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 A064476 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. 12

%I

%S 1,6,12,36,72,144,216,432,864,1296,1728,2592,5184,7776,10368,15552,

%T 20736,31104,46656,62208,93312,124416,186624,248832,279936,373248,

%U 559872,746496,1119744,1492992,1679616,2239488,2985984,3359232,4478976

%N 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.

%C Could be generalized by defining x* = (p_1+v)*(p_2+v) .. (p_n+v) where v is any integer.

%C 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

%H Harry J. Smith, <a href="/A064476/b064476.txt">Table of n, a(n) for n = 1..50</a>

%e 12 is in the sequence because 12 = 2 * 2 * 3, so 12* is 3 * 3 * 4 = 36 and 36 is divisible by 12.

%p with(numtheory); ListA064476:=proc(q) local a,b,i,n;

%p for n from 1 to q do a:=ifactors(n)[2]; b:=mul((a[i][1]+1)^a[i][2],i=1..nops(a)); if type(b/n,integer) then print(n); fi; od; end: ListA064476(10^6); # _Paolo P. Lava_, Jul 02 2013

%t diQ[n_]:=Divisible[Times@@(#+1&/@Flatten[Table[First[#],{Last[#]}]&/@ FactorInteger[n]]),n]; Select[Range[4500000],diQ] (* _Harvey P. Dale_, Aug 16 2011 *)

%o (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).

%o (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

%o a064476 n = a064476_list !! (n-1)

%o a064476_list = filter (\x -> a003959 x `mod` x == 0) [1..]

%o -- _Reinhard Zumkeller_, Feb 28 2013

%Y Cf. A064478, A064514, A064515, A064518, A064522, A003959.

%K nonn,easy,nice

%O 1,2

%A Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 06 2001

%E More terms from _Vladeta Jovovic_, Oct 07 2001

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Last modified October 21 21:46 EDT 2019. Contains 328315 sequences. (Running on oeis4.)