A247552 Least numbers x such that the ratio of the sum of all the cyclic permutations of x, plus the unpermuted number, and x itself is equal to n.

1, 11, 111, 1111, 11111, 111111, 428571, 11111111, 111111111, 1111111111, 1818, 111111111111, 230769, 428571428571, 111111111111111, 1111111111111111, 4705882352941176, 111111111111111111, 473684210526315789, 11111111111111111111, 142857, 18181818

Offset

1,2

Comments

Mainly repdigit numbers with n 1's.

If x has m digits with sum s then the sum of the m cyclic permutations of x (including x itself) is s*(10^m-1)/9, since each digit occurs once in each position. My program uses this to test potential (m, s) pairs. - Jens Kruse Andersen, Sep 23 2014

If appears that the number of digits of a(n) is n-1 if and only if n is a full reptend prime (A001913). - Michel Marcus, Sep 24 2014

There are 106 repdigit numbers with n 1's in the first 5000 terms. - Jens Kruse Andersen, Sep 30 2014

Links

Jens Kruse Andersen, Table of n, a(n) for n = 1..1000

Example

428571 is the minimum number such that 428571 + 142857 + 714285 + 571428 + 857142 + 285714 = 2999997 and 2999997 / 428571 = 7.
1818 is the minimum number such that 1818 + 8181 + 1818 + 8181 = 19998 and 19998 / 1818 = 11.

Maple

P:=proc(q) local a, b, c, d, j, n, t, v;
v:=array(1..100); for j from 1 to 100 do v[j]:=0; od; t:=0;
for n from 1 to q do a:=n; b:=a; c:=ilog10(a);
for k from 1 to c do a:=(a mod 10)*10^c+trunc(a/10); b:=b+a; od;
if type(b/n, integer) then if b/n=t+1
then t:=t+1; lprint(t, n); while v[t+1]>0 do t:=t+1; lprint(t, v[t]); od;
else if b/n>t+1 then if v[b/n]=0 then v[b/n]:=n; fi; fi;
fi; fi; od; end: P(10^6);

Prog

(PARI) isok(n, k) = {d = digits(k); nbd = #d; sp = 0; for (i=1, nbd, dpk = vector(nbd-1, j, d[j+1]); dpk = concat(dpk, d[1]); sp += subst(Pol(dpk, x), x, 10); d = dpk; ); sp == k*n; }
a(n) = {k = 1; while(! isok(n, k), k++; ); k ; } \\ Michel Marcus, Sep 21 2014
(PARI) a(n)=my(r=0, m, g, s, x); for(m=1, n, r=10*r+1; g=n/gcd(r, n); forstep(s=g, 9*m, g, x=s*r/n; if(#digits(x)==m && sumdigits(x)==s, return(x))))
vector(30, n, a(n)) \\ Faster program. Jens Kruse Andersen, Sep 23 2014

Crossrefs

Cf. A247315, A247316, A247317.

Sequence in context: A283604 A090103 A061851 * A138145 A165370 A134962

Adjacent sequences: A247549 A247550 A247551 * A247553 A247554 A247555

Keyword

nonn,base

Author

Paolo P. Lava, Sep 19 2014

Extensions

a(12)-a(22) from Jens Kruse Andersen, Sep 23 2014

Status

approved