

A274968


Even numbers n >= 4 which are not mgonal number for 3 <= m < n.


3



4, 8, 14, 20, 26, 32, 38, 44, 50, 56, 62, 68, 74, 80, 86, 98, 104, 110, 116, 122, 128, 134, 140, 146, 152, 158, 164, 170, 182, 188, 194, 200, 206, 212, 218, 224, 230, 236, 242, 248, 254, 266, 272, 278, 284, 290, 296, 302
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OFFSET

1,1


COMMENTS

An mgonal number, m >= 3, i.e., of the form n = (k/2)*[(m2)*k  (m4)], 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: 5gonal of order 8,
176: 5gonal of order 11, 8gonal of order 8,
260: 11gonal of order 8,
are not in this sequence.
Even numbers n for which A176948(n) = n.
Since we are looking for solutions of (m2)*k^2  (m4)*k  2*n = 0, with m >= 3 and k >= 3, the largest order k we need to consider is
k = {(m4) + sqrt[(m4)^2 + 8*(m2)*n]}/[2*(m2)] with m = 3, thus
k <= (1/2)*{1 + sqrt[1 + 8*n]}.
Or, since we are looking for solutions of 2n = m*k*(k1)  2*k*(k2), with m >= 3 and k >= 3, the largest m we need to consider is
m = [2n + 2*k*(k2)]/[k*(k1)] with k = 3, thus m <= (n+3)/3.
Composite numbers n which are divisible by 3 are mgonal 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: 4gonal with order k = 2.
All integers of the form n = 6j + 4, with j >= 1, are mgonal 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 20gonal number of order k = 2 (element of A051872) but not an mgonal 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*(k2)+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

Cf. A051872, A176948, A176949, A274967.
Sequence in context: A312684 A312685 A176949 * A317109 A173522 A049420
Adjacent sequences: A274965 A274966 A274967 * A274969 A274970 A274971


KEYWORD

nonn


AUTHOR

Daniel Forgues, Jul 12 2016


STATUS

approved



