A275449 -id:A275449 - cp's OEIS frontend

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

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

A285993 Largest odd abundant number (A005231) equal to the product of n consecutive primes.

Original entry on oeis.org

15015, 255255, 4849845, 111546435, 33426748355, 1236789689135, 50708377254535, 2180460221945005, 102481630431415235, 5431526412865007455, 320460058359035439845, 19548063559901161830545, 1309720258513377842646515, 1357656019974967471687377449, 107254825578022430263302818471

Offset: 5

M. F. Hasler, Apr 30 2017

nonn

The smallest term is a(5) = 3*5*7*11*13, there is no odd abundant number (A005231) equal to the product of less than 5 consecutive primes.
The smallest odd abundant number (A005231) equal to the product of n consecutive primes is equal (when it exists, i.e., for n >= 5) to the least odd number with n (distinct) prime divisors, equal to the product of the first n odd primes = A070826(n+1) = A002110(n+1)/2.
See A188342 = (945, 3465, 15015, 692835, 22309287, ...) for the least odd primitive abundant number (A006038) with n distinct prime factors, and A275449 for the least odd primitive abundant number with n prime factors counted with multiplicity.
The terms are in general not primitive abundant numbers (A091191), in particular this cannot be the case when a(n) is a multiple of a(n-1), as is the case for most of the terms, for which a(n) = a(n-1)*A117366(a(n-1)). In the other event, spf(a(n)) = nextprime(spf(a(n-1))), and a(n) is in A007741(2,3,4...). These are exactly the primitive terms in this sequence.

			For n < 5, there is no odd abundant number equal to the product of n distinct primes.
For 5 <= n <= 8, the largest odd abundant number equal to the product of n consecutive primes is 3*...*prime(n+1).
For 9 <= n <= 17, the largest odd abundant number equal to the product of n consecutive primes is 5*...*prime(n+2).
For 18 <= n <= 30, the largest odd abundant number equal to the product of n consecutive primes is 7*...*prime(n+3).
For 31 <= n <= 45, the largest odd abundant number equal to the product of n consecutive primes is 11*...*prime(n+4).
For 46 <= n <= 66, the largest odd abundant number equal to the product of n consecutive primes is 13*...*prime(n+5).

Amiram Eldar, Table of n, a(n) for n = 5..341

Cf. A005231, A006038, A007707, A007708, A007741.

A subsequence of A112643 (odd squarefree abundant numbers); see also A108227 (~ A107705) which give indices of primitive terms = those with smallest prime factor larger than that of earlier terms.

a(r,f=vector(r,i,prime(i+1)),o)={ while(sigma(factorback(f),-1)>2, o=f; f=concat(f[^1],nextprime(f[r]+1)));factorback(o)} \\ Intentionally throws an error when n < 5.

a(n) >= a(n-1)*p where p = A117366(a(n-1)) = A151800(A006530(a(n-1))) = nextprime(gpf(a(n-1))), an odd abundant number equal to the product of n consecutive primes. We have strict inequality for n = 9, 18, 31, 46, 67, ..., in which case a(n) = a(n-1)*p*p'/q, where p' = nextprime(p), q = least prime factor of a(n-1). This is the case if a(n) is in A007741.

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

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A285993 Largest odd abundant number (A005231) equal to the product of n consecutive primes.

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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