A054268 Sum of composite numbers between prime p and nextprime(p) is a repdigit.

3, 5, 109, 111111109, 259259257

Offset: 1

Author

Patrick De Geest, Apr 15 2000

Keywords

• nonn
• base
• hard

Comments

No additional terms below 472882027.

No additional terms below 10^58. - Chai Wah Wu, Jun 01 2024

Examples

			a(5) is ok since between 259259257 and nextprime 259259261 we get the sum 259259258 + 259259259 + 259259260 which yield repdigit 777777777.
		

Links

• Eric Weisstein's World of Mathematics, Repdigit

Crossrefs

Cf. A010785, A028987, A028988, A046933, A054264, A054265, A054266, A054267.

Programs

Mathematica

repQ[n_]:=Count[DigitCount[n],0]==9; Select[Prime[Range[2,14500000]], repQ[Total[Range[#+1,NextPrime[#]-1]]]&] (* Harvey P. Dale, Jan 29 2011 *)

Python

from sympy import prime
A054268 = [prime(n) for n in range(2,10**5) if len(set(str(int((prime(n+1)-prime(n)-1)*(prime(n+1)+prime(n))/2)))) == 1]
# Chai Wah Wu, Aug 12 2014

Python

from itertools import count, islice
from sympy import isprime, nextprime
from sympy.abc import x,y
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A054268_gen(): # generator of terms
    for l in count(1):
        c = []
        for m in range(1,10):
            k = m*(10**l-1)//9<<1
            for a, b in diop_quadratic((x-y-1)*(x+y)-k):
                if isprime(b) and a == nextprime(b):
                    c.append(b)
        yield from sorted(c)
A054268_list = list(islice(A054268_gen(),5)) # Chai Wah Wu, Jun 01 2024

Formula

Numbers A000040(n) for n > 1 such that A001043(n)*(A001223(n)-1)/2 is in A010785. - Chai Wah Wu, Aug 12 2014

Extensions

Offset changed by Andrew Howroyd, Aug 14 2024