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.

A368167 The largest unitary divisor of n that is a cubefull exponentially odd number (A335988).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 27, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 27, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Dec 14 2023

Keywords

Comments

First differs from A056191 and A366126 at n = 32, and from A367513 at n = 64.
Also, the largest exponentially odd unitary divisor of the powerful part on n.
Also, the powerful part of the largest exponentially odd unitary divisor of n.

Links

Crossrefs

Cf. A057521, A077610, A335275, A335988, A350389, A368168, A368169.
Cf. A056191, A366126, A367513.

Programs

  • Mathematica
    f[p_, e_] := If[e == 1 || EvenQ[e], 1, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1 || !(f[i, 2]%2), 1, f[i, 1]^f[i, 2]));}

Formula

Multiplicative with a(p^e) = p^e if e is odd that is larger than 1, and 1 otherwise.
a(n) = A350389(A057521(n)).
a(n) = A057521(A350389(n)).
a(n) >= 1, with equality if and only if n is in A335275.
a(n) <= n, with equality if and only if n is in A335988.

A382291 a(n) = A037445(n)/A034444(n).

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Mar 21 2025

Keywords

Comments

First differs from A368168 at n = 64, and from A359411, A367516 and A368979 at n = 128.

Links

Crossrefs

Cf. A000120, A034444, A037445, A138302, A243036, A382290, A382292.
Cf. A359411, A367516, A368168, A368979.

Programs

  • Mathematica
    f[p_, e_] := 2^(DigitCount[e, 2, 1] - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 105]
  • PARI
    a(n) = vecprod(apply(x -> 1 << (hammingweight(x)-1), factor(n)[, 2]));

Formula

a(n) = 2^A382290(n).
Multiplicative with a(p^e) = 2^(A000120(e)-1) = A048896(e-1) (= A243036(e) for e >= 2).
a(n) >= 1, with equality if and only if n is in A138302.
a(n) = 2 if and only if n is in A382292.

A368169 The number of divisors of the largest unitary divisor of n that is a cubefull exponentially odd number (A368167).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Dec 14 2023

Keywords

Links

Crossrefs

Cf. A000005, A013661, A098198, A335275, A335988, A368167, A368168.

Programs

  • Mathematica
    f[p_, e_] := If[e == 1 || EvenQ[e], 1, e+1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1 || !(f[i, 2]%2), 1, f[i, 2]+1));}

Formula

Multiplicative with a(p^e) = e+1 if e is odd that is larger than 1, and 1 otherwise.
a(n) = A000005(A368167(n)).
a(n) >= 1, with equality if and only if n is in A335275.
a(n) <= A000005(n), with equality if and only if n is in A335988.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2)^2 * Product_{p prime} (1 - 2/p^2 + 3/p^3 - 2/p^4 - 1/p^5 + 1/p^6) = 1.47140789970892803631... .

A384557 The number of exponential unitary (or e-unitary) divisors of n that are exponentially odd numbers (A268335).

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Jun 03 2025

Keywords

Comments

First differs from A359411 at n = 2097152 = 2^21: a(2097152) = 4, while A359411(2097152) = 2.
First differs from A368979 at n = 512 = 2^9: a(512) = 2, while A368979(512) = 3.
First differs from A367516 at n = 128 = 2^7: a(128) = 2, while A367516(128) = 1.
First differs from A382291 at n = 128 = 2^7: a(128) = 2, while A382291(128) = 4.
First differs from A368168 at n = 64 = 2^6: a(64) = 2, while A368168(64) = 1.
The sum of these divisors is A384559(n), and the largest of them is A331737(n).
The number of exponential unitary (or e-unitary) divisors of n is A278908(n) and the number of divisors of n that are exponentially odd numbers is A322483(n).
All the terms are powers of 2. The first term that is greater than 2 is a(32768) = 4.

Links

Crossrefs

Cf. A068068, A138302, A268335, A278908, A322483, A331737, A359411, A367516, A368168, A368979, A382291, A384559.

Programs

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

Formula

Multiplicative with a(p^e) = A068068(e).
a(n) >= 1, with equality if and only if n is in A138302.
a(n) <= A278908(n), with equality if and only if n is an exponentially odd number.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{k>=2} (d(k) - d(k-1))/p^k) = 1.13551542615965557947..., where d(k) is the number of odd unitary divisors of k (A068068).
Showing 1-4 of 4 results.

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