1,2

a(3) = 6 and a(4) = 50 as there are exactly 3 numbers 9, 25 and 49 with 3 divisors between 6 and 50.

Between 59 and 28562 there are {81, 625, 2401, 14641, 28561}, the fourth powers of the prime numbers from 3 to 13.

Between 28562 and 28709 there are {28577, 28604, 28629, 28636, 28683, 28708}, which are listed under A001222(.)=3.

Between 28709 and a(8) is 117649, the sixth power of 7. The "worst" case for a(8) is that A000040(4)^6 up to A000040(10)^6 are between a(7) and a(8), which yields the upper bound a(8) < A000040(10)^6+1=594823322.

Similar upper bound constructions with successive prime powers show that the sequence is infinite.

iscntdvs := proc(a, n) if nops(numtheory[divisors](a)) = n then 1 ; else 0 ; fi ; end: A079835 := proc(nmax) local a, n, anext, divcnt ; a := [0] ; while nops(a) < nmax do n := nops(a)+1 ; anext := op(-1, a)+1 ; divcnt := iscntdvs(anext, n-1) ; while divcnt < n-1 do anext := anext+1 ; divcnt := divcnt+iscntdvs(anext, n-1) ; od ; anext := anext+1 ; print(n, anext) ; a := [op(a), anext] ; od ; RETURN(a) ; end: A079835(100) ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 23 2007

(PARI) r=2; for(n=1, 10, s=0; for(i=r+1, prime(n+3)^n, if(numdiv(i)==n, s++; print(" ", s, ": ", i, " ", numdiv(i), " ", factor(i)); if(s==n, r=i+1; break))); print(n, " ", r); ) -Lambert Herrgesell (zero815(AT)googlemail.com), Feb 08 2008

Sequence in context: A086550 A080310 A103990 this_sequence A052332 A134047 A078464

Adjacent sequences: A079832 A079833 A079834 this_sequence A079836 A079837 A079838

nonn

Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Feb 15 2003

2 more terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 23 2007

a(8) onwards from Lambert Herrgesell (zero815(AT)googlemail.com), Feb 08 2008, Apr 10 2008