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.

A375934 Numbers whose prime factorization has a second-largest exponent that equals 1.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 204
Offset: 1

Views

Author

Amiram Eldar, Sep 03 2024

Keywords

Comments

First differs from A332785 at n = 112: A332785(112) = 360 = 2^3 * 3^2 * 5 is not a term of this sequence.
First differs from A317616 at n = 38: A317616(38) = 144 = 2*4 * 3^2 is not a term of this sequence.
Numbers k such that A375933(k) = 1.
Numbers of the form s1 * s2^e, where s1 and s2 are coprime squarefree numbers that are both larger than 1, and e >= 2.
The asymptotic density of this sequence is Sum_{e>=2} d(e) = 0.36113984820338109927..., where d(e) = Product_{p prime} (1 - 1/p^2 + 1/p^e - 1/p^(e+1)) - Product_{p prime} (1 - 1/p^(e+1)) is the asymptotic density of terms k with A051903(k) = e >= 2.

Links

Crossrefs

Equals A375142 \ A072777.
Subsequence of A013929.
Subsequences: A054753, A065036, A072357, A095990, A096156, A178739, A178740, A179664, A179668, A179692, A189987, A360793, A375076.
Cf. A001221, A005117, A051903, A051904, A072777, A317616, A332785, A375142, A375933.

Programs

  • Mathematica
    q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Max[0, Max[Select[e, # < Max[e] &]]] == 1]; Select[Range[300], q]
  • PARI
    is(n) = if(n == 1, 0, my(e = factor(n)[,2]); e = select(x -> x < vecmax(e), e); if(#e == 0, 0, vecmax(e) == 1));

Formula

A051904(a(n)) = 1.
A051903(a(n)) >= 2.
A001221(a(n)) = 2.

A384519 Numbers whose powerful part (A057521) is greater than 1 and is equal to a squarefree number raised to an even power (A384517).

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 25, 28, 36, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 75, 76, 80, 81, 84, 90, 92, 98, 99, 100, 112, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 162, 164, 169, 171, 172, 175, 176, 180, 188, 192, 196, 198, 204, 207, 208, 212, 220
Offset: 1

Views

Author

Amiram Eldar, Jun 01 2025

Keywords

Comments

Subsequence of A240112 and first differs from it at n = 30: A240112(30) = 108 is not a term of this sequence.
Subsequence of A368714 and differs from it by not having the terms 1, 144, 324, 400, 432, ... .
Numbers whose prime factorization has one distinct exponent that is larger than 1 and it is even.
Numbers that are a product of a squarefree number (A005117) and a coprime nonsquarefree number that is a squarefree number raised to an even power (A384517).
The asymptotic density of this sequence is Sum_{k>=1} (d(2*k)-1)/zeta(2) = 0.265530259454558018819..., where d(k) = zeta(k) * Product_{p prime} (1 + Sum_{i=k+1..2*k-1} (-1)^i/p^i).

Links

Crossrefs

Intersection of A335275 and A375142.
Intersection of A368714 and A375142.
Equals A375142 \ A384520.
Subsequence of A013929 and A240112.
Subsequences: A067259, A384517.
Cf. A005117, A013661, A057521.

Programs

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

A375342 The maximum exponent in the prime factorization of the numbers whose powerful part is a power of a squarefree number that is larger than 1.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Aug 12 2024

Keywords

Links

Crossrefs

Cf. A051903, A057521, A067259, A375142, A375341.

Programs

  • Mathematica
    s[n_] := Module[{e = Select[FactorInteger[n][[;; , 2]], # > 1 &]}, If[Length[e] > 0 && SameQ @@ e, e[[1]], Nothing]]; Array[s, 300]
  • PARI
    lista(kmax) = {my(e); for(k = 1, kmax, e = select(x -> x > 1, factor(k)[,2]); if(#e > 0 && vecmin(e) == vecmax(e), print1(e[1], ", ")));}

Formula

a(n) = A051903(A375142(n)).
a(n) = 2 if and only if A375142(n) is in A067259.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=2} k*d(k) / Sum_{k>=2} d(k) = 2.70113273169250927084..., where d(k) = (f(k)-1)/zeta(2) is the asymptotic density of terms m of A375142 with A051903(m) = k, f(k) = zeta(k) * Product_{p prime} (1 + Sum_{i=k+1..2*k-1} (-1)^i/p^i), if k is even, and f(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) if k is odd > 1.
Showing 1-4 of 4 results.

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