cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A362533 Decimal expansion of lim_{n->oo} ( Sum_{k=2..n} 1/(k * log(k) * log log(k)) - log log log(n) ).

Original entry on oeis.org

2, 6, 9, 5, 7, 4
Offset: 1

Views

Author

Bernard Schott, Apr 24 2023

Keywords

Comments

If u(n) = Sum_{k=2..n} ( 1/(k*log(k)*log log(k)) - log log log(n) ), then (u(n)) is convergent, while the series v(n) = Sum_{k=2..n} 1/(k*log(k)*log log log(k)) diverges (see link). This is an extension of A001620 and A361972.
Note that ( log log log(x) )' = 1 / ( x * log(x) * log log(x) ).

Examples

			2.69574...

Links

Crossrefs

Cf. A001620, A361972.

Formula

Limit_{n->oo} 1/( 2*log(2)*log log(2) ) + 1/( 3*log(3)*log log(3) ) + ... + 1/( n*log(n)*log log(n) ) - log log log(n).

A363368 Decimal expansion of Sum_{primes p} 1/(p*log(p)*log(log(p))).

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Jun 11 2023

Keywords

Comments

Value computed and communicated by Bill Allombert and confirmed by Pascal Sebah.

Examples

			1.9069738480349544...

Crossrefs

Cf. A077761, A086242, A137245, A138312, A221711, A303493, A319231, A319232, A354887, A354917, A354952, A361972, A362533.

Programs

  • PARI
    /* author Bill Allombert */
\p150
pz(x, ex=0)=
{
my(s=bitprecision(x));
my(B=s/real(polcoef(x, 0))+ex);
sum(n=1, B, my(a=moebius(n)); if(a!=0, a*log(zeta(n*x))/n));
}
my(P=primes([2, 61])); intnum(x=1, [oo, log(67)], (pz(x)-vecsum([p^-x|p<-P]))*intnum(s=0, [oo, 1], (x-1)^s/gamma(1+s))) + vecsum([1/p/log(p)/log(log(p))|p<-P])

A361089 a(n) = smallest integer x such that Sum_{k = 2..x} 1/(k*log(log(k))) > n.

Original entry on oeis.org

3, 5, 8, 21, 76, 389, 2679, 23969, 269777, 3717613, 61326301, 1188642478, 26651213526, 682263659097, 19720607003199, 637490095320530, 22857266906194526, 902495758030572213, 38993221443197045348, 1833273720522384358862
Offset: 2

Views

Author

Artur Jasinski, Jun 11 2023

Keywords

Comments

Because lim_{x->oo} (Sum_{k=2..x} 1 / (k*log(log(k)))) - li(log(x)) = 2.7977647035208... (see A363078) then a(n) = round(w) where w is the solution of the equation li(log(w)) + 2.7977647035208... = n.

Examples

			a(2) = 3 because Sum_{k=2..3} 1/(k*log(log(k))) = 2.18008755... > 2 and Sum_{k=2..2} 1/(k*log(log(k))) = -1.364208386450... < 2.
a(7) = 389 because Sum_{k=2..389} 1/(k*log(log(k))) = 7.000345... > 7 and Sum_{k=2..388} 1/(k*log(log(k))) = 6.99890560988... < 7.

Links

Crossrefs

Cf. A077761, A086242, A137245, A138312, A221711, A303493, A319231, A319232, A354887, A354917, A354952, A361972, A362533, A363368, A363078.

Programs

  • Mathematica
    (*slow procedure*)
lim = 2; sum = 0; aa = {}; Do[sum = sum + N[1/(k Log[Log[k]]), 100];
 If[sum >= lim, AppendTo[aa, k]; Print[{lim, sum, k}];
  lim = lim + 1], {k, 2, 269777}];aa
(*quick procedure *)
aa = {3}; cons = 2.79776470352080492766050456553352884330850083202326989577856315;
Do[ww = w /. NSolve[LogIntegral[Log[w]] + cons == n, w];
 AppendTo[aa, Round[ww][[1]]], {n, 3, 21}]; aa

Formula

For n >= 3, a(n) = round(w) where w is the solution of the equation li(log(w)) + 2.7977647035208... = n.

A363078 Decimal expansion of lim_{x->oo} (Sum_{k=2..x} 1 / (k*log(log(k)))) - li(log(x)).

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Jun 11 2023

Keywords

Comments

Value computed and communicated by Pascal Sebah.
For the smallest integer x such that Sum_{k = 2..x} 1/(k*log(log(k))) > n see A361089.

Examples

			2.7977647035208...

Crossrefs

Cf. A077761, A086242, A137245, A138312, A221711, A303493, A319231, A319232, A354887, A354917, A354952, A361972, A362533, A363368, A361089.

A366249 Decimal expansion of lim_{x->oo} (Sum_{primes p<=x} 1/(p*log(log(p)))) - log(log(log(x))).

Original entry on oeis.org

2, 9, 3, 8, 3, 2, 9, 0, 1
Offset: 1

Views

Author

Artur Jasinski, Oct 05 2023

Keywords

Comments

Value computed and communicated by Pascal Sebah.

Examples

			2.93832901...

Crossrefs

Cf. A077761, A086242, A137245, A138312, A221711, A303493, A319231, A319232, A354887, A354917, A354952, A361972, A362533, A363078, A363368.
Showing 1-5 of 5 results.

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