A274968 Even numbers n >= 4 which are not m-gonal number for 3 <= m < n.

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

Offset: 1

Author

Daniel Forgues, Jul 12 2016

Keywords

• nonn

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

Examples

			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]}.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000
• OEIS Wiki, Polygonal numbers

Crossrefs

Cf. A051872, A176948, A176949, A274967.

Programs

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

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

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