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.

A376171 Powerful numbers whose prime factorization has an odd maximum exponent.

Original entry on oeis.org

8, 27, 32, 72, 108, 125, 128, 200, 216, 243, 288, 343, 392, 500, 512, 675, 800, 864, 968, 972, 1000, 1125, 1152, 1323, 1331, 1352, 1372, 1568, 1800, 1944, 2048, 2187, 2197, 2312, 2592, 2700, 2744, 2888, 3087, 3125, 3200, 3267, 3375, 3456, 3528, 3872, 3888, 4000
Offset: 1

Views

Author

Amiram Eldar, Sep 13 2024

Keywords

Comments

Subsequence of A102834 and first differs from it at n = 14: A102834(14) = 432 = 2^4 * 3^3 is not a term of this sequence.
Powerful numbers k such that A051903(k) is odd.
Equivalently, numbers whose prime factorization exponents are all larger than 1 and their maximum is odd. The maximum exponent in the prime factorization of 1 is considered to be A051903(1) = 0, and therefore 1 is not a term of this sequence.
The numbers of terms that do not exceed the 10^k-powerful number (A376092(k)), for k = 1, 2, ..., are 3, 40, 416, 4255, 42829, 429393, 4299797, 43022803, ... . Apparently, the asymptotic density of this sequence within the powerful numbers (A001694) exists and approximately equals 0.43.

Links

Crossrefs

Complement of A376170 within A001694.
Intersection of A001694 and A376142.
Subsequence of A102834.
Subsequences: A030078, A050997, A079395, A092759, A138031, A179665, A335988 \ {1}.
Cf. A051903, A082695, A376092.

Programs

  • Mathematica
    seq[lim_] := Select[Union@ Flatten@ Table[i^2 * j^3, {j, 1, Surd[lim, 3]}, {i, 1, Sqrt[lim/j^3]}], # > 1 && OddQ[Max[FactorInteger[#][[;; , 2]]]] &]; seq[10^4]
  • PARI
    is(k) = {my(f = factor(k), e = f[,2]); #e && ispowerful(f) && vecmax(e) % 2;}

Formula

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Sum_{k>=2} (-1)^k * s(k) = 0.29116340833243888282..., where s(k) = Product_{p prime} (1 + Sum_{i=2..k} 1/p^i).

A377844 Numbers that have a single odd exponent larger than 1 in their prime factorization.

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 72, 88, 96, 104, 108, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 200, 224, 232, 243, 248, 250, 264, 270, 280, 288, 296, 297, 312, 328, 343, 344, 351, 352, 360, 375, 376, 378, 384, 392, 408, 416, 424, 432, 440, 456, 459, 472, 480, 486, 488, 500
Offset: 1

Views

Author

Amiram Eldar, Nov 09 2024

Keywords

Comments

First differs from A295661, A325990 and A376142 at n = 24: A295661(24) = A325990(24) = A376142(24) = 216 = 2^3 * 3^3 is not a term of this sequence.
Differs from A060476 by having the terms 432, 648, 1728, ..., and not having the terms 1, 216, 256, 768, 864, ... .
The asymptotic density of this sequence is Product_{p prime} (1 - 1/(p^2*(p+1))) * Sum_{p prime} (1/(p^3+p^2-1)) = 0.11498368544519741081... .

Links

Crossrefs

Subsequence of A295661.
Subsequences: A065036, A143610, A163569.
Cf. A060476, A065465, A325990, A376142, A377845.

Programs

  • Mathematica
    q[n_] := Count[FactorInteger[n][[;; , 2]], _?(# > 1 && OddQ[#] &)] == 1; Select[Range[500], q]
  • PARI
    is(k) = #select(x -> x>1 && x%2, factor(k)[, 2]) == 1;

A381951 Nonsquarefree odd numbers whose prime factorization has an odd maximum exponent.

Original entry on oeis.org

27, 125, 135, 189, 243, 297, 343, 351, 375, 459, 513, 621, 675, 783, 837, 875, 945, 999, 1029, 1107, 1125, 1161, 1215, 1269, 1323, 1331, 1375, 1431, 1485, 1593, 1625, 1647, 1701, 1715, 1755, 1809, 1917, 1971, 2079, 2125, 2133, 2187, 2197, 2241, 2295, 2375, 2403, 2457
Offset: 1

Views

Author

Amiram Eldar, Mar 11 2025

Keywords

Comments

Nonsquarefree odd numbers k such that A051903(k) is odd, or equivalently, odd numbers k such that A051903(k) is an odd number that is larger than 1.
The asymptotic density of this sequence is (1/2) * Sum_{k>=3} (-1)^(k+1) * (1 - 2^k/((2^k-1)*zeta(k))) = 0.019098071299657074975... .

Examples

			125 = 5^3 is a term since it is odd, divisible by a square (25, and thus it is nonsquarefree), and the maximum exponent in its prime factorization is 3, which is odd.

Links

Crossrefs

Intersection of A005408 and A376142.
Complement of A381950 within A013929.
Cf. A051903.

Programs

  • Mathematica
    q[n_] := n > 1 && OddQ[n]; Select[Range[1, 2500, 2], q[Max[FactorInteger[#][[;; , 2]]]] &]
  • PARI
    isok(k) = k > 1 && k % 2 && apply(x -> (x > 1 && x % 2), vecmax(factor(k)[, 2]));

A384520 Numbers whose powerful part (A057521) is greater than 1 and is equal to a squarefree number raised to an odd power (A384518).

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 88, 96, 104, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 216, 224, 232, 243, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 352, 375, 376, 378, 384, 408, 416, 424, 440, 456, 459, 472, 480, 486, 488, 512, 513, 520, 536
Offset: 1

Views

Author

Amiram Eldar, Jun 01 2025

Keywords

Comments

Subsequence of A301517 and A374459 and first differs from them at n = 85: A374459(85) = A374459(85) = 864 = 2^5 * 3^3 is not a term of this sequence.
First differs from its subsequence A381312 at n = 21: a(21) = 216 = 2^3 * 3^3 is not a term of A381312.
Numbers whose prime factorization has one distinct exponent that is larger than 1 and it is odd.
Numbers that are a product of a squarefree number (A005117) and a coprime nonsquarefree number that is a squarefree number raised to an odd power (A384518).
The asymptotic density of this sequence is Sum_{k>=1} (d(2*k+1)-1)/zeta(2) = 0.095609588748823080455..., where d(k) = (zeta(2*k)/zeta(k)) * Product_{p prime} (1 + 2/p^k + Sum_{i=k+1..2*k-1} (-1)^(i+1)/p^i).

Links

Crossrefs

Intersection of A268335 and A375142.
Intersection of A295661 and A375142.
Intersection of A376142 and A375142.
Equals A375142 \ A384519.
Subsequence of A301517 and A374459.
Subsequences: A381312, A384518.
Cf. A005117, A013661, A057521.

Programs

  • Mathematica
    q[n_] := Module[{u = Union[Select[FactorInteger[n][[;; , 2]], # > 1 &]]}, Length[u] == 1 && OddQ[u[[1]]]]; Select[Range[250], q]
  • PARI
    isok(k) = {my(e = select(x -> (x > 1), Set(factor(k)[, 2]))); #e == 1 && e[1] % 2;}
Showing 1-4 of 4 results.

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