Dumitru Damian - cp's OEIS frontend

A358559 Decimal expansion of Bi(0), where Bi is the Airy function of the second kind.

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

Offset: 0

Dumitru Damian, Nov 22 2022

			0.61492662744600073515092236909361355359472818864859650504087875301429651...

F. W. J. Olver, Asymptotics and Special Functions, Academic Press, ISBN 978-0-12-525856-2, 1974.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 56, page 559.

[DLMF] NIST Digital Library of Mathematical Functions, Eq. 9.2.5.
Eric Weisstein's World of Mathematics, Airy Functions.
Wikipedia, Airy Function.

Cf. A284867 (Ai(0)), A284868 (Ai'(0)), this sequence (Bi(0)), A358561 (Bi'(0)), A358564(Gi(0)).

RealDigits[AiryBi[0], 10, 120][[1]] (* Amiram Eldar, Nov 28 2022 *)

PARI
```
airy(0)[2]
```
PARI
```
airy(0)[1]*sqrt(3)
```
PARI
```
3^(1/3)*gamma(1/3)/(2*Pi)
```

airy_bi(0).n(algorithm='scipy', prec=250)

Bi(0) = A284867*A002194.
Bi(0) = A358564*3.
Bi(0) = 1/(3^(1/6)*A073006).
Bi(0) = A073005/(3^(1/6)*A186706).
Bi(0) = A073005/(3^(1/6)*2*A093602).
Bi(0) = 3^(1/3)*A073005/(2*A000796).
Bi(0) = A252799/(3^(1/6)*BarnesG[5/3]).
Bi(0) = 3^(1/4)/(2^(2/9) * Pi^(1/3) * AGM(2,(sqrt(2+sqrt(3))))^(1/3)), where AGM is the arithmetic-geometric mean.

A358561 Decimal expansion of the derivative Bi'(0), where Bi is the Airy function of the second kind.

Original entry on oeis.org

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

Offset: 0

Dumitru Damian, Nov 22 2022

			0.44828835735382635791482371039882839086622679921226206108280877837233075...

F. W. J. Olver, Asymptotics and Special Functions, Academic Press, ISBN 978-0-12-525856-2, 1974.

[DLMF] NIST Digital Library of Mathematical Functions, Eq. 9.2.6.
Eric Weisstein's World of Mathematics, Airy Functions.
Wikipedia, Airy Function.

Cf. A284867 (Ai(0)), A284868 (Ai'(0)), A358559 (Bi(0)), this sequence (Bi'(0)), A358564 (Gi(0)).

RealDigits[AiryBi'[0], 10, 120][[1]] (* Amiram Eldar, Nov 28 2022 *)

PARI
```
derivnum(x=0, airy(x)[2])
```

airy_bi_prime(0).n(algorithm='scipy', prec=250)

Bi'(0) = A284868*A002194.
Bi'(0) = 3*Gi'(0), where Gi' is the derivative of the inhomogeneous Airy function of the first kind.
Bi'(0) = 3^(1/6)/A073005.
Bi'(0) = A073006*3^(1/6)/A186706.
Bi'(0) = A073006*3^(1/6)/2*A093602.
Bi'(0) = 3^(2/3)*A073006/(2*A000796).
Bi'(0) = 3^(1/4)*AGM(2,(sqrt(2+sqrt(3))))^(1/3)/(2^(7/9) * Pi^(2/3)), where AGM is the arithmetic-geometric mean.

A358564 Decimal expansion of Gi(0), where Gi is the inhomogeneous Airy function of the first kind (also called Scorer function).

Original entry on oeis.org

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

Offset: 0

Dumitru Damian, Nov 22 2022

			0.204975542482000245050307456364537851198242729549532168346959584338098839...

Scorer, R. S., Numerical evaluation of integrals of the form Integral_{x=x1..x2} f(x)*e^(i*phi(x))dx and the tabulation of the function Gi(z)=(1/Pi)*Integral_{u=0..oo} sin(u*z+u^3/3) du, Quart. J. Mech. Appl. Math. 3 (1950), 107-112.

Amparo Gil, Javier Segura, and Nico Temme, On nonoscillating integrals for computing inhomogeneous Airy functions, Mathematics of Computation 70.235 (2001): 1183-1194.
[DLMF] NIST Digital Library of Mathematical Functions, Eq. 9.12.6.
Allan J. MacLeod, Computation of inhomogeneous Airy functions, Journal of Computational and Applied Mathematics, Volume 53, Issue 1, 1994, Pages 109-116, ISSN 0377-0427.
Wikipedia, Scorer's function.
Index entries for transcendental numbers.

Cf. A284867 (Ai(0)), A284868 (Ai'(0)), A358559 (Bi(0)), A358561 (Bi'(0)), this sequence (Gi(0)).

First[RealDigits[N[ScorerGi[0],90]]] (* Stefano Spezia, Nov 28 2022 *)

PARI
```
airy(0)[2]/3
```
PARI
```
1/(3^(7/6)*gamma(2/3))
```
PARI
```
sqrt(3)*gamma(1/3)/(3^(7/6)*2*Pi)
```

1/(3^(3/4)*2^(2/9)*Pi^(1/3)*sqrtn(agm(2,(sqrt(2+sqrt(3)))),3))

1/(3^(7/6)*gamma(2/3)).n(algorithm='scipy', prec=250)

Gi(0) = A358559/3.
Gi(0) = A284867/A002194.
Gi(0) = Hi(0)/2, where Hi is the inhomogeneous Airy function of the second kind.
Gi(0) = 1/(3^(7/6)*A073006).
Gi(0) = A073005/(3^(7/6)*A186706).
Gi(0) = A073005/(3^(7/6)*2*A093602).
Gi(0) = A073005/(3^(4/6)*2*A000796).
Gi(0) = A252799/(3^(7/6)*BarnesG(5/3)).
Gi(0) = 1/(3^(3/4) * 2^(2/9) * Pi^(1/3) * AGM(2,(sqrt(2+sqrt(3))))^(1/3)), where AGM is the arithmetic-geometric mean.

A348763 Decimal expansion of Sum_{n>=1} ((-1)^(n+1)*n)/(n+1)^2.

Original entry on oeis.org

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

Offset: 0

Dumitru Damian, Oct 31 2021

			0.12931985286416790881897546186483602653397481624314396474709910519161011...

Eric Weisstein's World of Mathematics, Dilogarithm
Index entries for zeta function.

Cf. A006752, A072691, A111003, A153071, A175571, A181983, A300707.

RealDigits[Pi^2/12 - Log[2], 10, 100][[1]] (* Amiram Eldar, Nov 30 2021 *)

-sumalt(n=1, (-1)^n*n/(n+1)^2) \\ Charles R Greathouse IV, Nov 01 2021

Pi^2/12-log(2) \\ Charles R Greathouse IV, Nov 01 2021

from scipy.special import zeta
from math import log
int(''.join(n for n in list(str(zeta(2)/2-log(2)))[2:-2]))

int(str(sum((-1)**(n+1)*n/(n+1)**2 for n in range(1,5000000)))[2:-2])

SageMath
```
(pi^2/12-log(2)).n(digits=100)
```

Equals Pi^2/12-log(2).
Equals Sum_{k>=2} (zeta(k)-zeta(k+1))/2^k. - Amiram Eldar, Mar 20 2022
Equals Integral_{x >= 0} x/(1 + exp(x))^2 dx = (1/2) * Integral_{x >= 0} x*(x - 2)*exp(x)/(1 + exp(x))^2 dx . - Peter Bala, Apr 26 2025

A348267 Primes of the form q^3+r^5+s^7, where q,r,s are consecutive primes.

Original entry on oeis.org

19504103, 410711297, 895293793, 19205982415663, 27139128435043, 122997897555661, 2351321783571193, 33026024797765183, 44544286011297461, 257023170905666323, 630639912549644209, 896737512757442999, 2267254920439040789, 2344105012311523369, 25786002910400593997

Offset: 1

Dumitru Damian, Oct 09 2021

nonn

Exponent values (3,5,7) given by the prime triplet of the form p,p+2,p+4.

			19504103 is a term because 5^3+7^5+11^7 = 19504103 is prime;
410711297 is a term because 11^3+13^5+17^7 = 410711297 is prime.

Cf. A000040, A348313, A259772, A304292.

Select[(#[[1]]^3 + #[[2]]^5 + #[[3]]^7) & /@ Partition[Select[Range[1000], PrimeQ], 3, 1], PrimeQ] (* Amiram Eldar, Oct 11 2021 *)

def Q3R5S7(x):
    return Primes().unrank(x)^3+Primes().unrank(x+1)^5+Primes().unrank(x+2)^7
A348267 = [Q3R5S7(x) for x in range(0,10^3) if Q3R5S7(x) in Primes()]

A348313 Primes q such that q^3+r^5+s^7 is also prime, where q,r,s are consecutive primes.

Original entry on oeis.org

5, 11, 13, 71, 73, 97, 149, 223, 229, 283, 337, 353, 401, 409, 577, 827, 887, 1051, 1277, 1321, 1489, 1543, 1627, 1787, 1931, 2237, 2467, 2903, 3137, 3181, 3559, 3917, 4243, 4357, 4363, 4441, 4583, 4723, 4933, 5113, 5693, 5839, 5857, 6007, 6043, 6053, 6121

Offset: 1

Dumitru Damian, Oct 11 2021

nonn

Exponent values (3,5,7) given by the prime triplet of the form p, p+2, p+4.

			5 is a term because 5^3+7^5+11^7 = 19504103 is prime;
11 is a term because 11^3+13^5+17^7 = 410711297 is prime.

Cf. A000040, A348267.

Select[Partition[Select[Range[6000], PrimeQ], 3, 1], PrimeQ[#[[1]]^3 + #[[2]]^5 + #[[3]]^7] &][[;; , 1]] (* Amiram Eldar, Oct 11 2021 *)

isok(p) = if (isprime(p), my(q=nextprime(p+1), r=nextprime(q+1)); isprime(p^3+q^5+r^7)); \\ Michel Marcus, Oct 11 2021

def Q(x):
    if Primes().unrank(x)^3+Primes().unrank(x+1)^5+Primes().unrank(x+2)^7 in Primes():
       return Primes().unrank(x)
A348313 = [Q(x) for x in range(0,10^3) if Q(x)!=None]

A348731 Decimal expansion of Integral_{x=0..1} x*log(x)/(1+x+x^2) dx (negated).

Original entry on oeis.org

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

Offset: 0

Dumitru Damian, Oct 31 2021

			-0.15766014916783233039054467406996221822374946546295676913413604497322566...

Cf. A072691, A086722, A086724, A111003, A222171.

RealDigits[Integrate[x*Log[x]/(1 + x + x^2), {x, 0, 1}], 10, 100][[1]] (* Amiram Eldar, Oct 31 2021 *)
RealDigits[Pi^2/54 - PolyGamma[1, 2/3]/9, 10, 100][[1]] (* Vaclav Kotesovec, Oct 31 2021 *)

intnum(x=0, 1, x*log(x)/(1+x+x^2)) \\ Michel Marcus, Oct 31 2021

RealField(25)(numerical_integral(x*log(x)/(1+x+x^2), 0, 1)[0])

Equals Pi^2/54 - PolyGamma(1, 2/3)/9. - Vaclav Kotesovec, Oct 31 2021

A348693 Decimal expansion of Integral_{x=0..oo} x*exp(-x)/(exp(x)+exp(-x)-1) dx.

Original entry on oeis.org

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

Offset: 0

Dumitru Damian, Oct 30 2021

			0.3118211318643269832...

Cf. A001620.

RealDigits[Integrate[x*Exp[-x]/(Exp[x] + Exp[-x] - 1), {x, 0, Infinity}], 10, 105][[1]] (* Amiram Eldar, Oct 30 2021 *)

RealField(45)(numerical_integral(x*exp(-x)/(exp(x)+exp(-x)-1), 0, +Infinity)[0])

A348607 Decimal expansion of BesselJ(1,2).

Original entry on oeis.org

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

Offset: 0

Dumitru Damian, Oct 25 2021

			0.5767248077568733872...

Cf. A010790, A054687, A058798, A058797, A133920, A245308, A296168, A301484, A318224.

Bessel function values: A334380 (J(0,1)), A091681 (J(0,2)), A334383 (J(0,sqrt(2))), this sequence (J(1,2)), A197036 (I(0,1)), A070910 (I(0,2)), A334381 (I(0,sqrt(2))), A096789 (I(1,2)).

RealDigits[BesselJ[1, 2], 10, 100][[1]] (* Amiram Eldar, Oct 25 2021 *)

besselj(1, 2) \\ Michel Marcus, Oct 25 2021

Sage
```
bessel_J(1, 2).n(digits=100)
```

Equals Sum_{k>=0} (-1)^k/(k!*(k+1)!).

A347114 Heptagonal pandigitals.

Original entry on oeis.org

1386470925, 1423809765, 1463095872, 1536942870, 1560837942, 1583406972, 1640538297, 1738402695, 1765403829, 1795023846, 1920538647, 2056743198, 2076149583, 2089571436, 2097384615, 2301546897, 2386051749, 2453718609, 2531869704, 2587063149, 2605431798

Offset: 1

Dumitru Damian, Aug 19 2021

There are 53 pandigital heptagonal numbers with no repeated digits, i.e., 10-digit pandigital heptagonal numbers. - Harvey P. Dale, Mar 26 2022

Cf. A000566, A171102.

h[n_] := n*(5*n - 3)/2; Select[h /@ Range[33000], Length @ DeleteDuplicates @ IntegerDigits[#] == 10 &] (* Amiram Eldar, Aug 19 2021 *)
Select[PolygonalNumber[7,Range[20234,62854]],Sort[IntegerDigits[#]] == Range[ 0,9]&] (* Harvey P. Dale, Mar 26 2022 *)

A000566 = list(int(n*(5*n-3)/2) for n in range(0,1000000))
def haspan(s): return any(len(set(s[i:i+10]))==10 for i in range(len(s)-9))
A347114 = list(elem for elem in A000566 if haspan(str(elem)))

Intersection of A000566 (heptagonal numbers) and A171102 (infinite pandigital numbers).

cp's OEIS Frontend

User: Dumitru Damian

A358559 Decimal expansion of Bi(0), where Bi is the Airy function of the second kind.

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A358561 Decimal expansion of the derivative Bi'(0), where Bi is the Airy function of the second kind.

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A358564 Decimal expansion of Gi(0), where Gi is the inhomogeneous Airy function of the first kind (also called Scorer function).

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A348763 Decimal expansion of Sum_{n>=1} ((-1)^(n+1)*n)/(n+1)^2.

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A348267 Primes of the form q^3+r^5+s^7, where q,r,s are consecutive primes.

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Sage

A348313 Primes q such that q^3+r^5+s^7 is also prime, where q,r,s are consecutive primes.

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