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.

A366742 The number of divisors of the least coreful infinitary divisor of n.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Oct 19 2023

Keywords

Comments

The sum of divisors of the least coreful infinitary divisor of n is A366744(n).

Links

Crossrefs

Cf. A000005, A006519, A138302, A365296, A366743, A366744.

Programs

  • Mathematica
    f[p_, e_] := 2^IntegerExponent[e, 2] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = vecprod(apply(x -> 2^valuation(x, 2) + 1, factor(n)[, 2]));

Formula

a(n) = A000005(A365296(n)).
a(n) = A000005(n) if and only if n is in A138302.
Multiplicative with a(p^e) = A006519(e) + 1.

A366743 The sum of infinitary divisors of the least coreful infinitary divisor of n.

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 3, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 12, 26, 42, 4, 40, 30, 72, 32, 3, 48, 54, 48, 50, 38, 60, 56, 18, 42, 96, 44, 60, 60, 72, 48, 68, 50, 78, 72, 70, 54, 12, 72, 24, 80, 90, 60, 120, 62, 96, 80, 5, 84, 144, 68, 90
Offset: 1

Views

Author

Amiram Eldar, Oct 19 2023

Keywords

Comments

Also, the sum of unitary divisors of the least coreful infinitary divisor of n, A365296(n), since A365296(n) is a term of A138302, which is also the sequence of numbers whose sets of unitary divisors (A077610) and infinitary divisors (A077609) coincide.
The number of infinitary divisors of the least coreful infinitary divisor of n is A034444(n).

Links

Crossrefs

Cf. A000203, A005117, A006519, A034444, A034448, A049417, A077609, A077610, A138302, A339597, A365296, A366742, A366744.

Programs

  • Mathematica
    f[p_, e_] := p^(2^IntegerExponent[e, 2]) + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^(2^valuation(f[i, 2], 2)));}

Formula

a(n) = A034448(A365296(n)).
a(n) = A049417(A365296(n)).
a(n) = A000203(n) if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = p^A006519(e) + 1.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 - 1/(p+1) + Sum_{e>=1} 1/p^f(e)-1/p^(f(e)+1)) = 0.61865169..., where f(k) = 2*k - A006519(k) = A339597(k-1).

A366744 The sum of divisors of the least coreful infinitary divisor of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 3, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 12, 31, 42, 4, 56, 30, 72, 32, 3, 48, 54, 48, 91, 38, 60, 56, 18, 42, 96, 44, 84, 78, 72, 48, 124, 57, 93, 72, 98, 54, 12, 72, 24, 80, 90, 60, 168, 62, 96, 104, 7, 84, 144, 68
Offset: 1

Views

Author

Amiram Eldar, Oct 19 2023

Keywords

Comments

The number of divisors of the least coreful infinitary divisor of n is A366742(n).

Links

Crossrefs

Cf. A000203, A006519, A138302, A339597, A365296, A366742, A366743.

Programs

  • Mathematica
    f[p_, e_] := (p^(2^IntegerExponent[e, 2]+1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^(2^valuation(f[i, 2], 2)+1) - 1)/(f[i, 1] -1 ));}

Formula

a(n) = A000203(A365296(n)).
a(n) = A000203(n) if and only if n is in A138302.
Multiplicative with a(p^e) = (p^(A006519(e)+1) - 1)/(p - 1).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 - p/(p^2-1) + Sum_{e>=1} 1/p^f(e)) = 0.696427154..., where f(k) = 2*k - A006519(k) = A339597(k-1).

A366075 The number of primes dividing the smallest coreful infinitary divisor of n, counted with multiplicity.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Sep 28 2023

Keywords

Links

Crossrefs

Cf. A001222, A006519, A246551, A365296.

Programs

  • Mathematica
    f[p_, e_] := 2^IntegerExponent[e, 2]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = vecsum(apply(x -> 2^valuation(x, 2), factor(n)[, 2]));

Formula

a(n) = A001222(A365296(n)).
Additive with a(p^e) = A006519(e).
a(n) = 1 if and only if n is in A246551.
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) = 0.42540262231508387576..., where f(x) = -x + (1-x) * Sum_{k>=0} (2^(k+1)-1)*x^(2^k)/(1+x^(2^k)).
Showing 1-4 of 4 results.

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