A085736 Numbers n such that all groups of order n are solvable.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84

Offset

1,2

Links

Table of n, a(n) for n=1..83.

Eric Weisstein's World of Mathematics, Solvable groups

Example

The symmetric and alternating groups on 5 elements are not solvable and have orders 60 and 120 respectively.

Prog

(Python)
from itertools import count, islice
from sympy import primerange
def A085736_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue, 1)):
        if n%5616 and n%60:
            flag = True
            for p in primerange(3, n+1):
                a, b, c, d = (1<<(p<<1))-1<<p, 3**p*(3**(p<<1)-1>>1), ((1<<(p<<1))+1)*((1<<p)-1)<<(p<<1), p*(p**2-1)>>1
                if min(a, b, c, d)>n:
                    break
                if not (n%a and n%b and n%c and (p<=3 or (p**2+1)%5 or n%d)):
                    flag = False
                    break
            if flag:
                yield n
A085736_list = list(islice(A085736_gen(), 83)) # Chai Wah Wu, Jan 05 2026

Crossrefs

See A056866, the complementary set of numbers, which is the main entry for this question.

Sequence in context: A166719 A272159 A227981 * A265711 A262065 A296765

Adjacent sequences: A085733 A085734 A085735 * A085737 A085738 A085739

Keyword

nonn

Author

Andrew Niedermaier, Jul 20 2003

Status

approved