A347818 - OEIS

%I #58 Jan 09 2024 19:22:01

%S 4,10,121,1003,10201,100013,1018081,10000043,100140049,1000000081,

%T 10000600009,100000000147,1000006000009,10000000000073,

%U 100000380000361,1000000000000003,10000001400000049,100000000000000831,1000000014000000049,10000000000000000049,100000000380000000361

%N Smallest n-digit brilliant number.

%C A brilliant number is a semiprime (products of two primes, A001358) whose two prime factors have the same number of decimal digits. For an n-digit brilliant number, the two prime factors must each have ceiling(n/2) decimal digits.

%C Since all brilliant numbers are semiprimes, a(n) >= A098449(n), also, a(n) = A098449(n) for n = 1, 2, 4, 16, 78, ..., are there infinitely many n such that a(n) = A098449(n)?

%H Dario Alejandro Alpern, <a href="https://www.alpertron.com.ar/BRILLIANT.HTM">Brilliant numbers</a>

%H World of Numbers, <a href="http://www.worldofnumbers.com/Smallest-n-digit-prp.pdf">Smallest n-digit prp</a>

%F a(n) = 10^(n-1) + A083289(n).

%F a(2*n) = 10^(2*n-1) + A084476(n).

%F a(2*n+1) = A003617(n+1)^2.

%F a(n) >= A098449(n).

%e a(6) =    100013 =   103 * 971.

%e a(7) =   1018081 =  1009 * 1009.

%e a(8) =  10000043 =  2089 * 4787.

%e a(9) = 100140049 = 10007 * 10007.

%t Join[{4,10},Table[Module[{k=1},While[PrimeOmega[10^n+k]!=2||Length[ Union[ IntegerLength/@ FactorInteger[ 10^n+k][[;;,1]]]]!=1,k+=2];10^n+k],{n,2,20}]] (* _Harvey P. Dale_, Jan 09 2024 *)

%o (PARI) isA078972(n)=my(f=factor(n)); (#f[, 1]==1 && f[1, 2]==2) || (#f[, 1]==2 && f[1, 2]==1 && f[2, 2]==1 && #Str(f[1, 1])==#Str(f[2, 1]))

%o A084476(n)=for(k=0,10^n,if(isA078972(10^(2*n-1)+k),return(k)))

%o a(n)=if(n%2,nextprime(10^((n-1)/2))^2,10^(n-1)+A084476(n/2)) \\ after _Charles R Greathouse IV_ in A078972

%Y Cf. A078972, A003617, A083289, A084475, A084476, A083128, A098449.

%K nonn,base

%O 1,1

%A _Eric Chen_, Sep 15 2021