A217090 -id:A217090 - cp's OEIS frontend

A138465 Non-optimus primes.

3, 23, 31, 137, 191, 239, 277, 359, 431, 439, 683, 719, 743, 911, 997, 1031, 1061, 1103, 1109, 1223, 1279, 1423, 1439, 1481, 1511, 1559, 1583, 1597, 1733, 1873, 2017, 2039, 2063, 2351, 2399, 2411, 2543, 2683, 2897, 2903, 3023, 3347, 3359, 3457, 3517, 3607, 3623, 3793, 3797

Offset: 1

N. J. A. Sloane, Feb 07 2009

nonn

A prime p is an optimus prime if (1 + sqrt( legendre(-1,p)*p ))^p - 1 = r + s*sqrt( legendre(-1,p)*p ) where gcd(r,s) = p.

			For p = 13, (1 + sqrt( legendre(-1,p)*p ))^p - 1 = 209588223+58200064*13^(1/2), and gcd(209588223,58200064) = 13, so 13 is an optimus prime.
For p = 23, (1 + sqrt( legendre(-1,p)*p ))^p - 1 = 7453766387236863-24397683359744*(-23)^(1/2), but gcd(7453766387236863,24397683359744) = 1081 != 23, so 23 is a non-optimus prime.

A. Slinko, Additive representability of finite measurement structures, in "The Mathematics of Preference, Choice and Order: Essays in Honor of Peter Fishburn", edited by Steven Brams, William V. Gehrlein and Fred S. Roberts, Springer, 2009, pp. 113-133.

Charles R Greathouse IV, Table of n, a(n) for n = 1..5000
S. Marshall, On the existence of extremal cones and comparative probability orderings, Proceedings of The 4th International Symposium on Imprecise Probabilities and Their Applications (ISIPTA 05), Pittsburg, Pennsylvania, 2005, pp. 246-255.
Arkadii Slinko, Additive Representability of Finite Measurement Structures, 2007, 26 pp.
Arkadii Slinko, Additive Representability of Finite Measurement Structures, 2007, 26 pp. [Cached copy]

Cf. A217090 (optimus primes).

is(p)=if(p<3 || !isprime(p),return(0)); my(t=(2*quadgen(kronecker(-1,p)*p))^p);gcd(imag(t),real(t)-1)!=p \\ Charles R Greathouse IV, Sep 26 2012

More terms from Charles R Greathouse IV, Sep 26 2012

A348699 Primes with a prime number of prime digits.

Original entry on oeis.org

23, 37, 53, 73, 127, 137, 157, 173, 223, 227, 229, 233, 239, 251, 257, 263, 271, 277, 283, 293, 307, 313, 317, 331, 337, 347, 353, 359, 367, 373, 379, 383, 397, 433, 457, 503, 521, 523, 547, 557, 563, 571, 577, 587, 593, 653, 673, 677, 727, 733, 739, 743, 751, 757, 773, 787, 797

Offset: 1

Jens Ahlström, Oct 30 2021

Jens Ahlström, Table of n, a(n) for n = 1..5853 (all terms < 10^5)
Hackerearth, Optimus prime

Intersection of A000040 and A092619.

Cf. A217090.

Select[Range[800], And @@ PrimeQ[{#, Count[IntegerDigits[#], ?PrimeQ]}] &] (* _Amiram Eldar, Nov 04 2021 *)

from sympy import isprime as i
from sympy import primerange as p
print([x for x in p(1, 800) if i(len([d for d in str(x) if i(int(d))]))])

from sympy import isprime
def ok(n): return isprime(n) and isprime(sum(1 for d in str(n) if d in "2357"))
print([k for k in range(800) if ok(k)]) # Michael S. Branicky, Jun 26 2025

cp's OEIS Frontend

A138465 Non-optimus primes.

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