/7 + c*n + O(n^(1/3)*log(n)), where c = gamma(2,7) - (1 - gamma)/7 = 0.188117..., gamma(2,7) = -(psi(2/7) + log(7))/7 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A363805 Number of divisors of n of the form 7*k + 3.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 2, 0, 0, 1, 0, 0, 2, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 2, 0, 0, 2, 0, 1, 2, 1, 0, 1, 0, 0, 1, 0, 1, 2, 0, 1, 1, 0, 0, 2, 0, 1, 1, 1, 0, 2, 1, 0, 1, 1, 0, 1, 0, 2, 1, 0, 0, 1, 1, 0, 2, 0, 0, 3, 0, 0, 2, 1, 0, 2, 0, 0, 1

Offset: 1

Seiichi Manyama, Jun 23 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A279061, A363795, A363806, A363807, A363808.
Cf. A001842, A001878.
Cf. A284444.
Cf. A001620, A016630, A354629 (psi(3/7)).

a[n_] := DivisorSum[n, 1 &, Mod[#, 7] == 3 &]; Array[a, 100] (* Amiram Eldar, Jun 23 2023 *)

PARI
```
a(n) = sumdiv(n, d, d%7==3);
```

G.f.: Sum_{k>0} x^(3*k)/(1 - x^(7*k)).
G.f.: Sum_{k>0} x^(7*k-4)/(1 - x^(7*k-4)).
Sum_{k=1..n} a(k) = n*log(n)/7 + c*n + O(n^(1/3)*log(n)), where c = gamma(3,7) - (1 - gamma)/7 = -0.0004108181..., gamma(3,7) = -(psi(3/7) + log(7))/7 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A363806 Number of divisors of n of the form 7*k + 4.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 1, 1, 1, 0, 0, 0, 2, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 2, 0, 0, 1, 2, 0, 0, 1, 2, 0

Offset: 1

Seiichi Manyama, Jun 23 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A279061, A363795, A363805, A363807, A363808.
Cf. A284445.
Cf. A001620, A016630, A354630 (psi(4/7)).

a[n_] := DivisorSum[n, 1 &, Mod[#, 7] == 4 &]; Array[a, 100] (* Amiram Eldar, Jun 23 2023 *)

PARI
```
a(n) = sumdiv(n, d, d%7==4);
```

G.f.: Sum_{k>0} x^(4*k)/(1 - x^(7*k)).
G.f.: Sum_{k>0} x^(7*k-3)/(1 - x^(7*k-3)).
Sum_{k=1..n} a(k) = n*log(n)/7 + c*n + O(n^(1/3)*log(n)), where c = gamma(4,7) - (1 - gamma)/7 = -0.102846..., gamma(4,7) = -(psi(4/7) + log(7))/7 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A363807 Number of divisors of n of the form 7*k + 5.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 2, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 2, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 2, 2, 0, 0, 1, 1

Offset: 1

Seiichi Manyama, Jun 23 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A279061, A363795, A363805, A363806, A363808.
Cf. A284446, A319995.
Cf. A001620, A016630, A354631 (psi(5/7)).

a[n_] := DivisorSum[n, 1 &, Mod[#, 7] == 5 &]; Array[a, 100] (* Amiram Eldar, Jun 23 2023 *)

PARI
```
a(n) = sumdiv(n, d, d%7==5);
```

G.f.: Sum_{k>0} x^(5*k)/(1 - x^(7*k)).
G.f.: Sum_{k>0} x^(7*k-2)/(1 - x^(7*k-2)).
Sum_{k=1..n} a(k) = n*log(n)/7 + c*n + O(n^(1/3)*log(n)), where c = gamma(5,7) - (1 - gamma)/7 = -0.169787..., gamma(5,7) = -(psi(5/7) + log(7))/7 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A363808 Number of divisors of n of the form 7*k + 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0

Offset: 1

Seiichi Manyama, Jun 23 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A279061, A363795, A363805, A363806, A363807.
Cf. A284105.
Cf. A001620, A016630, A354632 (psi(6/7)).

a[n_] := DivisorSum[n, 1 &, Mod[#, 7] == 6 &]; Array[a, 100] (* Amiram Eldar, Jun 23 2023 *)

PARI
```
a(n) = sumdiv(n, d, d%7==6);
```

G.f.: Sum_{k>0} x^(6*k)/(1 - x^(7*k)).
G.f.: Sum_{k>0} x^(7*k-1)/(1 - x^(7*k-1)).
Sum_{k=1..n} a(k) = n*log(n)/7 + c*n + O(n^(1/3)*log(n)), where c = gamma(6,7) - (1 - gamma)/7 = -0.218328..., gamma(6,7) = -(psi(6/7) + log(7))/7 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A016644 Decimal expansion of log(21).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			3.044522437723422996500597980365705434284575287404610640194084483575074....

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Index entries for transcendental numbers

Cf. A016449 (continued fraction).

RealDigits[Log[21], 10, 120][[1]] (* Harvey P. Dale, Sep 04 2012 *)

default(realprecision, 20080); x=log(21); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016644.txt", n, " ", d)); \\ Harry J. Smith, May 17 2009, corrected May 20 2009

Eqals A016630 + A002391. - R. J. Mathar, Jul 22 2025

A068460 Factorial expansion of log(7) = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 0, 3, 0, 8, 8, 2, 11, 1, 5, 11, 1, 7, 1, 11, 16, 12, 12, 13, 5, 4, 26, 19, 12, 20, 0, 18, 14, 22, 6, 29, 0, 27, 16, 23, 23, 23, 34, 27, 4, 27, 18, 0, 10, 27, 42, 24, 9, 16, 6, 52, 2, 38, 44, 30, 51, 61, 7, 19, 0, 45, 18, 51, 43, 54, 7, 15, 69, 44, 51, 9, 74, 5, 69

Offset: 1

Benoit Cloitre, Mar 10 2002

nonn

			log(7) = 1 + 1/2! + 2/3! + 2/4! + 3/5! + 3/6! + 0/7! + 3/8! + 0/9! + ...

Cf. A016630 (decimal expansion), A016735 (continued fraction).

Cf. A067882 (log(2)), A322334 (log(3)), A322333 (log(5)), A068461 (log(11)).

SetDefaultRealField(RealField(250));  [Floor(Log(7))] cat [Floor(Factorial(n)*Log(7)) - n*Floor(Factorial((n-1))*Log(7)) : n in [2..80]]; // G. C. Greubel, Dec 05 2018

Digits:=200: a:=n->`if`(n=1,floor(log(7)),floor(factorial(n)*log(7))-n*floor(factorial(n-1)*log(7))); seq[120](a(n),n=1..80); # Muniru A Asiru, Dec 06 2018

With[{b = Log[7]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]] (* G. C. Greubel, Dec 05 2018 *)

vector(30,n,if(n>1,t=t%1*n,t=log(7))\1) \\ Increase realprecision (e.g., \p500) to compute more terms. - M. F. Hasler, Nov 25 2018

default(realprecision, 250); b = log(7); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", ")) \\ G. C. Greubel, Dec 05 2018

def a(n):
    if (n==1): return floor(log(7))
    else: return expand(floor(factorial(n)*log(7)) - n*floor(factorial(n-1)*log(7)))
[a(n) for n in (1..80)] # G. C. Greubel, Dec 05 2018

Name edited and keywords cons,easy removed by M. F. Hasler, Nov 25 2018

cp's OEIS Frontend

A154158 Decimal expansion of log_7(10).

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Magma

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A020860 Decimal expansion of log(7)/log(2).

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A279061 Number of divisors of n of the form 7*k + 1.

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A363795 Number of divisors of n of the form 7*k + 2.

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A363805 Number of divisors of n of the form 7*k + 3.

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A363806 Number of divisors of n of the form 7*k + 4.

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A363807 Number of divisors of n of the form 7*k + 5.

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A363808 Number of divisors of n of the form 7*k + 6.

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