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A064476
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For an integer n with prime factorization p_1*p_2*p_3* ... *p_m let n* = (p_1+1)*(p_2+1)*(p_3+1)* ... *(p_m+1) (A064478); sequence gives n such that n* is divisible by n.
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12
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1, 6, 12, 36, 72, 144, 216, 432, 864, 1296, 1728, 2592, 5184, 7776, 10368, 15552, 20736, 31104, 46656, 62208, 93312, 124416, 186624, 248832, 279936, 373248, 559872, 746496, 1119744, 1492992, 1679616, 2239488, 2985984, 3359232, 4478976
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OFFSET
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1,2
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COMMENTS
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Could be generalized by defining x* = (p_1+v)*(p_2+v) .. (p_n+v) where v is any integer.
It is not difficult to show that these numbers have the form a(n) = 2^i*3^j with j <= i <= 2j. Hence 1 is the only odd term; also if n|n* then n*|n**. The values of i and j are given in A064514 and A064515. - Vladeta Jovovic and N. J. A. Sloane, Oct 07 2001
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LINKS
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EXAMPLE
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12 is in the sequence because 12 = 2 * 2 * 3, so 12* is 3 * 3 * 4 = 36 and 36 is divisible by 12.
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MATHEMATICA
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diQ[n_]:=Divisible[Times@@(#+1&/@Flatten[Table[First[#], {Last[#]}]&/@ FactorInteger[n]]), n]; Select[Range[4500000], diQ] (* Harvey P. Dale, Aug 16 2011 *)
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PROG
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(ARIBAS): function p2p3(stop:integer): array; var c, i, j, x: integer; b: boolean; ar: array; begin ar := alloc(array, stop); x := 0; c := 0; b := c < stop; while b do i := x; j := x - i; while b and i >= j do if i <= 2*j then ar[c] := (2^i * 3^j, i, j); inc(c); b := c < stop; end; dec(i); inc(j); end; inc(x); end; return sort(ar, comparefirst); end; function comparefirst(x, y: array): integer; begin return y[0] - x[0]; end; function a064476(maxarg: integer); var j: integer; ar: array; begin ar := p2p3(maxarg); for j := 0 to maxarg - 1 do write(ar[j][0], " "); end; end; a064476(35).
(PARI) ns(n)= { local(f, p=1); f=factor(n); for(i=1, matsize(f)[1], p*=(1 + f[i, 1])^f[i, 2]); return(p) } { n=0; for (m=1, 10^9, if (ns(m)%m == 0, write("b064476.txt", n++, " ", m); if (n==100, break)) ) } \\ Harry J. Smith, Sep 15 2009
(Haskell)
a064476 n = a064476_list !! (n-1)
a064476_list = filter (\x -> a003959 x `mod` x == 0) [1..]
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 06 2001
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EXTENSIONS
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STATUS
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approved
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