OFFSET
1,1
COMMENTS
According to the Conway article, every n <= 10000000 is the number of groups of order k for some k. - Eric M. Schmidt, Sep 14 2014
LINKS
John H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica, The Mathematical Intelligencer, Spring 2008, Volume 30, Issue 2, pp 6-15, doi:10.1007/BF02985731. See Table 2 but beware errors; see A046057.
FORMULA
a(n) = A046057(prime(n)). - Andrei Zabolotskii, May 11 2026
EXAMPLE
a(6)= 56 because prime(6) = 13 => there exists 13 groups of order 56.
PROG
(SageMath)
def A240007(n_range, search_limit=2000):
""" Returns -1 if no such k is found within the search_limit.
GAP's Small Groups library is complete only up to 2000. """
targets = {int(nth_prime(n)): n for n in n_range}
results = {n: -1 for n in n_range}
found = 0
bound = len(n_range)
for k in range(1, search_limit + 1):
try:
count = int(gap.NumberSmallGroups(k))
if count in targets:
val = targets[count]
if results[val] == -1:
results[val] = k
found += 1
if found == bound: break
# GAP might not have the order in its library
except Exception:
continue
return [results[n] for n in n_range]
lst = A240007(range(1, 20), 2000); print(lst) # Peter Luschny, May 11 2026
CROSSREFS
KEYWORD
nonn,more,changed
AUTHOR
Michel Lagneau, Mar 30 2014
EXTENSIONS
Values for 59, 71, 79, 89, and 97 filled in from Conway link by Eric M. Schmidt, Sep 14 2014
Edited by Andrei Zabolotskii, May 11 2026
STATUS
approved
