A354643 -id:A354643 - cp's OEIS frontend

A083917 Number of divisors of n that are congruent to 7 modulo 10.

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

Offset: 1

Reinhard Zumkeller, May 08 2003

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. A000005, A001227, A010879.
Cf. A001620, A002392, A354643 (psi(7/10)).
Cf. A083910, A083911, A083912, A083913, A083914, A083915, A083916, A083918, A083919.

Table[Count[Divisors[n],?(Mod[#,10]==7&)],{n,110}] (* _Harvey P. Dale, Sep 15 2023 *)
a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 7 &]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)

a(n) = sumdiv(n, d, d % 10 == 7); \\ Amiram Eldar, Dec 30 2023

a(n) = A000005(n) - A083910(n) - A083911(n) - A083912(n) - A083913(n) - A083914(n) - A083915(n) - A083916(n) - A083918(n) - A083919(n).
G.f.: Sum_{k>=1} x^(7*k)/(1 - x^(10*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/10 + c*n + O(n^(1/3)*log(n)), where c = gamma(7,10) - (1 - gamma)/10 = -0.150534..., gamma(7,10) = -(psi(7/10) + log(10))/10 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Dec 30 2023

A262246 Decimal expansion of Sum_{k>=0} (-1)^k/(5k+2).

Original entry on oeis.org

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

Offset: 0

Gheorghe Coserea, Oct 06 2015

			0.4069016342...

Gheorghe Coserea, Table of n, a(n) for n = 0..2015
H. Wilf, Accelerated series for universal constants, by the WZ method, Discrete Mathematics and Theoretical Computer Science 3(4) (1999), 189-192.

Cf. A003881, A024396, A113476, A181048, A181049, A181122, A193534.

N[(1/5)*((Sqrt[5]-1)*Log[2] + Sqrt[5]*Log[Sin[3*Pi/10]] + (Pi/2)*Sec[Pi/10]), 100] (* G. C. Greubel, Oct 07 2015 *) (* fixed by Vaclav Kotesovec, Dec 11 2017 *)

default(realprecision, 87);
eval(vecextract(Vec(Str(sumalt(n=0, (-1)^(n)/(5*n+2)))), "3..-2"))

Sum_{n>=0} (-1)^n/(5n+2) = Integral_{x=0..1} x/(1+x^5)dx.
From G. C. Greubel, Oct 07 2015: (Start)
Sum_{n>=0} (-1)^n/(5n+2) = (1/5)*(sqrt(5)*log(phi) - log(2) + Pi*(5*phi^2)^(-1/4)), where 2*phi=1+sqrt(5).
Sum_{n>=0} (-1)^n/(5n+2) = (1/5)*(sqrt(5)*log(2*sin(3*Pi/10)) - log(2) + (Pi/2)*sec(Pi/10)).
(End)
Sum_{n>=0} (-1)^n/(5n+2) = (Psi(1/5) - Psi(7/10))/10 , see A200135 and A354643. - Robert Israel, Oct 08 2015
From Peter Bala, Feb 19 2024: (Start)
Equals (1/2)*Sum_{n >= 0} n!*(5/2)^n/(Product_{k = 0..n} 5*k + 2) = (1/2)*Sum_{n >= 0} n!*(5/2)^n/A047055(n+1) (apply Euler's series transformation to Sum_{k >= 0} (-1)^k/(5*k + 2)).
Continued fraction: 1/(2 + 2^2/(5 + 7^2/(5 + 12^2/(5 + ... + (5*n + 2)^2/(5 + ... ))))).
The slowly converging series representation Sum_{n >= 0} (-1)^n/(5*n + 2) for the constant can be accelerated to give the following faster converging series
1/4 + (5/2)*Sum_{n >= 0} (-1)^n/((5*n + 2)*(5*n + 7)) and
19/56 + (5^2/2)*Sum_{n >= 0} (-1)^n/((5*n + 2)*(5*n + 7)*(5*n + 12)).
These two series are the cases r = 1 and r = 2 of the general result:
for r >= 0, the constant equals C(r) + ((5/2)^r)*r!*Sum_{n >= 0} (-1)^n/((5*n + 2)*(5*n + 7)*...*(5*n + 5*r + 2)), where C(r) is the rational number (1/2)*Sum_{k = 0..r-1} (5/2)^k*k!/(2*7*12*...*(5*k + 2)). The general result can be proved by the WZ method as described in Wilf. (End)
From Peter Bala, Mar 03 2024: (Start)
Equals (1/2)*hypergeom([2/5, 1], [7/5], -1).
Gauss's continued fraction: 1/(2 + 2^2/(7 + 5^2/(12 + 7^2/(17 + 10^2/(22 + 12^2/(27 + 15^2/(32 + 17^2/(37 + 20^2/(42 + 22^2/(47 + ... )))))))))). (End)

A354642 Decimal expansion of the negated digamma function at 3/10.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jun 01 2022

			psi(3/10) = -3.50252422220013...

Index entries for sequences related to the digamma function.

Cf. A306716, A354643, A354644.

RealDigits[PolyGamma[3/10], 10, 120][[1]] (* Amiram Eldar, Jun 14 2023 *)

PARI
```
psi(3/10) \\ Michel Marcus, Jun 02 2022
```

A354644 Decimal expansion of the negated digamma function at 9/10.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jun 01 2022

			psi(9/10) = -0.75492694994705...

Index entries for sequences related to the digamma function.

Cf. A306716, A354642, A354643.

RealDigits[PolyGamma[9/10], 10, 120][[1]] (* Amiram Eldar, Jun 14 2023 *)

PARI
```
psi(9/10) \\ Michel Marcus, Jun 02 2022
```

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A083917 Number of divisors of n that are congruent to 7 modulo 10.

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A262246 Decimal expansion of Sum_{k>=0} (-1)^k/(5k+2).

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A354642 Decimal expansion of the negated digamma function at 3/10.

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A354644 Decimal expansion of the negated digamma function at 9/10.

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