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A079835
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Smallest number such that there are exactly n numbers between a(n) and a(n+1) with n divisors.
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3
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0, 2, 6, 50, 59, 28562, 28709, 594823322, 594823354, 596971490, 596973489, 52599132235830050, 52599132235830326, 59091511031674153381442, 59091511031674153416513, 118313372322850068047045, 118313372322850068047119, 117257864492369852051862561201602
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OFFSET
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1,2
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LINKS
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EXAMPLE
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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.
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MAPLE
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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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PROG
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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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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a(8) onwards from Lambert Herrgesell (zero815(AT)googlemail.com), Feb 08 2008, Apr 10 2008
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STATUS
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approved
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