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.

Previous Showing 11-19 of 19 results.

A318315 The 2-adic valuation of A318314.

Original entry on oeis.org

0, 1, 0, 3, 0, 1, 0, 4, 1, 1, 0, 3, 0, 1, 0, 7, 0, 2, 0, 3, 0, 1, 0, 4, 1, 1, 1, 3, 0, 1, 0, 8, 0, 1, 0, 4, 0, 1, 0, 4, 0, 1, 0, 3, 1, 1, 0, 7, 1, 2, 0, 3, 0, 2, 0, 4, 0, 1, 0, 3, 0, 1, 1, 10, 0, 1, 0, 3, 0, 1, 0, 5, 0, 1, 1, 3, 0, 1, 0, 7, 3, 1, 0, 3, 0, 1, 0, 4, 0, 2, 0, 3, 0, 1, 0, 8, 0, 2, 1, 4, 0, 1, 0, 4, 0
Offset: 1

Views

Author

Antti Karttunen and Andrew Howroyd, Aug 29 2018

Keywords

Links

Crossrefs

Cf. A318314.
Cf. also A046645, A317946, A318451, A318455.

Programs

Formula

a(n) = A007814(A318314(n)).

A318451 The 2-adic valuation of A318450.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 0, 3, 1, 1, 1, 1, 1, 2, 0, 1, 3, 1, 1, 2, 1, 1, 1, 3, 1, 4, 1, 1, 2, 1, 0, 2, 1, 2, 3, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 1, 3, 3, 2, 1, 1, 4, 2, 1, 2, 1, 1, 2, 1, 1, 4, 0, 2, 2, 1, 1, 2, 2, 1, 3, 1, 1, 4, 1, 2, 2, 1, 1, 7, 1, 1, 2, 2, 1, 2, 1, 1, 4, 2, 1, 2, 1, 2, 1, 1, 3, 4, 3, 1, 2, 1, 1, 3
Offset: 1

Views

Author

Antti Karttunen and Andrew Howroyd, Aug 29 2018

Keywords

Links

Crossrefs

Cf. A007814, A318450.
Cf. also A046645, A317946, A318315, A318455.

Programs

Formula

a(n) = A007814(A318450(n)).

A318455 The 2-adic valuation of A318454(n).

Original entry on oeis.org

0, 1, 0, 3, 0, 1, 0, 4, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 4, 0, 1, 0, 3, 0, 1, 0, 8, 0, 1, 0, 3, 0, 1, 0, 4, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 4, 0, 1, 0, 3, 0, 1, 0, 10, 0, 1, 0, 3, 0, 1, 0, 4, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 4, 0, 1, 0, 3, 0, 1, 0, 8, 0, 1, 0, 3, 0, 1, 0, 4, 0
Offset: 1

Views

Author

Antti Karttunen and Andrew Howroyd, Aug 29 2018

Keywords

Links

Crossrefs

Cf. A001227, A007814, A318453, A318454.
Cf. also A046645, A317946, A318315, A318451.

Programs

  • Mathematica
    f[1] = 1; f[n_] := f[n] = 1/2 (Sum[Mod[d, 2], {d, Divisors[n]}] - Sum[f[d] f[n/d], {d, Divisors[n][[2 ;; -2]]}]);
a[n_] := IntegerExponent[Denominator[f[n]], 2];
Array[a, 105] (* Jean-François Alcover, Sep 13 2018 *)
  • PARI
    A318455(n) = valuation(A318454(n),2); \\ Needs also program from A318454.

Formula

a(n) = A007814(A318454(n)).

A318652 The 2-adic valuation of A046644(n)/A318512(n) (A318651).

Original entry on oeis.org

0, 1, 0, 3, 0, 2, 0, 4, 0, 2, 0, 4, 0, 2, 0, 7, 0, 2, 0, 4, 0, 2, 0, 5, 0, 2, 0, 4, 0, 2, 0, 8, 0, 2, 0, 4, 0, 2, 0, 5, 0, 2, 0, 4, 0, 2, 0, 8, 0, 2, 0, 4, 0, 2, 0, 5, 0, 2, 0, 4, 0, 2, 0, 10, 0, 2, 0, 4, 0, 2, 0, 6, 0, 2, 0, 4, 0, 2, 0, 8, 0, 2, 0, 4, 0, 2, 0, 5, 0, 2, 0, 4, 0, 2, 0, 9, 0, 2, 0, 4, 0, 2, 0, 5, 0
Offset: 1

Views

Author

Antti Karttunen, Aug 31 2018

Keywords

Links

Crossrefs

Cf. A046644, A318512, A318651.

Programs

  • PARI
    A318652(n) = valuation(A046644(n)/A318512(n),2); \\ Needs also code from those two respective entries.

Formula

a(n) = A007814(A318651(n)).
a(n) = A046645(n) - A318513(n).

A257089 a(n) = log_3 (A256689(n)).

Original entry on oeis.org

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

Views

Author

Wolfgang Hintze, Apr 16 2015

Keywords

Comments

a(n) is the logarithm to the base 3 of the denominator of the Dirichlet series of zeta(s)^(1/3). For details, see A256689.

Links

Crossrefs

Cf. A046645 (k = 2, log_2), A257089 (k = 3, log_3), A257090 (k = 4, log_2), A257091 (k = 5, log_5).

Formula

3^a(n) = A256689(n). a(n) = A007949(A256689(n)).

A257090 a(n) = log_2 (A256691(n)).

Original entry on oeis.org

0, 2, 2, 5, 2, 4, 2, 7, 5, 4, 2, 7, 2, 4, 4, 11, 2, 7, 2, 7, 4, 4, 2, 9, 5, 4, 7, 7, 2, 6, 2, 13, 4, 4, 4, 10, 2, 4, 4, 9, 2, 6, 2, 7, 7, 4, 2, 13, 5, 7, 4, 7, 2, 9, 4, 9, 4, 4, 2, 9, 2, 4, 7, 16, 4, 6, 2, 7, 4, 6, 2, 12, 2, 4, 7, 7, 4, 6, 2, 13, 11, 4, 2, 9, 4, 4, 4, 9, 2, 9, 4, 7, 4, 4, 4, 15, 2, 7, 7, 10
Offset: 1

Views

Author

Wolfgang Hintze, Apr 16 2015

Keywords

Comments

a(n) is the logarithm to the base 2 of the denominator of the Dirichlet series of zeta(s)^(1/4). For details, see A256691.

Links

Crossrefs

Cf. A046645 (k = 2, log_2), A257089 (k = 3, log_3), A257090 (k = 4, log_2), A257091 (k = 5, log_5).

Formula

2^a(n) = A256691(n).

A257091 a(n) = log_5 (A256693(n)).

Original entry on oeis.org

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

Views

Author

Wolfgang Hintze, Apr 16 2015

Keywords

Comments

a(n) is the logarithm to the base 5 of the denominator of the Dirichlet series of zeta(s)^(1/5). For details, see A256693.

Links

Crossrefs

Cf. A046645 (k = 2, log_2), A257089 (k = 3, log_3), A257090 (k = 4, log_2), A257091 (k = 5, log_5).

Programs

  • Maple
    F:= proc(n) local e,m;
add(add(floor(e/5^m),m=0..floor(log[5](e))),e=map(t-> t[2],ifactors(n)[2]));
end proc:
seq(F(i),i=1..100);
  • Mathematica
    F[n_] := Sum[Sum[Floor[e/5^m], {m, 0, Floor[Log[5, e]]}], {e, FactorInteger[n][[All, 2]]}];
F[1] = 0;
Array[F, 100] (* Jean-François Alcover, Jun 18 2020, after Maple *)

Formula

5^a(n) = A256693(n).
For n<=10000, if n = Product_i p_i^(e_i) is the prime factorization of n, a(n) = A001222(n) + Sum_i floor(e_i/5). - Robert Israel, May 13 2016
If n = Product_i p_i^(e_i) is the prime factorization of n, a(n) = Sum_{j >= 0} Sum_i floor(e_i/5^j). - Robert Israel, May 16 2016

A318659 The 2-adic valuation of A318658.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 3, 0, 4, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 4, 0, 1, 0, 3, 0, 2, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 4, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 4, 0, 2, 0, 1, 0, 7, 0, 1, 0, 2, 0, 2, 0, 1, 0, 2, 0, 2, 0, 2, 0, 1, 0, 4, 0, 1, 0, 1, 0, 3
Offset: 1

Views

Author

Antti Karttunen, Aug 31 2018

Keywords

Links

Crossrefs

Cf. A318658.
Cf. also A046645.

Programs

Formula

a(n) = A007814(A318658(n)).

A289619 Positions of ones in A289618.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 51, 55, 57, 58, 62, 65, 66, 69, 70, 72, 74, 77, 78, 82, 85, 86, 87, 91, 93, 94, 95, 96, 100, 102, 105, 106, 108, 110, 111, 114, 115, 118, 119, 122, 123, 129, 130, 133, 134, 138, 141, 142, 143, 145, 146, 154, 155, 158, 159, 160, 161, 165, 166, 170, 174
Offset: 1

Views

Author

Antti Karttunen, Jul 08 2017

Keywords

Comments

Numbers n such that A289617(n) = A005187(A001222(n)) is equal to 1 + A046645(n). Whether a number is included depends only on its prime signature, thus whenever any n is present in the sequence, so is also A046523(n).

Examples

			6 = 2^1 * 3^1, thus A001222(6) = 1+1 = 2, and A005187(2) = 3. On the other hand, A005187(1) = 1, and 1+1 = 2, which is one less than 3, thus 6 is included like all nonsquare semiprimes.
30 = 2^1 * 3^1 * 5^1, thus A001222(30) = 3, while A005187(3) = 4, thus 30 is included like all products of three distinct primes.
72 = 2^3 * 3^2, thus A001222(72) = 3+2 = 5, and A005187(5) = 8. On the other hand, A005187(3)+A005187(2) = 4+3 = 7, and 8 = 7+1, thus 72 is included in the sequence.

Links

Crossrefs

Cf. A001222, A005187, A046523, A046645, A289617, A289618.
Differs from A182853 for the first time at n=26, where a(26) = 72, while A182853(26) = 74.
Previous Showing 11-19 of 19 results.

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