Jason Bard - cp's OEIS frontend

A386440 Decimal expansion of Sum_{k>=1} 1/(8k)!.

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

Offset: 0

Jason Bard, Aug 25 2025

			0.00002480158734938207491278717784706929523828059685156788...

Jason Bard, Table of n, a(n) for n = 0..1003
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 21. (Erroneously misses a leading zero)

Cf. A091131, A195070.

RealDigits[-1 + HypergeometricPFQ[{}, Range[1, 7]/8, 2^-24], 10, 100][[1]]

A387206 Decimal expansion of Sum_{k>=1} (-1)^(k+1)/(6k-5)^6 + (-1)^(k+1)/(6k-1)^6.

Original entry on oeis.org

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

Offset: 1

Jason Bard, Aug 21 2025

			1.000055160794869463473756618844925625878197693652065...

Jason Bard, Table of n, a(n) for n = 1..1000
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 21. (Misprint contains erroneous minus sign)

c:= Re(sum((-1)^(k+1)/(6*k-5)^6+(-1)^(k+1)/(6*k-1)^6, k=1..infinity)):
evalf(c, 140);  # Alois P. Heinz, Aug 21 2025

RealDigits[(1/358318080)*(PolyGamma[5, 1/12] + PolyGamma[5, 5/12] - PolyGamma[5, 7/12] - PolyGamma[5, 11/12]), 10, 100][[1]]

(365/1492992) * (zetahurwitz(6, 1/4) - zetahurwitz(6, 3/4))

Equals (1/358318080) * (PolyGamma(5, 1/12) + PolyGamma(5, 5/12) - PolyGamma(5, 7/12) - PolyGamma(5, 11/12)).

Equals (73/35831808) * (PolyGamma(5, 1/4) - PolyGamma(5, 3/4)). - Amiram Eldar, Aug 22 2025

A386522 Decimal expansion of the number of radians in a minute of arc.

Original entry on oeis.org

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

Offset: 0

Jason Bard, Aug 21 2025

			0.00029088820866572159615394846141476878557381198142362...

Jason Bard, Table of n, a(n) for n = 0..10002
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 21.

Cf. A000796, A019673, A019679, A019685.

Join[{0,0,0},RealDigits[Pi/10800, 10, 100][[1]]]

Equals Pi/10800.

Equals A019685/60.

A386552 Concatenate powers of 10.

Original entry on oeis.org

1, 110, 110100, 1101001000, 110100100010000, 110100100010000100000, 1101001000100001000001000000, 110100100010000100000100000010000000, 110100100010000100000100000010000000100000000, 1101001000100001000001000000100000001000000001000000000

Offset: 0

Jason Bard, Jul 25 2025

Binary version of A045507. Base-2 representation of A164894.

Concatenate first A000217(n+1) terms of A010054.

Jason Bard, Table of n, a(n) for n = 0..43

Cf. A000096, A000217, A010054, A011557, A045507, A051639, A101305, A133082, A164894, A242646.

a:= proc(n) option remember;
      `if`(n<0, 0, parse(cat(a(n-1), 10^n)))
    end:
seq(a(n), n=0..10);  # Alois P. Heinz, Jul 28 2025

a[0] = 1; a[n_] := a[n - 1]*10^(n+1) + 10^n; Table[a[n], {n, 0, 9}]

def A386552(n): return 10**n*sum(10**(k*((n<<1)-k+1)>>1) for k in range(n+1)) # Chai Wah Wu, Aug 05 2025

a(n) = Sum_{k=1..n+1} 10^A133082(k,n+2).
a(n) = A101305(n) + 10^A000096(n).
For n >= 1, a(n) = 10^(n+1)*a(n-1)+10^n.
Number of digits in a(n) is A000217(n+1).

A386407 a(n) = -floor(log(Integral_{x=2n^3+2n+2...oo} n^(-x^3) dx)/log(n)).

Original entry on oeis.org

10658, 238337, 2628081, 17984736, 88716544, 345948416, 1131366096, 3228667360, 8266914656, 19378404864, 42216896176, 86468159456, 168013851840, 311941643776, 556629367184, 959168691936, 1602434480416, 2604149515520, 4128340746096, 6399632545504

Offset: 2

Jason Bard, Jul 20 2025

nonn

-a(10) = -2022^3 - 8 was the solution to the final problem of the 2022 MIT Integral Bee Finals; see MIT link.

Jason Bard, Table of n, a(n) for n = 2..36
MIT, 2022 Integration Bee Finals. See Problem 5.
Prime Newtons, I solved this using Upper Incomplete Gamma, YouTube video.

Cf. A071568.

Table[-Floor[(Log[Gamma[1/3, 8 (n^3 + n + 1)^3*Log[n]]] - Log[3] - (1/3) Log[Log[n]])/Log[n]], {n, 2, 36}]

a(n) = -floor((log(Gamma(1/3, 8 * log(n) * (n^3 + n + 1)^3)) - log(3) - (1/3) * log(log(n))) / log(n)).

A386732 Decimal expansion of Integral_{x>=2} 1/(x^12-1) dx.

Original entry on oeis.org

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

Offset: 0

Jason Bard, Jul 31 2025

			0.000044394388389732931619793708861045902941185047688518...

Jason Bard, Table of n, a(n) for n = 0..1003
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 21.

Cf. A016644, A105199, A123868, A304656.

Join[{0, 0, 0, 0}, RealDigits[1/72 (-4 (3 + Sqrt[3]) Pi + 3 (4 ArcTan[2] + 2 Sqrt[3] ArcTan[5/Sqrt[3]] + 2 ArcTan[4 - Sqrt[3]] + 2 ArcTan[4 + Sqrt[3]] + Log[21] - Sqrt[3] Log[5 - 2 Sqrt[3]] + Sqrt[3] Log[5 + 2 Sqrt[3]])), 10, 100][[1]]]
(* or *)
Join[{0, 0, 0, 0}, RealDigits[Integrate[1/(x^12 - 1), {x, 2, Infinity}], 10, 100][[1]]]
(* or *)
Join[{0, 0, 0, 0}, RealDigits[1/22528*Hypergeometric2F1[11/12, 1, 23/12, 1/4096], 10, 100][[1]]]

Equals (1/22528) * hypergeometric(11/12, 1; 23/12; 1/4096).

Equals (-6*Pi - 4*sqrt(3)*Pi + 12*arctan(2) - 3*arctan(12/5) + 6*sqrt(3) * arctan(5/sqrt(3)) + 6*sqrt(3) * arctanh((2*sqrt(3))/5) + log(9261))/72.

A383463 Decimal expansion of Sum_{k>=0} 1/(16*k+1)^3.

Original entry on oeis.org

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

Offset: 1

Jason Bard, Jul 24 2025

			1.00024930487675326174813579820380197902155271564286...

Jason Bard, Table of n, a(n) for n = 1..10000
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 21.

evalf(sum(1/(16*k+1)^3, k=0..infinity), 120);  # Alois P. Heinz, Jul 26 2025

RealDigits[-PolyGamma[2, 1/16]/8192, 10, 100][[1]]

zetahurwitz(3, 1/16)/16^3 \\ Amiram Eldar, Jul 28 2025

Equals -1/8192 * PolyGamma(2, 1/16).

A384949 Decimal expansion of Sum_{k>=0} 1/(2*k+1)^9.

Original entry on oeis.org

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

Offset: 1

Jason Bard, Jul 24 2025

			1.00005134518384377259281790054250500567996990246638...

Jason Bard, Table of n, a(n) for n = 1..10000
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 21.

Cf. A013667, A016761.

evalf(sum(1/(2*k+1)^9, k=0..infinity), 120);  # Alois P. Heinz, Jul 26 2025

RealDigits[511*Zeta[9]/512, 10, 100][[1]]

511*zeta(9)/512 \\ Amiram Eldar, Jul 28 2025

Equals 511*Zeta(9)/512 = 511/512 * A013667.

Equals Sum_{n>=0} 1/A016761(n). - Amiram Eldar, Jul 28 2025

A386358 Primes without {7, 9} as digits.

Original entry on oeis.org

2, 3, 5, 11, 13, 23, 31, 41, 43, 53, 61, 83, 101, 103, 113, 131, 151, 163, 181, 211, 223, 233, 241, 251, 263, 281, 283, 311, 313, 331, 353, 383, 401, 421, 431, 433, 443, 461, 463, 503, 521, 523, 541, 563, 601, 613, 631, 641, 643, 653, 661, 683, 811, 821, 823

Offset: 1

Jason Bard, Jul 20 2025

Intersection of A038615 and A038617.

Cf. A000040, A020471, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 1, 2, 3, 4, 5, 6, 8]];

Select[Prime[Range[120]], DigitCount[#, 10, 7] == 0 && DigitCount[#, 10, 9] == 0 &]

primes_with(, 1, [0, 1, 2, 3, 4, 5, 6, 8]) \\ uses function in A385776

print(list(islice(primes_with("01234568"), 41))) # uses function/imports in A385776

A386357 Primes without {7, 8} as digits.

Original entry on oeis.org

2, 3, 5, 11, 13, 19, 23, 29, 31, 41, 43, 53, 59, 61, 101, 103, 109, 113, 131, 139, 149, 151, 163, 191, 193, 199, 211, 223, 229, 233, 239, 241, 251, 263, 269, 293, 311, 313, 331, 349, 353, 359, 401, 409, 419, 421, 431, 433, 439, 443, 449, 461, 463, 491, 499, 503

Offset: 1

Jason Bard, Jul 20 2025

Intersection of A038615 and A038616.

Cf. A000040, A020470, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 1, 2, 3, 4, 5, 6, 9]];

Select[Prime[Range[120]], DigitCount[#, 10, 7] == 0 && DigitCount[#, 10, 8] == 0 &]

primes_with(, 1, [0, 1, 2, 3, 4, 5, 6, 9]) \\ uses function in A385776

print(list(islice(primes_with("01234569"), 41))) # uses function/imports in A385776

cp's OEIS Frontend

User: Jason Bard

A386440 Decimal expansion of Sum_{k>=1} 1/(8k)!.

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A387206 Decimal expansion of Sum_{k>=1} (-1)^(k+1)/(6*k-5)^6 + (-1)^(k+1)/(6*k-1)^6.

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A386522 Decimal expansion of the number of radians in a minute of arc.

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A386552 Concatenate powers of 10.

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A386407 a(n) = -floor(log(Integral_{x=2n^3+2n+2...oo} n^(-x^3) dx)/log(n)).

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A386732 Decimal expansion of Integral_{x>=2} 1/(x^12-1) dx.

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A383463 Decimal expansion of Sum_{k>=0} 1/(16*k+1)^3.

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A384949 Decimal expansion of Sum_{k>=0} 1/(2*k+1)^9.

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A387206 Decimal expansion of Sum_{k>=1} (-1)^(k+1)/(6k-5)^6 + (-1)^(k+1)/(6k-1)^6.