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 1-6 of 6 results.

A293442 Multiplicative with a(p^e) = A019565(e).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 6, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 12, 3, 4, 6, 6, 2, 8, 2, 10, 4, 4, 4, 9, 2, 4, 4, 12, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 12, 4, 12, 4, 4, 2, 12, 2, 4, 6, 15, 4, 8, 2, 6, 4, 8, 2, 18, 2, 4, 6, 6, 4, 8, 2, 10, 5, 4, 2, 12, 4, 4, 4, 12, 2, 12, 4, 6, 4, 4, 4, 20, 2, 6, 6, 9, 2, 8, 2, 12, 8
Offset: 1

Views

Author

Antti Karttunen, Oct 31 2017

Keywords

Comments

From Peter Munn, Apr 06 2021: (Start)
a(n) is determined by the prime signature of n.
Compare with the multiplicative, self-inverse A225546, which also maps 2^e to the squarefree number A019565(e). However, this sequence maps p^e to the same squarefree number for every prime p, whereas A225546 maps the e-th power of progressively larger primes to progressively greater powers of A019565(e).
Both sequences map powers of squarefree numbers to powers of squarefree numbers.
(End)

Links

Crossrefs

Sequences used in a definition of this sequence: A000188, A003961, A019565, A028234, A059895, A067029, A162642.
Sequences with related definitions: A225546, A293443, A293444.
Cf. also A293214.
Sequences used to express relationship between terms of this sequence: A006519, A007913, A008833, A064989, A334747.
Sequences related via this sequence: (A001222, A048675, A064547), (A007814, A162642), (A087207, A267116), (A248663, A268387).

Programs

  • Mathematica
    f[n_] := If[n == 1, 1, Apply[Times, Prime@ Flatten@ Position[Reverse@ IntegerDigits[Last@ #, 2], 1]] * f[n/Apply[Power, #]] &@ FactorInteger[n][[1]]]; Array[f, 105] (* Michael De Vlieger, Oct 31 2017 *)

Formula

a(1) = 1; for n > 1, a(n) = A019565(A067029(n)) * a(A028234(n)).
Other identities. For all n >= 1:
a(a(n)) = A293444(n).
A048675(a(n)) = A001222(n).
A001222(a(n)) = A064547(n) = A048675(A293444(n)).
A007814(a(n)) = A162642(n).
A087207(a(n)) = A267116(n).
A248663(a(n)) = A268387(n).
From Peter Munn, Mar 14 2021: (Start)
Alternative definition: a(1) = 1; a(2) = 2; a(n^2) = A003961(a(n)); a(A003961(n)) = a(n); if A059895(n, k) = 1, a(n*k) = a(n) * a(k).
For n >= 3, a(n) < n.
a(2n) = A334747(a(A006519(n))) * a(n/A006519(n)), where A006519(n) is the largest power of 2 dividing n.
a(2n+1) = a(A064989(2n+1)).
a(n) = a(A007913(n)) * a(A008833(n)) = 2^A162642(n) * A003961(a(A000188(n))).
(End)

A293439 Number of odious exponents in the prime factorization of n.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Nov 01 2017

Keywords

Examples

			For n = 2 = 2^1, the only exponent 1 is odious (that is, has an odd Hamming weight and thus is included in A000069), so a(2) = 1.
For n = 24 = 2^3 * 3^1, the exponent 3 (with binary representation "11") is evil (has an even Hamming weight and thus is included in A001969), while the other exponent 1 is odious, so a(24) = 1.

Links

Crossrefs

Cf. A000069, A010060, A077761, A262675, A293443.
Cf. A270428 (numbers such that a(n) = A001221(n)).
Differs from A144095 for the first time at n=24.

Programs

  • Mathematica
    a[n_] := Total@ ThueMorse[FactorInteger[n][[;; , 2]]]; a[1] = 0; Array[a, 100] (* Amiram Eldar, May 18 2023 *)
  • PARI
    A293439(n) = vecsum(apply(e -> (hammingweight(e)%2), factorint(n)[, 2]));
  • Python
    from sympy import factorint
def A293439(n): return sum(1 for e in factorint(n).values() if e.bit_count()&1) # Chai Wah Wu, Nov 23 2023

Formula

Additive with a(p^e) = A010060(e).
a(n) = A007814(A293443(n)).
From Amiram Eldar, Sep 28 2023: (Start)
a(n) >= 0, with equality if and only if n is an exponentially evil number (A262675).
a(n) <= A001221(n), with equality if and only if n is an exponentially odious number (A270428).
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) = -0.12689613844142998028..., where f(x) = 1/2 - x - ((1-x)/2) * Product_{k>=0} (1-x^(2^k)). (End)

A318469 Multiplicative with a(p^e) = A019565(A003714(e)).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 10, 2, 6, 2, 6, 4, 4, 2, 10, 3, 4, 5, 6, 2, 8, 2, 7, 4, 4, 4, 9, 2, 4, 4, 10, 2, 8, 2, 6, 6, 4, 2, 20, 3, 6, 4, 6, 2, 10, 4, 10, 4, 4, 2, 12, 2, 4, 6, 14, 4, 8, 2, 6, 4, 8, 2, 15, 2, 4, 6, 6, 4, 8, 2, 20, 10, 4, 2, 12, 4, 4, 4, 10, 2, 12, 4, 6, 4, 4, 4, 14, 2, 6, 6, 9, 2, 8, 2, 10, 8
Offset: 1

Views

Author

Antti Karttunen, Aug 30 2018

Keywords

Links

Crossrefs

Cf. A003714, A019565, A072649, A318464, A318465.
Cf. also A293442, A293443, A300834, A318470.

Programs

  • PARI
    A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A318469(n) = factorback(apply(e -> A019565(A003714(e)),factor(n)[,2]));

Formula

For all n >= 1, A001222(a(n)) = A318464(n).

A318498 Denominators of the sequence whose Dirichlet convolution with itself yields A061389, number of (1+phi)-divisors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 8, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 8, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 8, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 16, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 8, 8, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 8, 1, 2, 2, 4, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Antti Karttunen, Aug 30 2018

Keywords

Comments

The sequence seems to give the denominators of a few other similarly constructed rational valued sequences obtained as "Dirichlet Square Roots" (of possibly A092520 and A293443).

Links

Crossrefs

Cf. A061389, A318497 (numerators), A318499.
Cf. also A299150, A046644.

Programs

  • PARI
    up_to = 65537;
A061389(n) = factorback(apply(e -> (1+eulerphi(e)),factor(n)[,2]));
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA317937.
v318497_98 = DirSqrt(vector(up_to, n, A061389(n)));
A318497(n) = numerator(v318497_98[n]);
A318498(n) = denominator(v318497_98[n]);

Formula

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A061389(n) - Sum_{d|n, d>1, d 1.
a(n) = 2^A318499(n).

A318470 Multiplicative with a(p^e) = A260443(e).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 6, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 12, 3, 4, 6, 6, 2, 8, 2, 18, 4, 4, 4, 9, 2, 4, 4, 12, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 12, 4, 12, 4, 4, 2, 12, 2, 4, 6, 15, 4, 8, 2, 6, 4, 8, 2, 18, 2, 4, 6, 6, 4, 8, 2, 10, 5, 4, 2, 12, 4, 4, 4, 12, 2, 12, 4, 6, 4, 4, 4, 36, 2, 6, 6, 9, 2, 8, 2, 12, 8
Offset: 1

Views

Author

Antti Karttunen, Aug 30 2018

Keywords

Links

Crossrefs

Cf. A260443, A318306, A318307.
Differs from A293442 for the first time at n=32, where a(32) = 18, while A293442(32) = 10.
Cf. also A293216, A293217, A293443, A318469.

Programs

Formula

For all n >= 1, A001222(a(n)) = A318306(n).

A365810 Squareferee numbers ordered factorization-wise by Blue code: a(n) = A019565(A193231(n)).

Original entry on oeis.org

1, 2, 6, 3, 10, 5, 15, 30, 210, 105, 35, 70, 21, 42, 14, 7, 22, 11, 33, 66, 55, 110, 330, 165, 1155, 2310, 770, 385, 462, 231, 77, 154, 858, 429, 143, 286, 2145, 4290, 1430, 715, 5005, 10010, 30030, 15015, 2002, 1001, 3003, 6006, 39, 78, 26, 13, 390, 195, 65, 130, 910, 455, 1365, 2730, 91, 182, 546, 273, 1870, 935
Offset: 0

Views

Author

Antti Karttunen, Oct 06 2023

Keywords

Links

Crossrefs

Permutation of A005117.
Cf. A019565, A193231, A293231, A293443, A334205.
Cf. also A366263.

Programs

  • PARI
    A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) };
A365810(n) = A019565(A193231(n));

Formula

a(n) = A334205(A019565(n)).
Showing 1-6 of 6 results.

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