A354643 -id:A354643

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A083917

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

+10
12

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,77

LINKS

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.

FORMULA

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

MATHEMATICA

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 *)

PROG

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

CROSSREFS

Cf. A000005, A001227, A010879.

Cf. A001620, A002392, A354643 (psi(7/10)).

Cf. A083910, A083911, A083912, A083913, A083914, A083915, A083916, A083918, A083919.

KEYWORD

nonn,easy

AUTHOR

Reinhard Zumkeller, May 08 2003

STATUS

approved

A262246

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

+10
3

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

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.

FORMULA

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)

EXAMPLE

0.4069016342...

MATHEMATICA

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 *)

PROG

(PARI)

default(realprecision, 87);

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

CROSSREFS

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

KEYWORD

nonn,cons

AUTHOR

Gheorghe Coserea, Oct 06 2015

STATUS

approved

A354642

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

+10
3

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..92.

Index entries for sequences related to the digamma function.

EXAMPLE

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

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A306716, A354643, A354644.

KEYWORD

nonn,cons

AUTHOR

R. J. Mathar, Jun 01 2022

STATUS

approved

A354644

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

+10
3

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Table of n, a(n) for n=0..92.

Index entries for sequences related to the digamma function.

EXAMPLE

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

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A306716, A354642, A354643.

KEYWORD

nonn,cons

AUTHOR

R. J. Mathar, Jun 01 2022

STATUS

approved

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