A188439 -id:A188439 - cp's OEIS frontend

A006038 Odd primitive abundant numbers.

945, 1575, 2205, 3465, 4095, 5355, 5775, 5985, 6435, 6825, 7245, 7425, 8085, 8415, 8925, 9135, 9555, 9765, 11655, 12705, 12915, 13545, 14805, 15015, 16695, 18585, 19215, 19635, 21105, 21945, 22365, 22995, 23205, 24885, 25935, 26145, 26565, 28035, 28215

Offset: 1

N. J. A. Sloane

nonn

Dickson proves that there are only a finite number of odd primitive abundant numbers having n distinct prime factors. Sequence A188342 lists the smallest such numbers. - T. D. Noe, Mar 28 2011
Sequence A188439 sorts the numbers in this sequence by the number of distinct prime factors. Eight numbers have exactly three prime factors; 576 have exactly four prime factors. - T. D. Noe, Apr 04 2011
Any multiple of an abundant number (A005101) is again an abundant number. Primitive abundant numbers (A091191) are those not of this form, i.e., without an abundant proper divisor. We don't know any odd perfect number (A000396), so the (odd) terms here have only deficient proper divisors (A071395), and their prime factors p are less than sigma(n/p)/deficiency(n/p). See A005231 (odd abundant numbers) for an explanation why all terms have at least 3 distinct prime factors, and 5 prime factors when counted with multiplicity (A001222), whence a(1) = 3^3*5*7. All known terms are semiperfect (A005835, and thus in A006036): no odd weird number (A006037) is known, but if it exists, the smallest one is in this sequence. - M. F. Hasler, Jul 28 2016
So far, a(173) = 351351 is the only known term of A122036, i.e., which can't be written as sum of its proper divisors > 1. - M. F. Hasler, Jan 26 2020

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
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.
R. K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991
Jacob Liddy, An algorithm to determine all odd primitive abundant numbers with d prime divisors, Honors Research Projects (2018), 728.
Eric Weisstein's World of Mathematics, Primitive Abundant Number

Cf. A005101, A005231. Subsequence of A091191.

Cf. A000203, A027751, A379949 (subsequence of square terms).

a006038 n = a006038_list !! (n-1)
a006038_list = filter f [1, 3 ..] where
   f x = sum pdivs > x && all (<= 0) (map (\d -> a000203 d - 2 * d) pdivs)
         where pdivs = a027751_row x
-- Reinhard Zumkeller, Jan 31 2014

isA005101 := proc(n) is(numtheory[sigma](n) > 2*n ); end proc:
isA005100 := proc(n) is(numtheory[sigma](n) < 2*n ); end proc:
isA006038 := proc(n) local d; if type(n,'odd') and isA005101(n) then for d in numtheory[divisors](n) minus {1,n} do if not isA005100(d) then return false; end if; end do: return true;else false; end if; end proc:
n := 1 ; for a from 1 by 2 do if isA006038(a) then printf("%d %d\n",n,a) ; n := n+1 ; end if; end do: # R. J. Mathar, Mar 28 2011

t = {}; n = 1; While[Length[t] < 50, n = n + 2; If[DivisorSigma[1, n] > 2 n && Intersection[t, Divisors[n]] == {}, AppendTo[t, n]]]; t (* T. D. Noe, Mar 28 2011 *)

is(n)=n%2 && sumdiv(n,d,sigma(d,-1)>2)==1 \\ Charles R Greathouse IV, Jun 10 2013

is_A006038(n)=bittest(n,0) && sigma(n)>2*n && !for(i=1,#f=factor(n)[,1],sigma(n\f[i],-1)>2&&return) \\ More than 5 times faster. - M. F. Hasler, Jul 28 2016

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

A275449 Least odd primitive abundant number with n prime factors, counted with multiplicity.

Original entry on oeis.org

945, 7425, 81081, 78975, 1468935, 6375105, 85930875, 307879299, 1519691625, 8853249375, 17062700625, 535868474337, 2241870572475, 12759034818375, 64260996890625, 866566808687853, 2964430488515625, 23849823423763953, 100139192108634825, 772934641006640625, 2696807941801171875

Offset: 5

M. F. Hasler, Jul 27 2016

nonn

See A188342 = (945, 3465, 15015, 692835, 22309287, ...) for the least odd primitive abundant number (A006038) with n distinct prime factors.
At least up to a(11), the greatest prime factor gpf(a(n)) = Q(a(n)/gpf(a(n))), where Q(N) = floor(sigma(N)/(2N-sigma(N))). In general one has to apply the precprime() function A007917 to this integer.
The above holds also for a(12)-a(15). Lars Blomberg, Apr 09 2018

			We have:   a(5) = 945 = 3^3 * 5   * 7,
          a(6) = 7425 = 3^3 * 5^2 * 11,
         a(7) = 81081 = 3^4 *  7  * 11 * 13,
        a(8) =  78975 = 3^5 * 5^2 * 13,
       a(9) = 1468935 = 3^6 * 5   * 13 * 31,
      a(10) = 6375105 = 3^7 * 5   * 11 * 53,
     a(11) = 85930875 = 3^6 * 5^3 * 23 * 41,
    a(12) = 307879299 = 3^7 * 7^2 * 13^2 * 17,
   a(13) = 1519691625 = 3^8 * 5^3 * 17 * 109,
   a(14) = 8853249375 = 3^8 * 5^4 * 17 * 127,
  a(15) = 17062700625 = 3^9 * 5^4 * 19 * 73.

Cf. A006038, A188342, A188439.

a(n)=for(i=1,#A=A006038,bigomega(A[i])==n&&return(A[i])) \\ Provided that A006038 is defined as a set with enough elements. - M. F. Hasler, Jul 27 2016

generate(A, B, n) = A=max(A, 3^n); (f(m, p, k) = my(list=List()); if(sigma(m) > 2*m, return(list)); if(k==1, forprime(q=max(p, ceil(A/m)), B\m, my(t=m*q); if(sigma(t) > 2*t, my(F=factor(t)[,1], ok=1); for(i=1, #F, if(sigma(t\F[i], -1) > 2, ok=0; break)); if(ok, listput(list, t)))), forprime(q = p, sqrtnint(B\m, k), list=concat(list, f(m*q, q, k-1)))); list); vecsort(Vec(f(1, 3, n)));
a(n) = my(x=3^n, 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(12)-a(15) from Lars Blomberg, Apr 09 2018

a(16)-a(25) from Daniel Suteu, Feb 10 2024

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

A287646 Irregular triangle read by rows where row n lists all odd primitive abundant numbers with n prime factors, counted with multiplicity.

Original entry on oeis.org

945, 1575, 2205, 3465, 4095, 5355, 5775, 5985, 6435, 6825, 7245, 8085, 8415, 8925, 9135, 9555, 9765, 11655, 12705, 12915, 13545, 14805, 15015, 16695, 18585, 19215, 19635, 21105, 21945, 22365, 22995, 23205, 24885, 25935, 26145, 26565, 28035, 30555, 31395, 31815, 32445, 33495

Offset: 5

M. F. Hasler, May 30 2017

This triangle is the analog of A188439 for A001222 ("bigomega", total number of prime factors) instead of A001221 ("omega", distinct prime divisors). It starts with row 5, since there is no odd primitive abundant number, N in A006038, with less than A001222(N) = 5 prime factors (counted with multiplicity).
Sequence A287728 gives the row lengths: Row 5 has 121 terms (945, 1575, 2205, 3465, 4095, ..., 430815, 437745, 442365). This mostly equals the initial terms of A006038, except for those with indices {12, 39, 40, 45, 48, 54, ..., 87}. These are in turn mostly (except for the 17th and 18th term) those of the subsequent row 6 which has 15772 terms, (7425, 28215, 29835, 33345, 34155, ..., 13443355695, 13446051465, 13455037365).
Sequences A275449 and A287581 give the smallest and largest* element of each row (*assuming that the largest term in the row is squarefree). Accordingly, row 7 starts with A275449(7) = 81081, and ends with A287581(7) = 1725553747427327895.

Cf. A188439, A001222, A006038, A287728, A275449, A287581, A338133.

A287646_row( r, p=3, a=2, n=1/(a-1))={ r>1 || return(if(n>=p, primes([p,n]))); p(p-1)*a && p-1/p^(r-1)<(p-1)*a,[p^r],[]),ap=1,np=nextprime(p+1)); until( 0, if( (1+1/np)^(r-e) > (aa = a/ap += 1/p^e) && aa > 1, if(n=A287646_row(r-e,np,aa), if(e>1, my(aaa=a/(ap-1/p^e)); n=select(t->sigma(t,-1)1 || n || break; np=nextprime((e=ap=1)+p=np)); S}

A303933 a(n) is the number of odd primitive abundant numbers with n distinct prime divisors.

Original entry on oeis.org

0, 0, 8, 576, 3913172

Offset: 1

Jacob Liddy, May 02 2018

The 6th term is possibly 59687996404445, but this is at present unverified. - Jacob Liddy, Oct 16 2018

Valdas Diciunas, On the number of odd primitive abundant numbers with five and six distinct prime factors, Vilnius Conference in Combinatorics and Number Theory, page 12, 2017.

Valdas Diciunas, On the number of odd primitive abundant numbers with five and six distinct prime factors, slides, 2017.
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.
Jacob Liddy, OPANS (Odd Primitive Abundant NumberS), 2018. The downloadable file is a Zip-archive, 640 Megabytes, uncompresses to 10 Gigabytes.

Cf. A005231 (odd abundant), A006038 (odd primitive abundant).

Row lengths of A188439.

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A006038 Odd primitive abundant numbers.

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

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A275449 Least odd primitive abundant number with n prime factors, counted with multiplicity.

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

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A287646 Irregular triangle read by rows where row n lists all odd primitive abundant numbers with n prime factors, counted with multiplicity.

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A303933 a(n) is the number of odd primitive abundant numbers with n distinct prime divisors.

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