OFFSET
1,1
COMMENTS
An m-gonal number, m >= 3, i.e., of the form n = (k/2)*[(m-2)*k - (m-4)], yields a nontrivial factorization of n if and only if of order k >= 3.
Except for a(1) = 4, all a(n) are congruent to 2 (mod 6), although from 8 to 302, the numbers
92: 5-gonal of order 8,
176: 5-gonal of order 11, 8-gonal of order 8,
260: 11-gonal of order 8,
are not in this sequence.
Even numbers n for which A176948(n) = n.
Since we are looking for solutions of (m-2)*k^2 - (m-4)*k - 2*n = 0, with m >= 3 and k >= 3, the largest order k we need to consider is
k = {(m-4) + sqrt[(m-4)^2 + 8*(m-2)*n]}/[2*(m-2)] with m = 3, thus
k <= (1/2)*{-1 + sqrt[1 + 8*n]}.
Or, since we are looking for solutions of 2n = m*k*(k-1) - 2*k*(k-2), with m >= 3 and k >= 3, the largest m we need to consider is
m = [2n + 2*k*(k-2)]/[k*(k-1)] with k = 3, thus m <= (n+3)/3.
Composite numbers n which are divisible by 3 are m-gonal numbers of order 3, with m = (n + 3)/3. Thus all a(n) are coprime to 3.
a(1) = 4 is the only square number: 4-gonal with order k = 2.
All integers of the form n = 6j + 4, with j >= 1, are m-gonal numbers of order k = 4, with m = j + 2, which means that none are in this sequence. - Daniel Forgues, Aug 01 2016
LINKS
Chai Wah Wu, Table of n, a(n) for n = 1..10000
OEIS Wiki, Polygonal numbers
EXAMPLE
20 is in this sequence because 20 is trivially a 20-gonal number of order k = 2 (element of A051872) but not an m-gonal number for 3 <= k <= (1/2)*{-1 + sqrt[1 + 8*20]}.
PROG
(Sage)
def is_A274968(n):
if is_odd(n): return False
for m in (3..(n+3)//3):
if pari('ispolygonal')(n, m):
return False
return True
print([n for n in (3..302) if is_A274968(n)]) # Peter Luschny, Jul 28 2016
(Python)
A274968_list = []
for n in range(4, 10**6, 2):
k = 3
while k*(k+1) <= 2*n:
if not (2*(k*(k-2)+n)) % (k*(k - 1)):
break
k += 1
else:
A274968_list.append(n) # Chai Wah Wu, Jul 28 2016
(PARI) lista(nn) = {forstep(n=4, nn, 2, sp = n; forstep(k=n, 3, -1, if (ispolygonal(n, k), sp=k); ); if (sp == n, print1(n, ", ")); ); } \\ Michel Marcus, Sep 06 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
Daniel Forgues, Jul 12 2016
STATUS
approved