A084475 -id:A084475 - cp's OEIS frontend

A179975 Smallest k such that k*10^n is a sum of two successive primes.

5, 3, 1, 6, 6, 6, 14, 6, 9, 19, 21, 21, 42, 93, 21, 6, 11, 2, 12, 111, 37, 39, 63, 38, 42, 24, 15, 15, 60, 6, 39, 82, 47, 58, 337, 49, 72, 25, 34, 21, 6, 107, 128, 96, 20, 2, 63, 231, 70, 7, 62, 144, 28, 151, 157, 33, 98, 55, 134, 162, 87, 201, 124, 303, 64, 106, 130, 13, 43

Offset: 0

Zak Seidov, Aug 04 2010

nonn

From Robert G. Wilson v, Aug 11 2010: (Start)
A179975 n's such that a(n)=1: 3, 335, ..., .
A179975 First occurrence of k: 3, 18, 2, ???, 1, 4, 50, 162, 9, 335, 17, 19, 68, 7, 27, ..., .
Records: 5, 6, 14, 19, 21, 42, 93, 111, 337, 449, 862, 1049, 1062, 1122, 1280, 2278, 3168, 4290, ..., . (End)

			a(0)=5 because 5=2+3
a(1)=3 because 30=13+17
a(2)=1 because 100=47+53
a(3)=6 because 6000=2999+3001.

Robert G. Wilson v, Table of n, a(n) for n = 0..400.
Dario Alejandro Alpern, Brilliant numbers

Cf. A064397, A071220, A074924, A074925.

Cf. A033873, A033874, A005235, A055211, A038804, A084475.

Join[{5,3},Reap[Do[Do[n=10^m k; If[n==PreviousPrime[n/2]+NextPrime[n/2],Sow[k];Break[]],{k,2000}],{m,2,50}]][[2,1]]]
f[n_] := Block[{k = 1, tn = 10^n}, While[h = k*tn/2; NextPrime[h, -1] + NextPrime@h != k*tn, k++ ]; k]; f[1] = 3; Array[f, 70, 0] (* Robert G. Wilson v, Aug 11 2010 *)

More terms from Robert G. Wilson v, Aug 11 2010

A084476 Least k such that 10^(2n-1)+k is a brilliant number.

Original entry on oeis.org

0, 3, 13, 43, 81, 147, 73, 3, 831, 49, 987, 691, 183, 4153, 279, 667, 709, 277, 1687, 997, 1207, 91, 1411, 393, 951, 9793, 2217, 6229, 2317, 213, 399, 19, 2317, 609, 2607, 11901, 10563, 5473, 3, 5923, 17527, 8569, 16701, 11989, 9757, 6489, 3489, 2899

Offset: 1

Robert G. Wilson v, Jun 27 2003

Least brilliant number greater than 10^(2n) is {10^n+A033873(n)}^2. The web site also lists the two prime factors.

			a(3)=13 because 10^5+13 = 100013 = 103*971.

Harry Metrebian, Table of n, a(n) for n = 1..65
Dario Alejandro Alpern, Brilliant numbers

Cf. A078972, A084475.

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[2n + 1], {n, 1, 24}]

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

A374350 Least n-digit reversible prime whose difference from its reversal is minimal.

Original entry on oeis.org

2, 11, 101, 1231, 10301, 105601, 1003001, 10012001, 100030001, 1007457001, 10000500001, 100124521001, 1000008000001, 10000523500001, 100000323000001, 1000034344300001, 10000000500000001, 100000188981000001, 1000000008000000001, 10000001189110000001, 100000000212000000001

Offset: 1

Robert G. Wilson v, Jul 05 2024

Inspired by A084475 and A373349.

For n > 1, a(2n) has a difference of 9*10^n and a(2n-1) has a difference of 0.

			a(3) = 101 since its reversal is also 101;
a(4) = 1231 since its reversal is 1321 which is also prime and their difference is minimal at 90;
a(6) = 105601 since its reversal is 106501 which is also prime and their difference is minimal at 900;
a(8) = 10012001 since its reversal is 10021001 which is also prime and their difference is minimal at 9000; etc.

Cf. A028989, A084475, A100027, A114018, A373349, A374377.

fe[n_] := Block[{k = 1, j, p, q}, While[ j = k(10^IntegerLength[k]) + IntegerReverse[k +1]; p = 10^(2 n -1) + j(10^(n - IntegerLength[j]/2)) + 1; q = IntegerReverse@ p; !PrimeQ@ p || !PrimeQ@ q, k++; If[ Mod[k, 10] == 9, k++]]; p]; fe[1] = 11;
fo[n_] := Block[{k = 0, j, p}, While[ j = k(10^(IntegerLength[k] -1)) + IntegerReverse@ Quotient[k, 10]; p = 10^(2n -2) + j(10^(n - (IntegerLength[j] + 1)/2)) +1; !PrimeQ@ p, k++]; p];
a[n_] := If[ OddQ@ n, fo[(n +1)/2], fe[n/2]]; Array[a, 21]

a(2n-1) = A100027(n) = A028989(n).

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

A179975 Smallest k such that k*10^n is a sum of two successive primes.

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A084476 Least k such that 10^(2n-1)+k is a brilliant number.

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

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A374350 Least n-digit reversible prime whose difference from its reversal is minimal.

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A347818 Smallest n-digit brilliant number.

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