A084476 -id:A084476 - cp's OEIS frontend

A084475 a(n) defines the first brilliant number, b_n, greater than 10^n. If n is odd or zero, then b_n is 10^n+a(n); and if n is a positive even number, then b_n is {10^(n/2)+a(n)}^2.

3, 0, 1, 3, 1, 13, 9, 43, 7, 81, 3, 147, 3, 73, 19, 3, 7, 831, 7, 49, 19, 987, 3, 691, 39, 183, 37, 4153, 31, 279, 37, 667, 61, 709, 3, 277, 3, 1687, 51, 997, 39, 1207, 117, 91, 9, 1411, 117, 393, 7, 951, 13, 9793, 67, 2217, 103, 6229, 331, 2317, 319, 213, 57, 399, 33, 19

Offset: 0

Jason Earls, Jun 03 2003

			a(5)=13 because 10^5+13 = 100013 = 103*971 and a(6)=9 because (10^3+9)^2 = 1009^2. For n>0, a(2n) = A033873(n).

Harry Metrebian, Table of n, a(n) for n = 0..130
Dario Alejandro Alpern, Brilliant numbers

Cf. A078972, A084476.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; LengthBase10[n_] := Floor[ Log[10, n] + 1]; f[n_] := Block[{k = 0}, If[ EvenQ[n] && n > 1, NextPrim[ 10^(n/2)]^2 - 10^(n/2), While[fi = FactorInteger[10^n + k]; Plus @@ Flatten[ Table[ # [[2]], {1}] & /@ fi] != 2 || Length[ Union[ LengthBase10 /@ Flatten[ Table[ # [[1]], {1}] & /@ fi]]] != 1, k++ ]; k]]; Table[ f[n], {n, 0, 63}]

Edited and extended by Robert G. Wilson v, Jun 27 2003

A083289 Least k such that 10^n+k is a brilliant number (cf. A078972).

Original entry on oeis.org

3, 0, 21, 3, 201, 13, 18081, 43, 140049, 81, 600009, 147, 6000009, 73, 380000361, 3, 1400000049, 831, 14000000049, 49, 380000000361, 987, 600000000009, 691, 78000000001521, 183, 740000000001369, 4153, 6200000000000961, 279

Offset: 0

Jason Earls, Jun 03 2003

If n is an even positive exponent, then a(n) is the first prime greater than 10^(n/2) squared less 10^n.

Chai Wah Wu, Table of n, a(n) for n = 0..42
Dario Alejandro Alpern, Brilliant numbers

Cf. A078972, A084475, A084476.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; LengthBase10[n_] := Floor[ Log[10, n] + 1]; f[n_] := Block[{k = 0}, If[ EvenQ[n] && n > 1, NextPrim[ 10^(n/2)]^2 - 10^n, While[fi = FactorInteger[10^n + k]; Plus @@ Flatten[ Table[ # [[2]], {1}] & /@ fi] != 2 || Length[ Union[ LengthBase10 /@ Flatten[ Table[ # [[1]], {1}] & /@ fi]]] != 1, k++ ]; k]]; Table[ f[n], {n, 0, 30}]

from sympy import nextprime, factorint
def A083289(n):
    a, b = divmod(n,2)
    c, d = 10**n, 10**a
    if b == 0: return nextprime(d)**2-c
    k = 0
    while True:
        fs = factorint(c+k,multiple=True)
        if len(fs) == 2 and min(fs) >= d:
            return k
        k += 1 # Chai Wah Wu, Sep 28 2021

Edited and extended by Robert G. Wilson v, Jun 27 2003

A347818 Smallest n-digit brilliant number.

Original entry on oeis.org

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

Offset: 1

Eric Chen, Sep 15 2021

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)?

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

Dario Alejandro Alpern, Brilliant numbers
World of Numbers, Smallest n-digit prp

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

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 *)

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

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).

cp's OEIS Frontend

A084475 a(n) defines the first brilliant number, b_n, greater than 10^n. If n is odd or zero, then b_n is 10^n+a(n); and if n is a positive even number, then b_n is {10^(n/2)+a(n)}^2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A083289 Least k such that 10^n+k is a brilliant number (cf. A078972).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A347818 Smallest n-digit brilliant number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula