A078400 -id:A078400 - cp's OEIS frontend

A078403 Primes whose digital root (A038194) is prime.

2, 3, 5, 7, 11, 23, 29, 41, 43, 47, 59, 61, 79, 83, 97, 101, 113, 131, 137, 149, 151, 167, 173, 191, 223, 227, 239, 241, 257, 263, 277, 281, 293, 311, 313, 317, 331, 347, 349, 353, 367, 383, 389, 401, 419, 421, 439, 443, 457, 461, 479, 491, 509, 547, 563, 569

Offset: 1

N. J. A. Sloane, Dec 26 2002

Union of A061238, A061240, A061241 and 3. - Ya-Ping Lu, Jan 03 2024

			59 is a term because 5+9=14, giving (final) iterated sum 1+4=5 and 5 is prime.

Cf. A038194, A061238, A061240, A061241, A070027, A078400.

Select[ Range[580], PrimeQ[ # ] && PrimeQ[Mod[ #, 9]] &]
Select[Prime[Range[200]],PrimeQ[Mod[#,9]]&] (* Harvey P. Dale, Aug 20 2017 *)

forprime(p=2,997,if(isprime(p%9),print1(p,",")))

from sympy import isprime, primerange; [print(p, end = ', ') for p in primerange(2, 570) if isprime(p%9)] # Ya-Ping Lu, Jan 03 2024

a(n) ~ 2n log n. - Charles R Greathouse IV, May 14 2025

Extended by Robert G. Wilson v, Klaus Brockhaus and Rick L. Shepherd, Dec 27 2002

A079155 The number of primes less than 10^n whose digital root (A038194) is also prime.

Original entry on oeis.org

4, 15, 85, 619, 4800, 39266, 332276, 2880818, 25423985, 227527467

Offset: 1

Robert G. Wilson v, Dec 27 2002

			a(2) = 15 because the only primes less than 100 whose have digital roots are also prime are {2,3,5,7,11,23,29,41,43,47,59,61,79,83,97}.

The primes are in A078403, their digital roots are in A078400.

c = 0; k = 1; Do[ While[ k < 10^n, If[ PrimeQ[k] && PrimeQ[ Mod[k, 9]], c++ ]; k++ ]; Print[c], {n, 1, 8}]

# use primerange (slower) vs. sieve.primerange (>> memory) for larger terms
from sympy import isprime, sieve
def afind(terms):
  s = 0
  for n in range(1, terms+1):
    s += sum(isprime(p%9) for p in sieve.primerange(10**(n-1), 10**n))
    print(s, end=", ")
afind(7) # Michael S. Branicky, Apr 15 2021

a(9)-a(10) from Michael S. Branicky, Apr 15 2021

cp's OEIS Frontend

A078403 Primes whose digital root (A038194) is prime.

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