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.

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A098391 a(n) = Log2(Log2(prime(n))), where Log2 = A000523.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3
Offset: 1

Views

Author

Reinhard Zumkeller, Sep 06 2004

Keywords

Comments

a(n) = A000523(A098388(n)).
a(n) = A000523(A000523(A000040(n))).

Crossrefs

Cf. A000040, A000523, A098388.

A098392 Prime(n)-Log2(Log2(prime(n))), where Log2=A000523.

Original entry on oeis.org

2, 3, 4, 6, 10, 12, 15, 17, 21, 27, 29, 35, 39, 41, 45, 51, 57, 59, 65, 69, 71, 77, 81, 87, 95, 99, 101, 105, 107, 111, 125, 129, 135, 137, 147, 149, 155, 161, 165, 171, 177, 179, 189, 191, 195, 197, 209, 221, 225, 227, 231, 237, 239, 249, 254, 260, 266, 268, 274, 278
Offset: 1

Views

Author

Reinhard Zumkeller, Sep 06 2004

Keywords

Comments

a(n) = A000040(n) - A098391(n).

Examples

			a(10) = A000040(10) - A098391(10) = 29 - 2 = 27.

Crossrefs

Cf. A098390, A014689, A098386, A098389.

A098393 Prime(n)+Log2(Log2(prime(n))), where Log2=A000523.

Original entry on oeis.org

2, 3, 6, 8, 12, 14, 19, 21, 25, 31, 33, 39, 43, 45, 49, 55, 61, 63, 69, 73, 75, 81, 85, 91, 99, 103, 105, 109, 111, 115, 129, 133, 139, 141, 151, 153, 159, 165, 169, 175, 181, 183, 193, 195, 199, 201, 213, 225, 229, 231, 235, 241, 243, 253, 260, 266, 272, 274, 280
Offset: 1

Views

Author

Reinhard Zumkeller, Sep 06 2004

Keywords

Comments

a(n) = A000040(n) + A098391(n).

Examples

			a(10) = A000040(10) + A098391(10) = 29 + 2 = 31.

Crossrefs

Cf. A098389, A014688, A098387, A098390.

Programs

  • Mathematica
    #+Floor[Log[2,Floor[Log[2,#]]]]&/@Prime[Range[60]] (* Harvey P. Dale, May 07 2017 *)

A098396 Number of primes that are not less than prime(n)-Log2(n) and not greater than prime(n)+Log2(n), where Log2=A000523.

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 3, 2, 2, 2, 1, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 3, 3, 2, 1, 2, 3, 3, 2, 2, 2, 1, 1, 1, 2, 2, 2, 3, 2, 1, 2, 4, 4, 3, 2, 2, 3, 3, 4, 2, 2, 3, 3, 3, 2, 2, 2, 1, 2, 2, 2, 3, 3, 3, 2, 3, 4, 4, 3, 1, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3
Offset: 1

Views

Author

Reinhard Zumkeller, Sep 06 2004

Keywords

Examples

			a(10) = #{p prime: A098386(10) <= p <= A098387(10)} =
= #{p prime: 26 <= p <= 32} = #{29,31} = 2.

Links

Programs

  • Maple
    f:= proc(n) local p,d;
  p:= ithprime(n); d:= ilog2(n);
  numtheory:-pi(p+d)-numtheory:-pi(p-d-1)
end proc:
map(f, [$1..200]); # Robert Israel, Aug 13 2018
  • Mathematica
    a[n_] := With[{p = Prime[n], d = BitLength[n]-1}, PrimePi[p+d] - PrimePi[p-d-1]];
Table[a[n], {n, 1, 200}] (* Jean-François Alcover, Feb 07 2023 *)

Formula

a(n) = A000720(A098386(n)) - A000720(A098387(n)-1).
A098398(n) <= a(n) <= A098397(n) <= A097935(n).

A285324 a(n) = A000523(A285327(n)-n).

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 1, 4, 1, 3, 1, 3, 1, 2, 1, 5, 1, 2, 1, 4, 1, 2, 1, 4, 1, 3, 1, 3, 1, 2, 1, 6, 1, 2, 1, 5, 1, 2, 1, 5, 1, 3, 1, 3, 1, 2, 1, 5, 1, 2, 1, 4, 1, 2, 1, 4, 1, 3, 1, 3, 1, 2, 1, 7, 1, 2, 1, 6, 1, 2, 1, 6, 1, 3, 1, 3, 1, 2, 1, 6, 1, 2, 1, 4, 1, 2, 1, 4, 1, 3, 1, 3, 1, 2, 1, 6, 1, 2, 1, 5, 1, 2, 1, 5, 1, 3, 1, 3, 1, 2, 1, 5, 1, 2, 1, 4, 1, 2, 1, 4
Offset: 1

Views

Author

Antti Karttunen, Apr 19 2017

Keywords

Links

Crossrefs

Cf. A000523, A285110, A285327.

Programs

Formula

a(n) = A000523(A285327(n)-n). [Note that A285327(n)-n is always a power of 2.]

A339823 a(n) = A056239(n) - A000523(n).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 0, 1, 1, 2, 1, 3, 2, 2, 0, 3, 1, 4, 1, 2, 2, 5, 1, 2, 3, 2, 2, 6, 2, 7, 0, 2, 3, 2, 1, 7, 4, 3, 1, 8, 2, 9, 2, 2, 5, 10, 1, 3, 2, 4, 3, 11, 2, 3, 2, 5, 6, 12, 2, 13, 7, 3, 0, 3, 2, 13, 3, 5, 2, 14, 1, 15, 7, 2, 4, 3, 3, 16, 1, 2, 8, 17, 2, 4, 9, 6, 2, 18, 2, 4, 5, 7, 10, 5, 1, 19, 3, 3, 2, 20, 4, 21, 3, 3
Offset: 1

Views

Author

Antti Karttunen, Dec 18 2020

Keywords

Comments

a(n) is the difference at n between the value of a PrimePi-based pseudo-logarithmic function (A056239) and log_2 floored down (A000523).
All terms are nonnegative. (Cf. Bertrand's postulate).

Links

Crossrefs

Cf. A000523, A056239.

Programs

Formula

a(n) = A056239(n) - A000523(n).

A339893 a(n) = A000523(n) - A001222(n); floor(log_2(n)) minus the number of prime divisors of n, counted with multiplicity.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 0, 2, 1, 1, 0, 3, 1, 3, 1, 2, 2, 3, 0, 2, 2, 1, 1, 3, 1, 3, 0, 3, 3, 3, 1, 4, 3, 3, 1, 4, 2, 4, 2, 2, 3, 4, 0, 3, 2, 3, 2, 4, 1, 3, 1, 3, 3, 4, 1, 4, 3, 2, 0, 4, 3, 5, 3, 4, 3, 5, 1, 5, 4, 3, 3, 4, 3, 5, 1, 2, 4, 5, 2, 4, 4, 4, 2, 5, 2, 4, 3, 4, 4, 4, 0, 5, 3, 3, 2, 5, 3, 5, 2, 3
Offset: 1

Views

Author

Antti Karttunen, Dec 21 2020

Keywords

Links

Crossrefs

Cf. A000523, A001222, A029744 (positions of 0's), A339895.
Cf. also A339823, A342657 [= a(A156552(n))].

Programs

  • PARI
    A339893(n) = (#binary(n) - 1 - bigomega(n));

Formula

a(n) = A000523(n) - A001222(n).
a(n) = A339895(A122111(n)).

A339894 a(n) = A000523(A122111(n)).

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 2, 3, 3, 5, 3, 6, 4, 4, 2, 7, 3, 8, 4, 5, 5, 9, 3, 4, 6, 4, 5, 10, 4, 11, 3, 6, 7, 5, 4, 12, 8, 7, 4, 13, 5, 14, 6, 5, 9, 15, 4, 6, 5, 8, 7, 16, 5, 6, 5, 9, 10, 17, 5, 18, 11, 6, 3, 7, 6, 19, 8, 10, 6, 20, 5, 21, 12, 6, 9, 7, 7, 22, 5, 5, 13, 23, 6, 8, 14, 11, 6, 24, 6, 8, 10, 12, 15, 9, 4, 25
Offset: 1

Views

Author

Antti Karttunen, Dec 21 2020

Keywords

Links

Crossrefs

Cf. A000523, A122111, A339895, A339896.

Programs

  • PARI
    A000523(n) = if( n<1, 0, #binary(n) - 1);
A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
A339894(n) = A000523(A122111(n));

Formula

a(n) = A000523(A122111(n)).

A098397 Number of primes that are not less than prime(n)-Log2(prime(n)) and not greater than prime(n)+Log2(prime(n)), where Log2=A000523.

Original entry on oeis.org

2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 3, 2, 1, 2, 2, 4, 3, 4, 3, 3, 2, 3, 4, 5, 5, 4, 3, 2, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 2, 3, 4, 4, 3, 1, 3, 4, 4, 4, 3, 2, 2, 3, 4, 4, 4, 5, 3, 3, 1, 3, 4, 4, 3, 2, 2, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 3, 3, 4, 3, 3, 4, 4, 4, 3, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 4, 4, 4
Offset: 1

Views

Author

Reinhard Zumkeller, Sep 06 2004

Keywords

Comments

a(n) = A000720(A098389(n)) - A000720(A098390(n)-1);
A098398(n) <= A098396(n) <= a(n) <= A097935(n).

Examples

			a(10) = #{p prime: A098389(10) <= p <= A098390(10)} =
= #{p prime: 25 <= p <= 33} = #{29,31} = 2.

A098398 Number of primes that are not less than prime(n)-Log2(Log2(prime(n))) and not greater than prime(n)+Log2(Log2(prime(n))), where Log2=A000523.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2
Offset: 1

Views

Author

Reinhard Zumkeller, Sep 06 2004

Keywords

Comments

a(n) = A000720(A098392(n)) - A000720(A098393(n)-1);
a(n) <= A098396(n) <= A098397(n) <= A097935(n);
a(n)<=2 for n<=6543; a(6544)=#{2^16+1=65537,65539,65543}=3.

Examples

			a(10) = #{p prime: A098392(10) <= p <= A098393(10)} =
= #{p prime: 27 <= p <= 31} = #{29,31} = 2.
Previous Showing 21-30 of 303 results. Next

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