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 1-4 of 4 results.

A229168 Define a sequence of real numbers by b(1)=2, b(n+1) = b(n) + log_2(b(n)); a(n) = smallest i such that b(i) >= 2^n.

Original entry on oeis.org

1, 3, 5, 7, 11, 17, 27, 44, 74, 127, 225, 402, 728, 1333, 2459, 4566, 8525, 15993, 30122, 56936, 107953, 205253, 391223, 747369, 1430648, 2743721, 5270959, 10141978, 19542806, 37708232, 72849931, 140905791, 272836175, 528832794, 1026008203, 1992390617
Offset: 1

Views

Author

N. J. A. Sloane, Sep 27 2013

Keywords

Examples

			The initial terms of the b(n) sequence are approximately
2, 3.00000000000000000000000, 4.58496250072115618145375, 6.78187243514238888864578, 9.54355608312733448665509, 12.7980830210090262451102, 16.4759388461842196480290, 20.5182276175427023220954, 24.8770618274970204646817, 29.5138060245244394221195, 34.3971240984210783617324, ...
b(5) is the first term >= 8, so a(3) = 5.

Links

Crossrefs

Cf. A229169, A229170, A010062, A229167, A155921, A229177; also A004207, A229171, A229172, A229173.

Programs

  • Maple
    # A229168, A229169, A229170.
Digits:=24;
log2:=evalf(log(2));
lis:=[2]; a:=2;
t1:=[1]; l:=2;
for i from 2 to 128 do
a:=evalf(a+log(a)/log2);
if a >= 2^l then
l:=l+1; t1:=[op(t1),i]; fi;
lis:=[op(lis),a];
od:
lis;
map(floor,lis);
map(ceil,lis);
t1;
  • PARI
    n=1; p2=2^n; m=2; lg2=log(2); for(i=1, 1992390617, if(m>=p2, print(n " " i); n++; p2=2^n); m=m+log(m)/lg2) /* Donovan Johnson, Oct 04 2013 */

Extensions

a(11)-a(36) from Donovan Johnson, Oct 04 2013

A229170 Define a sequence of real numbers by b(1)=2, b(n+1) = b(n) + log_2(b(n)); a(n) = ceiling( b(n) ).

Original entry on oeis.org

2, 3, 5, 7, 10, 13, 17, 21, 25, 30, 35, 40, 45, 51, 56, 62, 68, 74, 80, 87, 93, 100, 106, 113, 120, 127, 134, 141, 148, 155, 162, 170, 177, 185, 192, 200, 207, 215, 223, 231, 239, 246, 254, 262, 270, 278, 287, 295, 303, 311, 319, 328, 336, 345, 353, 361, 370, 378, 387, 396, 404, 413, 422, 430, 439, 448, 457, 465, 474
Offset: 1

Views

Author

N. J. A. Sloane, Sep 27 2013

Keywords

Examples

			Comment from _N. J. A. Sloane_, Jun 07 2023: (Start)
b(1) to b(4) are:
  [2, 3, 3+ln(3)/ln(2), 3+ln(3)/ln(2)+ln(3+ln(3)/ln(2))/ln(2)]
b(1) to b(6) computed exactly and THEN converted to floating point are:
  2., 3., 4.58496250072115618145373894394, 6.78187243514238888864576363676, 9.54355608312733448665507868324, 12.7980830210090262451101642020. (End)

Links

Crossrefs

Cf. A229168, A229169, A155921, A229177.

Programs

  • Mathematica
    Ceiling[NestList[N[#+Log2[#],1000]&,2,70]] (* Caution: this program uses an arbitrary cut-off at 1000 digits precision. - N. J. A. Sloane, Jun 07 2023 *) (* Harvey P. Dale, Jun 10 2023 *)

A229171 Define a sequence of real numbers by b(1)=e, b(n+1) = b(n) + log(b(n)); a(n) = smallest i such that b(i) >= e^n.

Original entry on oeis.org

1, 5, 10, 20, 41, 86, 192, 441, 1039, 2493, 6072, 14960, 37199, 93193, 234957, 595562, 1516639, 3877905, 9950908, 25615654, 66127187, 171144672, 443966371, 1154115393, 3005950908
Offset: 1

Views

Author

N. J. A. Sloane, Sep 27 2013

Keywords

Examples

			The initial terms of the b(n) sequence are approximately
2.71828182845904523536029, 3.71828182845904523536029, 5.03154351597726806940929, 6.64727031503970856301384, 8.54147660649653209023621, 10.6864105040926911986276, 13.0553833920216929230460, 15.6245839611886549261305, 18.3734295299727029212384, 21.2843351036624388705641, 24.3423064646657059114213, 27.5345223079930416816192, 30.8499628820185220765989, ...
b(5) is the first term >= e^2, so a(2) = 5.

Links

Crossrefs

Cf. A229172, A229173, A229175; A229168, A229169, A229170, A010062, A229167; also A004207.

Programs

  • Maple
    # A229171, A229172, A229173.
Digits:=24;
e:=evalf(exp(1));
lis:=[e]; a:=e;
t1:=[1]; l:=2;
for i from 2 to 128 do
a:=evalf(a+log(a));
if a >= e^l then
l:=l+1; t1:=[op(t1),i]; fi;
lis:=[op(lis),a];
od:
lis;
map(floor,lis);
map(ceil,lis);
t1;
  • PARI
    n=1; m=exp(1); mn=m^n; for(i=1, 3005950908, if(m>=mn, print(n " " i); n++; mn=exp(1)^n); m=m+log(m)) /* Donovan Johnson, Oct 04 2013 */

Extensions

a(7)-a(25) from Donovan Johnson, Oct 04 2013

A229177 Decimal expansion of 3 + log_2(3) + log_2(3 + log_2(3)).

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Sep 28 2013

Keywords

Comments

This is the fourth term in the sequence of real numbers discussed in A229168-A229170.

Examples

			6.7818724351423888886457636367782053029041818834546993272707...

Links

Crossrefs

Cf. A229168, A229169, A229170, A155921, A020857.

Programs

  • Mathematica
    RealDigits[3 + Log2[3] + Log2[3 + Log2[3]], 10, 120][[1]] (* Amiram Eldar, Jun 06 2023 *)
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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