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A085736
Numbers n such that all groups of order n are solvable.
1
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
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
KEYWORD
nonn
AUTHOR
Andrew Niedermaier, Jul 20 2003
STATUS
approved