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.

A374459 Nonsquarefree exponentially odd numbers.

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, Jul 09 2024

Keywords

Comments

First differs from A301517 at n = 1213. A301517(1213) = 12500 = 2^2 * 5^5 is not an exponentially odd number.
Numbers whose exponents in their prime factorization are all odd and at least one of them is larger than 1.
All the squarefree numbers (A005117) are exponentially odd. Therefore, the sequence of exponentially odd numbers (A268335) is a disjoint union of the squarefree numbers and this sequence.
The asymptotic density of this sequence is A065463 - A059956 = 0.096515099145... .

Examples

			8 = 2^3 is a term since 3 is odd and larger than 1.

Links

Crossrefs

Intersection of A013929 (or A046099) and A268335.
Subsequence of A301517.
Subsequences: A062838 \ {1}, A065036, A102838, A113850, A113852, A179671, A190011, A335988 \ {1}.
Cf. A005117, A374460.

Programs

  • Mathematica
    q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, AllTrue[e, OddQ] && ! AllTrue[e, # == 1 &]]; Select[Range[1000], q]
  • PARI
    is(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!(e[i] %2), return(0))); for(i = 1, #e, if(e[i] >1, return(1))); 0;}

Formula

a(n) = A268335(A374460(n)).
Sum_{n>=1} 1/a(n)^s = zeta(2*s) * (Product_{p prime} (1 + 1/p^s - 1/p^(2*s))) - zeta(s)/zeta(2*s) for s > 1.

A190012 Numbers with prime factorization pq^4r^4.

Original entry on oeis.org

6480, 9072, 14256, 16848, 22032, 24624, 29808, 30000, 37584, 40176, 47952, 53136, 55728, 60912, 68688, 70000, 76464, 79056, 86832, 92016, 94608, 101250, 102384, 107568, 110000, 115248, 115344, 125712, 130000, 130896, 133488, 138672
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, May 04 2011

Keywords

Links

Crossrefs

Cf. A179665, A179668, A190011.

Programs

  • Mathematica
    f[n_]:=Sort[Last/@FactorInteger[n]]=={1,4,4};Select[Range[250000],f]
  • PARI
    list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\2)^(1/8), t1=p^4;forprime(q=p+1, (lim\t1)^(1/4), t2=t1*q^4;forprime(r=2, lim\t2, if(p==r||q==r, next);listput(v,t2*r)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2011

A377845 Numbers that have more than one odd exponent larger than 1 in their prime factorization.

Original entry on oeis.org

216, 864, 1000, 1080, 1512, 1944, 2376, 2744, 2808, 3000, 3375, 3456, 3672, 4000, 4104, 4320, 4968, 5400, 6048, 6264, 6696, 6750, 7000, 7560, 7776, 7992, 8232, 8856, 9000, 9261, 9288, 9504, 9720, 10152, 10584, 10648, 10976, 11000, 11232, 11448, 11880, 12000, 12744, 13000
Offset: 1

Views

Author

Amiram Eldar, Nov 09 2024

Keywords

Comments

The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 1/(p^2*(p+1))) * (1 + Sum_{p prime} (1/(p^3+p^2-1))) = 0.0035024748296318122535... .

Links

Crossrefs

Complement of the union of A335275 and A377844.
Subsequence of A295661.
Subsequences: A162142, A179671, A190011.
Cf. A065465.

Programs

  • Mathematica
    q[n_] := Count[FactorInteger[n][[;; , 2]], _?(# > 1 && OddQ[#] &)] > 1; Select[Range[13000], q]
  • PARI
    is(k) = #select(x -> x>1 && x%2, factor(k)[, 2]) > 1;

A190114 Numbers with prime factorization p^2*q^2*r^5 where p, q, and r are distinct primes.

Original entry on oeis.org

7200, 14112, 24300, 34848, 39200, 47628, 48672, 83232, 96800, 103968, 112500, 117612, 135200, 152352, 164268, 189728, 231200, 242208, 264992, 276768, 280908, 288800, 297675, 350892, 394272, 423200, 453152, 484128, 514188, 532512, 566048
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, May 04 2011

Keywords

Links

Crossrefs

Cf. A179665, A190011, A190012.
Cf. A085548, A085964, A085965, A085967, A085968, A085969.

Programs

  • Mathematica
    f[n_]:=Sort[Last/@FactorInteger[n]]=={2,2,5};Select[Range[900000],f]
With[{upto=600000},Select[#[[1]]^2 #[[2]]^2 #[[3]]^5&/@ Flatten[ Permutations/@ Subsets[Prime[Range[Ceiling[Surd[upto,5]+1]]],{3}],1]// Union,#<=upto&]] (* Harvey P. Dale, Jul 29 2018 *)
  • PARI
    list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\36)^(1/5), t1=p^5;forprime(q=2, sqrt(lim\t1), if(p==q, next);t2=t1*q^2;forprime(r=q+1, sqrt(lim\t2), if(p==r,next);listput(v,t2*r^2)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

Formula

Sum_{n>=1} 1/a(n) = P(2)^2*P(5)/2 - P(2)*P(8)/2 - P(4)*P(5)/2 - P(2)*P(7) + P(9) = 0.00053812627050585644544..., where P is the prime zeta function. - Amiram Eldar, Mar 07 2024
Showing 1-4 of 4 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.