A079835 Smallest number such that there are exactly n numbers between a(n) and a(n+1) with n divisors.
0, 2, 6, 50, 59, 28562, 28709, 594823322, 594823354, 596971490, 596973489, 52599132235830050, 52599132235830326, 59091511031674153381442, 59091511031674153416513, 118313372322850068047045, 118313372322850068047119, 117257864492369852051862561201602
Offset: 1
Keywords
Examples
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.
Programs
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Maple
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, Mar 23 2007
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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
Extensions
2 more terms from R. J. Mathar, Mar 23 2007
a(8) onwards from Lambert Herrgesell (zero815(AT)googlemail.com), Feb 08 2008, Apr 10 2008