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 1-6 of 6 results.

A381950 Odd numbers whose prime factorization has an even maximum exponent.

Original entry on oeis.org

1, 9, 25, 45, 49, 63, 75, 81, 99, 117, 121, 147, 153, 169, 171, 175, 207, 225, 245, 261, 275, 279, 289, 315, 325, 333, 361, 363, 369, 387, 405, 423, 425, 441, 475, 477, 495, 507, 525, 529, 531, 539, 549, 567, 575, 585, 603, 605, 625, 637, 639, 657, 693, 711, 725
Offset: 1

Views

Author

Amiram Eldar, Mar 11 2025

Keywords

Comments

Odd numbers k such that A051903(k) is even.
The asymptotic density of this sequence is (1/2) * Sum_{k>=2} (-1)^k * (1 - 2^k/((2^k-1)*zeta(k))) = 0.075617194130991839249... .

Examples

			9 = 3^2 is a term since it is odd and 2 is even.
45 = 3^2 * 5 is a term since it is odd and 2 is even.
125 = 5^3 is not a term since 3 is odd.

Links

Crossrefs

Intersection of A005408 and A368714.
Subsequence of A381956.
A381823 is a subsequence.
Cf. A051903, A375039, A381951.

Programs

  • Mathematica
    Select[Range[1, 1000, 2], # == 1 || EvenQ[Max[FactorInteger[#][[;; , 2]]]] &]
  • PARI
    isok(k) = if(k == 1, 1, k % 2 && !(vecmax(factor(k)[, 2]) % 2));

A375040 The maximum exponent in the prime factorization of 2*n.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Jul 28 2024

Keywords

Links

Crossrefs

Bisection of A051903.
Cf. A033150, A375039.

Programs

  • Mathematica
    a[n_] := Max[FactorInteger[2*n][[;; , 2]]]; Array[a, 100]
  • PARI
    a(n) = vecmax(factor(2*n)[,2]);

Formula

a(n) = A051903(2*n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{k>=2} (1 - (2^k-2)/((2^k-1)*zeta(k))) = 2.15062559388175538361... .

A375667 The maximum exponent in the prime factorization of the 5-rough numbers (A007310).

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Aug 23 2024

Keywords

Links

Crossrefs

Cf. A007310, A051903, A375039, A375668.

Programs

  • Mathematica
    a[n_] := Max[FactorInteger[6*Floor[n/2] - (-1)^n][[;; , 2]]]; a[1] = 0; Array[a, 100]
  • PARI
    a(n) = if(n == 1, 0, vecmax(factor(n\2*6-(-1)^n)[,2]));

Formula

a(n) = A051903(A007310(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{k>=2} (1 - 1/((1-1/2^k) * (1-1/3^k) * zeta(k))) = 1.1034178389191320571029... .

A375668 The maximum exponent in the prime factorization of the 7-rough numbers (A007775).

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Aug 23 2024

Keywords

Links

Crossrefs

Cf. A007775, A051903, A375039, A375667.

Programs

  • Mathematica
    If[# == 1, 0, Max[FactorInteger[#][[;; , 2]]]] & /@ Select[Range[300], CoprimeQ[#, 30] &]
  • PARI
    lista(nmax) = print1(0, ", "); for(n = 2, nmax, if(gcd(n, 30) == 1, print1(vecmax(factor(n)[,2]), ", ")));

Formula

a(n) = A051903(A007775(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{k>=2} (1 - 1/((1-1/2^k) * (1-1/3^k) * (1-1/5^k) * zeta(k))) = 1.05546104674564363968... .
In general, the asymptotic mean of the maximum exponent in the prime factorization of the p-rough numbers (numbers that are not divisible by any prime smaller than p) is 1 + Sum_{k>=2} (1 - 1/(zeta(k) * Product_{primes q < p} (1-1/q^k))).

A381823 Odd cubefree numbers that are not squarefree.

Original entry on oeis.org

9, 25, 45, 49, 63, 75, 99, 117, 121, 147, 153, 169, 171, 175, 207, 225, 245, 261, 275, 279, 289, 315, 325, 333, 361, 363, 369, 387, 423, 425, 441, 475, 477, 495, 507, 525, 529, 531, 539, 549, 575, 585, 603, 605, 637, 639, 657, 693, 711, 725, 735, 747, 765, 775
Offset: 1

Views

Author

Amiram Eldar, Mar 08 2025

Keywords

Comments

Numbers whose prime factorization has only odd primes, exponents that are smaller than 3 and at least one exponent that equals 2.
Odd numbers k such that A051903(k) = A375039((k+1)/2) = 2.
The asymptotic density of this sequence is 4/(7*zeta(3)) - 2/(3*zeta(2)) = 0.070090906905338896329... .
In general, the asymptotic density of odd k-free numbers (numbers that are not divisible by a k-th power other than 1) that are not (k-1)-free, for k >= 2, is 2^(k-1)/((2^k-1) * zeta(k)) - 2^(k-2)/((2^(k-1)-1) * zeta(k-1)).

Links

Crossrefs

Intersection of A005408 and A067259.
Complement of A056911 within A381822.
Cf. A002117, A013661, A051903, A375039.
Subsequence of A048103.

Programs

  • Mathematica
    Select[Range[1, 1000, 2], Max[FactorInteger[#][[;;, 2]]] == 2 &]
  • PARI
    isok(k) = k % 2 && if(k == 1, 0, vecmax(factor(k)[, 2]) == 2);

A372960 Concatenation of the exponents in the prime factorization of 2*n-1.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 11, 1, 1, 11, 1, 2, 3, 1, 1, 11, 11, 1, 11, 1, 1, 21, 1, 2, 11, 1, 11, 11, 1, 1, 21, 11, 1, 11, 1, 1, 12, 11, 1, 4, 1, 11, 11, 1, 11, 11, 11, 1, 21, 1, 1, 111, 1, 1, 11, 1, 11, 21, 11, 2, 11, 3, 1, 11, 1, 11, 31, 1, 1, 11, 11, 11, 12, 1, 1, 21
Offset: 1

Views

Author

Jean-Marc Rebert, Aug 02 2024

Keywords

Examples

			a(8) = 11, because 2*8 - 1 = 15 = 3^1 * 5^1 and the concatenation of the exponents of the prime factorization is 11.

Links

Crossrefs

Cf. A005408, A037916, A375039.

Programs

  • Mathematica
    Join[{0},Table[FromDigits[Flatten[IntegerDigits/@FactorInteger[2n-1][[;;,2]]]],{n,2,80}]] (* Harvey P. Dale, Jun 04 2025 *)

Formula

a(n) = A037916(2*n-1).
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.