A287581 -id:A287581 - 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

A287590 Number of squarefree odd primitive abundant numbers with n prime factors.

Original entry on oeis.org

0, 0, 0, 0, 87, 14172, 101053625, 3475496953795289

Offset: 1

M. F. Hasler, May 26 2017

See A287581 for the largest squarefree odd primitive abundant number (A249263) with n prime factors.

Squarefree odd primitive abundant numbers (SOPAN) with r prime factors are of the form N = p_1 * ... * p_r with 3 <= p_1 < ... < p_r and such that the abundancy A(p_1 * ... * p_k) < 2 for k < r and > 2 for k = r, where A(N) = sigma(N)/N. For r < 5 this can never be satisfied, the largest possible value is A(3*5*7*11) = 2 - 2/385.

			From _M. F. Hasler_, Jun 26 2017: (Start)
All squarefree odd primitive abundant numbers (SOPAN) have at least 5 prime factors, since the abundancy of a product of 4 distinct odd primes cannot be larger than that of N = 3*5*7*11, with A000203(N)/N = 4/3 * 6/5 * 8/7 * 12/11 = 768/385 = 2 - 2/385 < 2.
The 87 SOPAN with 5 prime factors range from A249263(1) = 15015 = 3*5*7*11*13 to  A287581(5) = A249263(87) = 442365 = 3*5*7*11*383.
The 14172 SOPAN with 6 prime factors range from A188342(6) = A249263(88) = 692835 = 3*5*11*13*17*19 to A287581(6) = 13455037365  = 3*5*7*11*389*29947.
The 101053625 SOPAN with 7 prime factors range from A188342(7) = A249263(608) = 22309287 = 3*7*11*13*17*19*23 to A287581(7) = 1725553747427327895 = 3*5*7*11*389*29959*128194559. (End)

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

Cf. A249263, A287581.

A287590(r,p=2,a=2,s=0,n=precprime(1\(a-1)))={ r>1 || return(primepi(n)-primepi(p)); (pa && while( 0A287590(r-1,p=nextprime(p+1),a/(1+1/p)),s+=n); s}

Added a(8) calculated by Gianluca Amato. - M. F. Hasler, Jun 26 2017

Example for 101053625 corrected by Peter Munn, Jul 23 2017

A338427 a(n) is the largest prime(n)-smooth primitive nondeficient number.

Original entry on oeis.org

6, 20, 2205, 12705, 117234117, 42840834309, 2792098376579421, 674431969285588989475, 21526530767769616227341527825, 292210459765634328314801626540200511773, 292210459765634328314801626540200511773

Offset: 2

David A. Corneth and Peter Munn, Oct 26 2020

nonn

See A006039 for a definition and list of primitive nondeficient numbers.
The first prime being 2, the prime(1)-smooth numbers are the powers of 2, which are all deficient. So a(1) is undefined, and the sequence offset is 2.
Omitting the initial "6" gives us the largest prime(n)-smooth primitive abundant numbers (based on their A071395 definition). Using the variant definition of primitive abundant from A091191, the equivalent sequence starts 18, 30, 2205, 12705, 117234117, ... .
If m is a prime(n)-smooth primitive nondeficient number, the odd part of m divides a member of one of the first (n - 1) finite sets described in the Dickson reference and the even part of m is less than 2^A035100(n). This provides an upper bound for such numbers, meaning there is a largest prime(n)-smooth primitive nondeficient number for all n >= 2.

			Initial terms, showing factorization:
   n          a(n)
   2             6 = 2 * 3,
   3            20 = 2^2 * 5,
   4          2205 = 3^2 * 5 * 7^2,
   5         12705 = 3 * 5 * 7 * 11^2,
   6     117234117 = 3^2 * 7^2 * 11^2 * 13^3,
   7   42840834309 = 3^4 * 7^2 * 13^3 * 17^3,
   ...
The largest primitive nondeficient (and primitive abundant) number that has prime(12) = 37 as largest prime factor is 29504726357465429322218597476548828125, which is one digit shorter than the largest 31-smooth primitive nondeficient (and primitive abundant) number, 292210459765634328314801626540200511773. So a(12) = a(11).

Peter Munn, Table of n, a(n) for n = 2..18
L. E. Dickson, Finiteness of the Odd Perfect and Primitive Abundant Numbers with n Distinct Prime Factors, Amer. J. Math., 35 (1913), 413-426.
Peter Munn, PARI program
Eric Weisstein's World of Mathematics, Smooth Number.
Index entries for sequences related to sigma(n)

After removing duplicate terms we get a subsequence of A006039, A338133.
Cf. A000040, A006530, A035100, A071395, A091191.
The largest prime(n)-smooth numbers meeting other divisor-related criteria: A211198, A273057.
Largest primitive nondeficient numbers meeting other criteria: A287581.

a(n) = Max_{m <= n, k >= 1} A338133(m, k).

a(n) = max( {m in A006039 : A006530(m) <= A000040(n)} ).

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}

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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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A287590 Number of squarefree odd primitive abundant numbers with n prime factors.

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A338427 a(n) is the largest prime(n)-smooth primitive nondeficient number.

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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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PARI