A371900 Numbers k such that k+1 is composite and A371641(k) != p^2 where p = A020639(k+1) is the smallest prime factor of k+1.

406, 766, 988, 1036, 1072, 1138, 1246, 1396, 1402, 1456, 1500, 1642, 1738, 1762, 1768, 1816, 1918, 1926, 1942, 2076, 2116, 2158, 2182, 2278, 2506, 2716, 2746, 2812, 2866, 2920, 2992, 3076, 3148, 3172, 3286, 3316, 3382, 3496, 3568, 3682, 3706, 3712, 3742, 3762

Offset: 1

Author

Chai Wah Wu, Apr 11 2024

Keywords

• nonn
• base

Comments

If k+1 is composite, then A371641(k) <= A020639(k+1)^2. This sequence lists numbers k where the inequality is strict.

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A020639, A027746, A259047, A322843, A248915, A371641.

Programs

Python

from itertools import count, islice
from sympy import isprime, primefactors, factorint, integer_log
def A371900_gen(startvalue=2): # generator of terms >= startvalue
    for n in count(max(startvalue,2)):
        if not isprime(n+1):
            q = min(primefactors(n+1))
            for m in range(4,q**2):
                f = factorint(m)
                if sum(f.values()) > 1:
                    c = 0
                    for p in sorted(f):
                        a = pow(n,integer_log(p,n)[0]+1,m)
                        for _ in range(f[p]):
                            c = (c*a+p)%m
                    if not c:
                        yield n
                        break
A371900_list = list(islice(A371900_gen(),30))