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.

A249263 Primitive, odd, squarefree abundant numbers.

Original entry on oeis.org

15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, 39585, 41055, 42315, 42735, 45885, 47355, 49665, 50505, 51765, 54285, 55965, 58695, 61215, 64155, 68145, 70455, 72345, 77385, 80535, 82005, 83265, 84315, 91245, 95865, 102795, 112035, 116655, 118965
Offset: 1

Views

Author

Derek Orr, Oct 23 2014

Keywords

Comments

The subsequence of primitive terms (not multiples of smaller terms) of A112643.
The subsequence of squarefree terms of A006038.
The subsequence of odd terms of A249242.
Not the same as A129485. Does not contain, for example, 195195, 255255, 285285, 333795, 345345, 373065, which are in A129485. - R. J. Mathar, Nov 09 2014
Sequences A287590, A188342 and A287581 list the number, smallest* and largest of all squarefree odd primitive abundant numbers with n prime factors. (*At least whenever A188342(n) is squarefree, which appears to be the case for all n >= 5.) - M. F. Hasler, May 29 2017

Crossrefs

Intersection of A112643 and A006038.
Cf. A188342 (least with n factors), A287581 (largest with n factors), A287590 (number of terms with n factors).

Programs

  • Maple
    # see A112643 and A006038 for the coding of isA112643 and isA006038
    isA249263 := proc(n)
        isA112643(n) and isA006038(n) ;
    end proc:
    for n from 1 do
        if isA249263(n) then
            print(n);
        end if;
    end do: # R. J. Mathar, Nov 10 2014
  • Mathematica
    PrimAbunQ[n_] := Module[{x, y},
       y = Most[Divisors[n]]; x = DivisorSigma[1, y];
       DivisorSigma[1, n] > 2 n  &&  AllTrue[x/y, # <= 2  &]];
    Select[Range[1, 120000, 2], PrimAbunQ[#] &&
    AllTrue[FactorInteger[#][[All, 2]], # == 1 &]  &] (* Robert Price, Sep 26 2019 *)
  • PARI
    v=[]; for(k=1, 10^5, n=2*k+1; if(issquarefree(n) && sigma(n)>2*n, for(i=1, #v, n%v[i] || next(2)); print1(n, ", "); v=concat(v, n))) \\ Improved (from 20 sec to 0.2 sec) by M. F. Hasler, May 27 2017

A295369 Number of squarefree primitive abundant numbers (A071395) with n prime factors.

Original entry on oeis.org

0, 0, 1, 18, 610, 216054, 12566567699
Offset: 1

Views

Author

Gianluca Amato, Feb 12 2018

Keywords

Comments

Here primitive abundant number means an abundant number all of whose proper divisors are deficient numbers (A071395). The alternative definition (an abundant number having no abundant proper divisor, see A091191) would yield an infinite count for a(3): since 2*3 = 6 is perfect, all numbers of the kind 2*3*p with p > 3 would be primitive abundant.
See A287590 for the number of squarefree ODD primitive abundant numbers with n prime factors.
The actual numbers are listed in A298973. - M. F. Hasler, Feb 16 2018

Examples

			For n=3, the only squarefree primitive abundant number (SFPAN) is 2*5*7 = 70, which is also a primitive weird number, see A002975.
For n=4, the 18 SFPAN range from 2*5*11*13 = 1430 to 2*5*11*53 = 5830.
For n=5, the 610 SFPAN range from 3*5*7*11*13 = 15015 to 2*5*11*59*647 = 4199030.
		

Crossrefs

Cf. A071395 (primitive abundant numbers), A287590 (counts of odd SFPAN), A298973, A249242 (using A091191).

Programs

  • PARI
    A295369(n, p=1, m=1, sigmam=1) = {
      my(centerm = sigmam/(2*m-sigmam), s=0);
      if (n==1,
        if (centerm > p, primepi(ceil(centerm)-1) - primepi(p), 0),
        p = max(floor(centerm),p); while (0A295369(n-1, p=nextprime(p+1), m*p, sigmam*(p+1)), s+=c); s
      )
    }
    
  • SageMath
    def A295369(n, p=1, m=1, sigmam=1):
      centerm = sigmam/(2*m-sigmam)
      if n==1:
        return prime_pi(ceil(centerm)-1) - prime_pi(p) if centerm > p else 0
      else:
        p = max(floor(centerm), p)
        s = 0
        while True:
           p = next_prime(p)
           c = A295369(n-1, p, m*p, sigmam*(p+1))
           if c <= 0: return s
           s+=c

A298973 Squarefree primitive abundant numbers (first definition: having only deficient proper divisors).

Original entry on oeis.org

70, 1430, 1870, 2002, 2090, 2210, 2470, 2530, 2990, 3190, 3230, 3410, 3770, 4030, 4070, 4510, 4730, 5170, 5830, 15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, 39585, 41055, 42315, 42735, 45885, 47355, 49665, 49742, 50505, 51765, 54285, 55965, 58695, 58786, 60214, 61215, 64155, 67298
Offset: 1

Views

Author

M. F. Hasler, Feb 16 2018

Keywords

Comments

Squarefree numbers (A005117) in A071395. The number of terms with n prime factors are counted in A295369. The subsequence of odd terms is A249263.
Two variants of the present sequence are possible: the terms listed by size, or as a table whose n-th row gives all those with n prime factors (so that A295369 would be the row lengths). They would differ only from a(322) = 692835 on, which is the first term with 6 prime factors, while a(755) = 4199030 is the last term with 5 prime factors.
A subsequence of the variant A249242, squarefree primitive abundant numbers using the 2nd definition, A091191, i.e., having no abundant proper divisors.
These numbers are also primitive unitary abundant numbers: unitary abundant numbers (A034683) that are also primitive abundant numbers (A071395). A unitary abundant number k is primitive if and only if usigma(k) - 2*k < 2*k/p^e, where p^e is the largest prime power dividing k and usigma is the sum of unitary divisors function (A034448). For numbers k in this sequence limsup_{k->oo} usigma(k)/k = 2. (Prasad and Reddy, 1990). - Amiram Eldar, Jul 18 2020

Examples

			The only squarefree primitive abundant number (SFPAN) with only 3 prime factors is a(1) = 2*5*7 = 70. Indeed, this number is abundant (sigma(70) - 70 = 1 + 2 + 5 + 7 + 10 + 14 + 35 = 74) but all of 2*5, 2*7 and 5*7 are deficient. This is also the smallest (thus primitive) weird number, see A002975.
The A295369(4) = 18 SFPAN with 4 prime factors range from a(2) = 2*5*11*13 = 1430 to a(19) = 2*5*11*53 = 5830.
The A295369(5) = 610 SFPAN with 5 prime factors range from a(20) = 3*5*7*11*13 = 15015 to a(755) = 2*5*11*59*647 = 4199030, but the first term with 6 prime factors occurs already at a(322) =  3*5*11*13*17*19 = 692835.
		

References

  • József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter III, p. 115.

Crossrefs

Cf. A005117 (squarefree numbers), A071395 (primitive abundant numbers, first definition), A091191 (idem, second definition), A249242 (squarefree numbers in A091191).

Programs

  • Mathematica
    spaQ[n_] := SquareFreeQ[n] && DivisorSigma[1, n] > 2*n && AllTrue[Most @ Divisors[n], DivisorSigma[1, #] < 2*# &]; Select[Range[70000], spaQ] (* Amiram Eldar, Jul 18 2020 *)
  • PARI
    is_A298973(n)=issquarefree(n)&&is_A071395(n)

A357696 Cubefree primitive abundant numbers: cubefree abundant numbers having no abundant proper divisor.

Original entry on oeis.org

12, 18, 20, 30, 42, 66, 70, 78, 102, 114, 138, 174, 186, 196, 222, 246, 258, 282, 308, 318, 354, 364, 366, 402, 426, 438, 474, 476, 498, 532, 534, 550, 572, 582, 606, 618, 642, 644, 650, 654, 678, 748, 762, 786, 812, 822, 834, 836, 868, 894, 906, 942, 978, 1002
Offset: 1

Views

Author

Amiram Eldar, Oct 10 2022

Keywords

Crossrefs

Intersection of A004709 and A091191.
Subsequence of A357695.
A249242 is a subsequence.
Cf. A308618.

Programs

  • Mathematica
    cubeFreeQ[n_] := Max[FactorInteger[n][[;; , 2]]] < 3; primQ[n_] := DivisorSigma[-1, n] > 2 && AllTrue[n/FactorInteger[n][[;; , 1]], DivisorSigma[-1, #] <= 2 &]; Select[Range[1500], cubeFreeQ[#] && primQ[#] &]
  • PARI
    is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 2] > 2, return(0))); if(sigma(f, -1) <= 2, return(0)); for(i = 1, #f~, if(sigma(n/f[i,1], -1) > 2, return(0))); 1};
Showing 1-4 of 4 results.