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.

A353352 Number of divisors d of n for which A048675(d) is a multiple of 3.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Apr 15 2022

Keywords

Comments

a(n) is the number of terms of A332820 that divide n.

Links

Crossrefs

Inverse Möbius transform of A353350.
Cf. A000005, A003961, A048675, A059269, A332820, A348717, A353328, A353329, A353351, A353470.
Cf. also A353332, A353354, A353362.

Programs

  • Mathematica
    f[p_, e_] := e*2^(PrimePi[p] - 1); q[1] = True; q[n_] := Divisible[Plus @@ f @@@ FactorInteger[n], 3]; a[n_] := DivisorSum[n, 1 &, q[#] &]; Array[a, 100] (* Amiram Eldar, Apr 15 2022 *)
  • PARI
    A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A353350(n) = (0==(A048675(n)%3));
A353352(n) = sumdiv(n,d,A353350(d));

Formula

a(n) = Sum_{d|n} A353350(d).
a(n) = A000005(n) - A353351(n).
a(p) = 1 for all primes p.
a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.
From Peter Munn, Apr 22 2022: (Start)
a(n) = A353328(n) = A353329(n) iff 3|A000005(n) [i.e., A353470(n) = 1].
Otherwise a(n) = A353328(n) iff A048675(n) == 1 (mod 3); a(n) = A353329(n) iff A048675(n) == 2 (mod 3).
(End)

A353328 Number of divisors d of n for which A332823(d) is positive (+1).

Original entry on oeis.org

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

Views

Author

Antti Karttunen and Peter Munn, Apr 16 2022

Keywords

Comments

Number of divisors of n such that A048673(d) == +1 (mod 3).

Links

Crossrefs

Cf. A003961, A048675, A332823, A353329, A353351, A353352, A353354.
Cf. A353355 [a(n) == A353329(n)], A353356 [a(n) > A353329(n)], A353357 [a(n) < A353329(n)].

Programs

  • PARI
    A332823(n) = { my(f = factor(n),u=(sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3); if(2==u,-1,u); };
A353328(n) = sumdiv(n,d,(A332823(d)>0));

Formula

a(n) = Sum_{d|n} [A332823(d) > 0], where [ ] is the Iverson bracket, giving 1 only if A332823 computed for the divisor d is strictly positive, and 0 otherwise.
a(n) = A353354(n) + A353329(n).
a(n) = A353351(n) - A353329(n).
a(n) = A353329(A003961(n)).

A353329 Number of divisors d of n for which A332823(d) is negative (-1).

Original entry on oeis.org

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

Views

Author

Antti Karttunen and Peter Munn, Apr 16 2022

Keywords

Comments

Number of divisors of n such that A048673(d) == -1 (mod 3).

Links

Crossrefs

Cf. A003961, A048675, A332823, A353328, A353351, A353352, A353354.
Cf. A353355 [a(n) == A353328(n)], A353356 [a(n) < A353328(n)], A353357 [a(n) > A353328(n)].

Programs

  • PARI
    A332823(n) = { my(f = factor(n),u=(sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3); if(2==u,-1,u); };
A353329(n) = sumdiv(n,d,(A332823(d)<0));

Formula

a(n) = Sum_{d|n} [A332823(d) < 0], where [ ] is the Iverson bracket, giving 1 only if A332823 computed for the divisor d is strictly negative, and 0 otherwise.
a(n) = A353328(n) - A353354(n).
a(n) = A353351(n) - A353328(n).
a(n) = A353328(A003961(n)).

A353361 Number of divisors d of n for which A156552(d) is not a multiple of 3.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Apr 15 2022

Keywords

Links

Crossrefs

Cf. A000005, A003961, A156552, A348717, A353269, A353350, A353362.
Cf. also A353351.

Programs

  • PARI
    A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A353269(n) = (!(A156552(n)%3));
A353361(n) = sumdiv(n,d,!A353269(d));

Formula

a(n) = Sum_{d|n} (1-A353269(d)).
a(n) = A000005(n) - A353362(n).
a(p) = 1 for all primes p.
a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.
Showing 1-4 of 4 results.

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