A243162 -id:A243162 - cp's OEIS frontend

A147553 Numbers k such that k^2 divides k.k where dot "." means concatenation.

1, 143, 142857143, 142857142857143, 142857142857142857143, 142857142857142857142857143, 142857142857142857142857142857143, 142857142857142857142857142857142857143

Offset: 1

Farideh Firoozbakht, Dec 23 2008

I proved that for n > 0, a(n+1) = (10^(6n-3) + 1)/7. Namely for n > 1, a(n) is of the form 142857.142857. ... .142857.143. Except for a(1), 11 divides all terms, so there is no prime p such that p^2 divides p.p. For n > 1, a(n).a(n)/(a(n)*a(n))=7.

			143*143|143.143 (143143/(143*143)=7) so 143 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..167
Index entries for linear recurrences with constant coefficients, signature (1000001,-1000000).

Cf. A147554, A243162 (k^2 divides k.k.k).

1, seq((10^(6*n-3)+1)/7, n=1..20); # Robert Israel, Sep 26 2016

a[0]=1;a[n_]:=(10^(6n-3)+1)/7;Table[a[k],{k,0,8}]
Do[d=Divisors[10^i+1];s=Select[d,Length[IntegerDigits[#]]==i&];If[Length[s]>0,Do[Print[s[[j]]],{j,Length[s]}]],{i,69}] (* Hans Havermann, May 31 2014 *)
LinearRecurrence[{1000001,-1000000},{1,143,142857143},20] (* Harvey P. Dale, Apr 02 2018 *)

A147554 Primes p such that p^2 divides p.p.p where dot "." means concatenation.

Original entry on oeis.org

3, 13, 37, 9901, 333667, 99990001, 999999000001, 9999999900000001, 13168164561429877, 130654897808007778425046117

Offset: 1

Farideh Firoozbakht, Dec 26 2008

Primes p dividing 10^(2*d)+10^d+1 where d=ceiling(log(p)/log(10)) is the number of decimal digits in p. - Max Alekseyev
There is no prime p such that p^2 divides p.p.
All primes of the forms 10^(2m) - 10^m + 1 or (1/3)*(10^(2m) + 10^m + 1) are in the sequence.
Primes in A243162. - Hans Havermann, May 31 2014
a(11) > 10^158. - Max Alekseyev, Sep 11 2024

Cf. A147553, A243162.

cp's OEIS Frontend

A147553 Numbers k such that k^2 divides k.k where dot "." means concatenation.

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