A287590 -id:A287590 - cp's OEIS frontend

A249263 Primitive, odd, squarefree abundant numbers.

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

Derek Orr, Oct 23 2014

nonn

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

Giovanni Resta, Table of n, a(n) for n = 1..10000

Intersection of A112643 and A006038.
Cf. A249242, A129485.
Cf. A188342 (least with n factors), A287581 (largest with n factors), A287590 (number of terms with n factors).

# 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

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 *)

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

A188342 Smallest odd primitive abundant number (A006038) having n distinct prime factors.

Original entry on oeis.org

945, 3465, 15015, 692835, 22309287, 1542773001, 33426748355, 1635754104985, 114761064312895, 9316511857401385, 879315530560980695, 88452776289145528645, 2792580508557308832935, 428525983200229616718445, 42163230434005200984080045, 1357656019974967471687377449

Offset: 3

T. D. Noe, Mar 28 2011

nonn

Dickson proves that there are only a finite number of odd primitive abundant numbers having n distinct prime factors. For n=3, there are 8 such numbers: 945, 1575, 2205, 7425, 78975, 131625, 342225, 570375. See A188439.
a(14) <= 88452776289145528645. - Donovan Johnson, Mar 31 2011
a(15) <= 2792580508557308832935, a(16) <= 428525983200229616718445, a(17) <= 42163230434005200984080045. If these a(n) are squarefree and don't have a greatest prime factor more than 3 primes away from that of the preceding term, then these bounds are the actual values of a(n). The PARI code needs only fractions of a second to compute further bounds, which under the given hypotheses are the actual values of a(n). - M. F. Hasler, Jul 17 2016
It appears that the terms are squarefree for n >= 5, so they yield also the smallest term of A249263 with n factors; see A287581 for the largest such, and A287590 for the number of such terms with n factors. (For nonsquarefree odd abundant numbers, this seems to be known only for n = 3 and n = 4 prime factors (8 respectively 576 terms), cf. A188439.) - M. F. Hasler, May 29 2017
Comment from Don Reble, Jan 17 2023: (Start)
"If these a(n) are squarefree and don't have a greatest prime factor more than 3 primes away from that of the preceding term, then these bounds are the actual values of a(n)."
This conjecture is correct up to a(50). (End)

			From _M. F. Hasler_, Jul 17 2016: (Start)
               945 = 3^3 * 5 * 7
              3465 = 3^2 * 5 * 7 * 11
             15015 = 3 * 5 * 7 * 11 * 13
            692835 = 3 * 5 * 11 * 13 * 17 * 19     (n=6: gpf increases by 2 primes)
          22309287 = 3 * 7 * 11 * 13 * 17 * 19 * 23
        1542773001 = 3 * 7 * 11 * 17 * 19 * 23 * 29 * 31
       33426748355 = 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 31
     1635754104985 = 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 37 * 41     (here too)
   114761064312895 = 5 * 7 * 11 * 13 * 17 * 23 * 29 * 31 * 37 * 41 * 43
  9316511857401385 = 5 * 7 * 13 * 17 * 19 * 23 * 29 * 31 * 37 * 41 * 43 * 47
879315530560980695 = 5 * 7 * 13 * 17 * 19 * 23 * 29 * 31 * 37 * 41 * 53 * 59 * 61 (n=13: gpf increases for the first time by 3 primes) (End)

Daniel Suteu, Table of n, a(n) for n = 3..27
L. E. Dickson, Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors, American Journal of Mathematics 35 (1913), pp. 413-422.
H. N. Shapiro, Note on a theorem of Dickson, Bull Amer. Math. Soc. 55 (4) (1949), 450-452

Cf. A006038, A275449.

PrimAbunQ[n_] := Module[{x, y},
   y = Most[Divisors[n]]; x = DivisorSigma[1, y];
   DivisorSigma[1, n] > 2 n  &&  AllTrue[x/y, # <= 2  &]];
Table[k = 1;
 While[! PrimAbunQ[k] || Length[FactorInteger[k][[All, 1]]] != n,
k += 2]; k, {n, 3, 6}] (* Robert Price, Sep 26 2019 *)

A188342=[0,0,945,3465]; a(n,D(n)=n\6+1)={while(n>#A188342, my(S=#A188342, T=factor(A188342[S])[,1], M=[primepi(T[1]),primepi(T[#T])+D(S++)], best=prime(M[2])^S); forvec(v=vector(S,i,M), best>(T=prod(i=1,#v,prime(v[i]))) && (S=prod(i=1,#v,prime(v[i])+1)-T*2)>0 && S*prime(v[#v])A188342=concat(A188342,best));A188342[n]} \\ Assuming a(n) squarefree for n>4, search is exhaustive within the limit primepi(gpf(a(n))) <= primepi(gpf(a(n-1)))+D(n), with D(n) given as optional 2nd arg. - M. F. Hasler, Jul 17 2016

generate(A, B, n) = A=max(A, vecprod(primes(n+1))\2); (f(m, p, j) = my(list=List()); if(sigma(m) > 2*m, return(list)); forprime(q=p, sqrtnint(B\m, j), my(v=m*q); while(v <= B, if(j==1, if(v>=A && sigma(v) > 2*v, my(F=factor(v)[,1], ok=1); for(i=1, #F, if(sigma(v\F[i], -1) > 2, ok=0; break)); if(ok, listput(list, v))), if(v*(q+1) <= B, list=concat(list, f(v, q+1, j-1)))); v *= q)); list); vecsort(Vec(f(1, 3, n)));
a(n) = my(x=vecprod(primes(n+1))\2, y=2*x); while(1, my(v=generate(x, y, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Feb 10 2024

a(8)-a(12) from Donovan Johnson, Mar 29 2011
a(13) from Donovan Johnson, Mar 31 2011
a(14)-a(17) confirmed and a(18) from Daniel Suteu, Feb 10 2024

A287581 Largest squarefree odd primitive abundant number with n prime factors.

Original entry on oeis.org

442365, 13455037365, 1725553747427327895, 977844705701880720314685634538055, 29094181301361888360228876470808927597684302024968488289496445

Offset: 5

M. F. Hasler, May 26 2017

nonn

There is no squarefree odd abundant number with fewer than 5 prime factors: the largest abundancy an odd squarefree number with 4 prime factors can have is that of N = 3*5*7*11 with sigma_{-1}(N) = sigma(N)/N = 2 - 2/385.
See A287590 for the number of squarefree odd primitive abundant numbers (A249263) with n prime factors.
The next term, a(10), is too large to display.
It appears that the largest odd primitive abundant number with a given number of prime factors counted with multiplicity (bigomega = A001222), is always squarefree. Whenever this holds for a given n, then a(n) is also equal to the last term in row n of A287646 which lists odd primitive abundant numbers with n prime factors.

			a(5) = 442365 = 3 * 5 * 7 * 11 * 383 is the largest squarefree odd primitive abundant number (SOPAN). Here, 3*5*7*11 is the smallest possibility to produce a squarefree odd deficient number with 4 prime factors, and it is the one with the largest possible abundancy, and 383 is the largest prime by which this can be multiplied to yield an abundant number. One can increase 11 up to 19 to get more SOPAN (for a total of 71 + 12 + 3 + 1 = 87 = A287590(5) SOPAN with 5 factors), none of which is larger. One can see that increasing the 3rd prime factor 7 to 11 yields no further possibilities, and therefore also the second and third factor can't be increased.
a(6) = 13455037365 = 3 * 5 * 7 * 11 * 389 * 29947,
a(7) = 1725553747427327895 = 3 * 5 * 7 * 11 * 389 * 29959 * 128194559,
a(8) = 3 * 5 * 7 * 11 * 389 * 29959 * 128194589 * 566684450325179,
a(9) = a(8)/gpf(a(8)) * 566684450325197 * 29753376105337343078941364893,
a(10) = a(9)/gpf(a(9)) * 29753376105337343078941364947 * 30082232218581187462432471034748868284388270918928732059.

Cf. A287590, A249263.

A287581(n,p=3,P=p,s=2)={forstep(i=n,2,-1,n=max(1\(-1+s/=1+1/p),p+1); P*=p=if(i>2,nextprime(n),precprime(n)));P}

a(n+1) = (a(n)/p(n))*p'(n)*q(n), where p(n) = gpf(a(n)), p'(n) = nextprime(p(n)+1), q(n) = precprime(1/(2/sigma[-1](a(n)/p(n)*p'(n))-1)), sigma[-1](x) = sigma(x)/x; conjectured to hold for all n >= 5.

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

Gianluca Amato, Feb 12 2018

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

			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.

Gianluca Amato, Primitive Weirds and Abundant Numbers, GitHub.
Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi, and Maurizio Parton, Primitive abundant and weird numbers with many prime factors, arXiv:1802.07178 [math.NT], 2018.

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

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
  )
}

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

A287728 Number of odd primitive abundant numbers with n prime factors, counted with multiplicity.

Original entry on oeis.org

0, 0, 0, 0, 121, 15772, 102896101, 3475842606319962

Offset: 1

M. F. Hasler, May 30 2017

There is no odd abundant number (A005231) with less than 5 prime factors counted with multiplicity (cf. A001222).
Sequence A188439 lists the odd primitive abundant numbers (A006038) sorted by increasing number of distinct prime factors. It is known that there are 576 such terms with r = 3 distinct prime factors, but their number for any larger r = omega(x) appears to be unknown as of today.
It appears that a(n) is just slightly larger than A287590(n), the number of squarefree odd primitive abundant numbers (A249263) with n prime factors. Those with a prime factor to a higher power become less frequent because there are increasingly many terms of the form m*p_r where m has abundancy slightly less than 2, and p_r can be any prime between gpf(m) and 1/(2/A(m)-1) which becomes very large as A(m) -> 2. This also makes difficult the computation of a(n) for n >= 8: The lexicographic smallest choice of (p_1,...,p_8) has p_7 = 128194589 and then 128194601 <= p_8 <= 566684450325179, and calculation of primepi(566'684'450'325'179) takes very long.

Gianluca Amato, Primitive Weirds and Abundant Numbers, GitHub.
Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi, and Maurizio Parton, Primitive abundant and weird numbers with many prime factors, arXiv:1802.07178 [math.NT], 2018.

Cf. A005231, A006038, A188439, A249263, A287590, A001222.

SageMath
```
# See GitHub link.
```

a(7) from Gianluca Amato, Jun 26 2017

a(8) from Gianluca Amato, Feb 26 2018

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A249263 Primitive, odd, squarefree abundant numbers.

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A188342 Smallest odd primitive abundant number (A006038) having n distinct prime factors.

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A287581 Largest squarefree odd primitive abundant number with n prime factors.

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

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A287728 Number of odd primitive abundant numbers with n prime factors, counted with multiplicity.

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