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.

A010062 a(0)=1; thereafter a(n+1) = a(n) + number of 1's in binary representation of a(n).

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 12, 14, 17, 19, 22, 25, 28, 31, 36, 38, 41, 44, 47, 52, 55, 60, 64, 65, 67, 70, 73, 76, 79, 84, 87, 92, 96, 98, 101, 105, 109, 114, 118, 123, 129, 131, 134, 137, 140, 143, 148, 151, 156, 160, 162, 165, 169, 173, 178, 182, 187, 193, 196, 199, 204
Offset: 0

Views

Author

Leonid Broukhis, Mar 15 1996

Keywords

Comments

Sequence A230297 (and A157845 without initial term) converted from binary to decimal, cf. formula. - M. F. Hasler, Nov 18 2019

Examples

			a(7) = 14 because a(6) = 12, which is 1100 in binary (having 2 on bits), and 12 + 2 = 14.
a(8) = 17 because a(7) = 14, which is 1110 in binary (having 3 on bits), and 14 + 3 = 17.

Links

Crossrefs

First row of A228083.
For the base-10 analog see A004207.
Cf. A000120, A010061, A092391, A229167, A096303, A229743, A229744, A230297 (this sequence written in binary), A230298 (read mod 2).
See A230088 for partial sums.
Equals A028897 o A230297 = A028897 o A157845 (up to offset); see also A007088.

Programs

Formula

a(n) = (n/2)*log n + O(n*sqrt(log n * loglog n)), where log means log_2. In particular, a(n) ~ (n/2)*log n. [Stolarsky]
a(n + 1) = A092391(a(n)) = a(n) + A000120(a(n)). - Reinhard Zumkeller, May 27 2012, May 08 2004; corrected thanks to a notice by Lambert Herrgesell
a(n) = A028897(A230297(n)) = A028897(A157845(n+1)). - M. F. Hasler, Nov 18 2019

Extensions

More terms from Benoit Cloitre, Jun 02 2002
Stolarsky reference from Matthew C. Russell, Oct 08 2013

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

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

A228952 A010062(2^n-1).

Original entry on oeis.org

1, 2, 5, 14, 38, 92, 216, 518, 1165, 2641, 5981, 13215, 28880, 62481, 133539, 281878, 595867, 1257995, 2656439, 5585174, 11751388, 24708442, 51644779, 107729838, 224507391, 467923765, 971364960, 2016542071
Offset: 0

Views

Author

M. F. Hasler, Oct 05 2013

Keywords

Comments

Arises in studying the asymptotics of A010062.

Crossrefs

Cf. A229167.

Programs

  • PARI
    s=1;for(n=0,30,for(i=2^n+1,2^(n+1),s+=hammingweight(s));print1(s","))

Formula

A010062(2^n) = A092391(a(n)).
Showing 1-4 of 4 results.

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

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