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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
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

Harry J. Smith, Table of n, a(n) for n = 1..50

EXAMPLE

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

MAPLE

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

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

MATHEMATICA

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

PROG

(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..]

-- Reinhard Zumkeller, Feb 28 2013

CROSSREFS

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

Sequence in context: A096932 A212976 A176681 * A324483 A239171 A264955

Adjacent sequences:  A064473 A064474 A064475 * A064477 A064478 A064479

KEYWORD

nonn,easy,nice

AUTHOR

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

EXTENSIONS

More terms from Vladeta Jovovic, Oct 07 2001

STATUS

approved

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Last modified July 21 19:57 EDT 2019. Contains 325199 sequences. (Running on oeis4.)