A347818 Smallest n-digit brilliant number.

4, 10, 121, 1003, 10201, 100013, 1018081, 10000043, 100140049, 1000000081, 10000600009, 100000000147, 1000006000009, 10000000000073, 100000380000361, 1000000000000003, 10000001400000049, 100000000000000831, 1000000014000000049, 10000000000000000049, 100000000380000000361

Offset

1,1

Comments

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.

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)?

Links

Table of n, a(n) for n=1..21.

Dario Alejandro Alpern, Brilliant numbers

World of Numbers, Smallest n-digit prp

Formula

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

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

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

a(n) >= A098449(n).

Example

a(6) =    100013 =   103 * 971.
a(7) =   1018081 =  1009 * 1009.
a(8) =  10000043 =  2089 * 4787.
a(9) = 100140049 = 10007 * 10007.

Mathematica

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 *)

Prog

(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]))
A084476(n)=for(k=0, 10^n, if(isA078972(10^(2*n-1)+k), return(k)))
a(n)=if(n%2, nextprime(10^((n-1)/2))^2, 10^(n-1)+A084476(n/2)) \\ after Charles R Greathouse IV in A078972

Crossrefs

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

Sequence in context: A118378 A203224 A198156 * A273517 A370687 A003086

Adjacent sequences: A347815 A347816 A347817 * A347819 A347820 A347821

Keyword

nonn,base

Author

Eric Chen, Sep 15 2021

Status

approved