Decimal expansion of Integral_{x=0..1} arcsin(x)^2/x dx.

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

0,1

George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 122.

Chenli Li, Wenchang Chu, <a href="http://dx.doi.org/10.3390/math10162980">Improper integrals involving powers of Inverse Trigonometric and hyperbolic Functions</a>, Mathematics 10 (16) (2022) 2980

D. Cvijovic, H. M. Srivastava, <a href="https://dx.doi.org/10.1016/j.jmaa.2008.10.017">Evaluations of some classes of the trigonometric moment intgrals</a>, J. Math. Anal. Applic. 351 (2009) 244-256, example set 1.

J. C. Tanner, <a href="http://www.jstor.org/stable/3213813">The Proportion of Quadrilaterals Formed by Random Lines in a Plane</a>, Journal of Applied Probability, Vol. 20, No. 2 (Jun., 1983), pp. 400-404.

J. C. Tanner, <a href="http://www.jstor.org/stable/3213589">Polygons Formed by Random Lines in a Plane: Some Further Results</a>, Journal of Applied Probability, Vol. 20, No. 4 (Dec., 1983), p. 778

Equals 1/8*(Pi^2*log(4) - 7*zeta(3)).

Also equals Sum_{n>=1} 4^(n-1)/(n^3*binomial(2*n, n)).

Also equals 1/2*hypergeometric4F3([1, 1, 1, 1], [3/2, 2, 2], 1).

Also equals Integral_{x=0..Pi/2} x^2*cot(x) dx. - Michel Marcus, Aug 29 2015

0.65847232569963413648709889716600527590558175624904185726279535120517...

1/8*(Pi^2*Log[4] - 7*Zeta[3]) // RealDigits[#, 10, 100]& // First

(PARI) (Pi^2*log(4) - 7*zeta(3))/8 \\ Michel Marcus, Sep 04 2015

Cf. A002117 (zeta(3)).

nonn,cons,easy,changed

Jean-François Alcover, Apr 29 2013

approved

Triangle read by rows: Eulerian numbers T(n,k) = A008292(n,k) reduced mod n+1.

0, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 3, 2, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 5, 3, 3, 5, 7, 1, 1, 7, 1, 7, 4, 7, 1, 7, 1, 1, 3, 0, 2, 4, 4, 2, 0, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 11, 9, 6, 6, 6

0,8

The row sums are:

{0, 2, 3, 8, 5, 12, 7, 32, 36, 20, 11, 72, 13,...}

Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EulerianNumber.html">Eulerian Number</a>

T(n,m) = Eulerian(n+1,m) mod (n+1).

Triangle begins:

{0},

{1,1},

{1,1,1},

{1,3,3,1},

{1,1,1,1,1},

{1,3,2,2,3,1},

{1,1,1,1,1,1,1},

{1,7,5,3,3,5,7,1},

{1,7,1,7,4,7,1,7,1},

{1,3,0,2,4,4,2,0,3,1},

{1,1,1,1,1,1,1,1,1,1,1},

{1,3,11,9,6,6,6,6,9,11,3,1},

{1,1,1,1,1,1,1,1,1,1,1,1,1},

...

Flatten[Table[Table[Mod[Eulerian[n+1, m], n+1], {m, 0, n}], {n, 0, 12}]]

Cf. A008292, A225043.

nonn,tabl,changed

Roger L. Bagula, Apr 26 2013

approved

Decimal expansion of Pi^3/8.

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

1,1

<a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

Equals Integral_{x>0} log(x)^2/(1+x^2) dx.

Equals Integral_{x=0..Pi/2} log(tan(x))^2 dx.

Equals Integral_{x=0..Pi/2} log(sin(x)^3)*log(sin(x))-(3*Pi/2)*log(2)^2 dx.

Equals (27/7) * Sum_{k>=0} binomial(2*k, k)/((2*k+1)^3*16^k);

Equals (27/7) * 4F3([1/2, 1/2, 1/2, 1/2], [3/2, 3/2, 3/2], 1/4), where pFq() is the generalized hypergeometric function.

From Amiram Eldar, Aug 21 2020: (Start)

Equals Integral_{x=0..oo} x^2/cosh(x) dx.

Equals 2 + Integral_{x=0..oo} x^2 * exp(-x) * tanh(x) dx. (End)

From Gleb Koloskov, Jun 15 2021: (Start)

Equals 2*Integral_{x=0..1} log(x)^2/(1+x^2) dx.

Equals 2*Integral_{x=1..oo} log(x)^2/(1+x^2) dx.

Equals 2*(-1)^n*Integral_{x=-1/e..0} W(n,x)*(1-W(n,x))*log(-W(n,x))^2/x/(1-W(n,x)^4) dx, where W=LambertW, for n=0 and n=-1. (End)

3.875784585037477521934539383387674400278161070735638461768067262975799364683...

RealDigits[Pi^3/8, 10, 100][[1]]

(PARI) Pi^3/8 \\ Charles R Greathouse IV, Oct 01 2022

Cf. A000796, A002388, A019669, A091476, A091925.

nonn,cons,easy,changed

Jean-François Alcover, Apr 24 2013

Offset corrected by Rick L. Shepherd, Jan 01 2014

approved

Product of remainders of prime(n) mod k, for k = 2,3,4,...,prime(n)-1

1, 1, 2, 6, 720, 2160, 2419200, 65318400, 754427520000, 32953394073600000, 311409573995520000, 37269497815783833600000, 7890485108998805913600000000, 1096106738916569123487744000000, 4067286739206415827555188736000000000, 7924734685010508814047938347008000000000000

1,3

Nonzero entries in A180491. Note that this sequence, while increasing in general, is not strictly increasing.

a(n) is divisible by (n-1)!. - Robert G. Wilson v, Sep 09 2010

a(n) = A173392(A000040(n)) = A180491(A000040(n)). - Ridouane Oudra, Nov 01 2024

Since prime(4) = 7, a(4) = (7 mod 2) * (7 mod 3) * (7 mod 4) * (7 mod 5) * (7 mod 6) = 1 * 1 * 3 * 2 * 1 = 6.

a:= n-> (p-> mul(irem(p, k), k=2..p-1))(ithprime(n)):

seq(a(n), n=1..17); # Alois P. Heinz, Jul 16 2022

f[n_]:=Times@@(Mod[n, # ]&/@ Range[2, n-1]); Table[f[Prime[i]], {i, 20}] (* Harvey P. Dale, Sep 18 2010 *)

f[n_] := Times @@ Mod[n, Range[2, n - 1]]; Table[ f@ Prime@ n, {n, 10}] (* Robert G. Wilson v, Sep 09 2010 *)

Cf. A034386, A000142, A004125, A180491, A180493.

Cf. A173392, A000040.

nonn,changed

Carl R. White, Sep 08 2010

approved

Product of remainders of n mod k, for k = 2,3,4,...,n-1.

1, 1, 1, 0, 2, 0, 6, 0, 0, 0, 720, 0, 2160, 0, 0, 0, 2419200, 0, 65318400, 0, 0, 0, 754427520000, 0, 0, 0, 0, 0, 32953394073600000, 0, 311409573995520000, 0, 0, 0, 0, 0, 37269497815783833600000, 0, 0, 0, 7890485108998805913600000000, 0

1,5

a(n) is zero where n is composite and is trivially less than or equal to n! when n is prime or 1.

a(n)=0 iff n is composite. See A180492. - Robert G. Wilson v, Sep 09 2010

a(n) = A080339(n)*A173392(n). - Ridouane Oudra, Nov 01 2024

a(7) = (7 mod 2) * (7 mod 3) * (7 mod 4) * (7 mod 5) * (7 mod 6) = 1 * 1 * 3 * 2 * 1 = 6.

a:=proc(n) if n=1 then 1; elif isprime(n)=true then mul(n mod i, i=2..n-1); else 0; fi: end: seq(a(n), n=1..60); # Ridouane Oudra, Nov 01 2024

f[n_] := Times @@ Mod[n, Range[2, n - 1]]; Array[f, 42] (* Robert G. Wilson v, Sep 09 2010 *)

Cf. A034386, A000142, A004125, A180492, A180493.

Cf. A080339, A173392.

nonn,changed

Carl R. White, Sep 08 2010

approved

Abundant numbers k such that k^2 + A033880(k)^2 is a perfect square.

336, 1080, 3078, 6048, 6552, 19845, 47616, 239760, 435708, 599400, 760320, 873180, 997920, 1468800, 1602300, 2004480, 4312440, 4612608, 4713984, 10181808, 10665984, 11554816, 12160512, 24149664, 31244850, 46431744, 56439504, 64995840, 116958492

1,1

These abundant numbers along with their abundances form the legs of an integral Pythagorean triangle.

Odd terms are very rare: 19845 is the only one up to 10^9.

336 is a term because its abundance is 320 and 320^2 + 336^2 = 464^2.

l={}; Do[a=DivisorSigma[1, n]-2*n; If[a>0&&IntegerQ@Sqrt[n^2+a^2], AppendTo[l, n]], {n, 12, 2*10^8}]; l

(PARI) for(n=12, 2*10^8, a=sigma(n)-2*n; a>0&&issquare(n^2+a^2)&&print1(n", "))

(Python)

import sympy as sp

for i in range(12, 200000000):

a=sp.ntheory.factor_.divisor_sigma(i) - 2*i

if a>0 and sp.ntheory.primetest.is_square(i*i+a*a):

print(i, end=", ")

Cf. A005101, A033880, A000290.

nonn,new

Waldemar Puszkarz, Oct 17 2024

approved

Numbers k such that 3*k+1 divides 3^k+1.

0, 1, 9, 3825, 6561, 102465, 188505, 190905, 1001385, 1556985, 3427137, 5153577, 5270625, 5347881, 13658225, 14178969, 20867625, 23828049, 27511185, 29400657, 48533625, 80817009, 83406609, 89556105, 108464265, 123395265, 127558881, 130747689, 133861905

1,3

This is to 3 as A224486 is to 2

Displayed terms complete up to 200*10^6. - _Joerg Arndt_, Apr 08 2013

Joerg Arndt, <a href="/A222948/b222948.txt">Table of n, a(n) for n = 1..64</a> (all terms <= 10^9)

{n such that (1+A000244(n))/A016777(n) is an integer}.

0 is a term because (3^0+1)/(3*0+1) = 2.

1 is a term because (3^1+1)/(3*1+1) = 1.

9 is a term because (3^9+1)/(3*9+1) = 703.

(PARI) for(n=0, 10^9, if((3^n+1)%(3*n+1)==0, print1(n, ", "))); /* Joerg Arndt, Apr 08 2013 */

/* the following program is significantly faster; it gives terms >=1: */

(PARI) for(n=0, 10^12, my(m=3*n+1); if( Mod(3, m)^n==Mod(-1, m), print1(n, ", ") ) ); /* Joerg Arndt, Apr 08 2013 */

Cf. A224486 (k such that 2*k+1 divides 2^k+1).

Cf. A000244, A016777.

nonn,changed

Jonathan Vos Post, Apr 07 2013

Terms > 9 from Joerg Arndt, Apr 08 2013

approved

Conjectured lower bounds for the Riemann hypothesis function floor(H(k) + exp(H(k))*log(H(k))) - sigma(k).

0, 1, 2, 6, 23, 33, 34, 78, 105, 207, 492, 1536, 1667, 3036, 5155, 5206, 7682, 8748, 9051, 15895, 21295, 22160, 36300, 58331, 58657, 71186, 81276, 91902, 126789, 142721, 143828, 240466, 291217, 306310, 471093, 743434, 872803, 963860, 1652806, 1742555

1,3

Here H(k) is the k-th harmonic number, Sum_{i=1..k} 1/i, and sigma(k) is the sum of the divisors of k. This sequence gives the conjectured values of A057641 that form a lower bound. For instance,

all numbers x <= 0 appear for n <= 12 = A176679(3);

all numbers x <= 1 appear for n <= 24 = A176679(4);

all numbers x <= 2 appear for n <= 60 = A176679(5);

all numbers x <= 6 appear for n <= 120 = A176679(6);

all numbers x <= 23 appear for n <= 180 = A176679(7).

The pattern continues.

Cf. A057641, A176679.

nonn,changed

T. D. Noe, Mar 28 2013

approved

Numbers m such that m^2 + (m+3)^2 is prime.

1, 2, 5, 7, 10, 11, 16, 20, 22, 25, 37, 40, 41, 46, 50, 55, 61, 62, 65, 77, 85, 91, 92, 101, 106, 107, 116, 122, 125, 127, 130, 131, 142, 145, 146, 152, 155, 161, 172, 181, 182, 187, 196, 197, 206, 220, 221, 232, 235, 241, 242, 257, 260, 262, 265, 271, 275, 280, 281, 286, 295, 310, 317, 325, 326, 346, 356, 362, 380, 382, 386, 391

1,2

Zak Seidov, <a href="/A224870/b224870.txt">Table of n, a(n) for n = 1..1000</a>

a(n) = (1/2)*(sqrt(2*A076727(n) - k^2) - k), k = 3.

A224870:=n->`if`(isprime(n^2 + (n+3)^2), n, NULL): seq(A224870(n), n=1..10^3); # Wesley Ivan Hurt, Feb 11 2017

k = 3; Select[Range[500], PrimeQ[#^2 + (# + k)^2]&]

(PARI) isok(n) = isprime(n^2 + (n+3)^2); \\ Michel Marcus, Feb 13 2017

Cf. A027861, A027862, A076727.

nonn,changed

Zak Seidov, Jul 22 2013

approved

A finite set of numbers relevant for the representation of numbers as primitive distinct sums of three squares (0 squared allowed).

1, 2, 3, 6, 9, 11, 18, 19, 22, 27, 33, 43, 51, 57, 67, 99, 102, 123, 163, 177, 187, 267, 627

1,2

This set of 23 numbers, possibly with one more number a >= 5*10^10, appears in a corollary of the Halter-Koch reference (Korollar 1.(c), p. 13 with the first line of r_3(n) on p. 11). A number is representable as a^2 + b^2 + c^2 with a,b, and c integers, 0 <= a < b < c, and gcd(a,b,c) = 1 if and only if n is not congruent 0, 4, 7 (mod 8) and not one of the numbers {a(k), k = 1 .. 23}, and, if it exists at all, a further number >= 5*10^10.

For the multiplicities of these representable numbers see A224447, and for the numbers themselves see A224448.

For a similar set of numbers relevant for sums of three nonzero squares see A051952.

F. Halter-Koch, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa42/aa4212.pdf">Darstellung natürlicher Zahlen als Summe von Quadraten</a>, Acta Arith. 42 (1982) 11-20, pp. 13 and 11.

representableQ[n_] := Length[ Select[ PowersRepresentations[n, 3, 2], Unequal @@ # && GCD @@ # == 1 & ]] > 0; Select[ Range[1000], Not[ representableQ[#] || MatchQ[ Mod[#, 8], 0 | 4 | 7]] &] (* Jean-François Alcover, Apr 10 2013 *)

Cf. A224447, A224448, A051952.

nonn,fini,changed

Wolfdieter Lang, Apr 09 2013

approved