A274967 Odd composite numbers n which are not m-gonal number for 3 <= m < n.

77, 119, 143, 161, 187, 203, 209, 221, 299, 319, 323, 329, 371, 377, 391, 407, 413, 437, 473, 493, 497, 517, 527, 533, 539, 551, 581, 583, 589, 611, 623, 629, 649, 667, 689, 707, 713, 731, 737, 749, 767, 779, 791, 799, 803, 817, 851, 869, 893, 899, 901, 913

Offset: 1

Author

Daniel Forgues, Jul 12 2016

Keywords

• nonn
• easy

Comments

An m-gonal number, m >= 3, i.e. of form n = (k/2)*[(m-2)*k - (m-4)], yields a nontrivial factorization of n if and only if of order k >= 3.

Odd composite numbers n for which A176948(n) = n.

All odd composite n are coprime to 30 (see next comment) and have smallest prime factor >= 7, e.g.

77 = 7*11, 119 = 7*17, 143 = 11*13, 161 = 7*23,

187 = 11*17, 203 = 7*29, 209 = 11*19, 221 = 13*17,

299 = 13*23, 319 = 11*29, 323 = 17*19, 329 = 7*47,

371 = 7*53, 377 = 13*29, 391 = 17*23, 407 = 11*37,

413 = 7*59, 437 = 19*23, 473 = 11*43, 493 = 17*29,

497 = 7*71, 517 = 11*47, 527 = 17*31, 533 = 13*41,

539 = 7*7*11, 551 = 19*29, 581 = 7*83, 583 = 11*53,

589 = 19*31, 611 = 13*47, 623 = 7*89, 629 = 17*37,

649 = 11*59, 667 = 23*29, 689 = 13*53, 707 = 7*101,

713 = 23*31, 731 = 17*43, 737 = 11*67, 749 = 7*107,

767 = 13*59, 779 = 19*41, 791 = 7*113, 799 = 17*47,

803 = 11*73, 817 = 19*43, 851 = 23*37, 869 = 11*79,

893 = 19*47, 899 = 29*31, 901 = 17*53, 913 = 11*83.

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.

Odd composite numbers n which are divisible by 5 are m-gonal numbers of order 5, with m = (n + 15)/10. Thus all a(n) are coprime to 5.

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 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.

Examples

			77 is in this sequence because 77 is trivially a 77-gonal number of order k = 2, but not an m-gonal number for 3 <= k <= (1/2)*{-1 + sqrt[1 + 8*77]}.
		

Links

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

Crossrefs

Cf. A176774, A176948, A176949, A274968.

Programs

Mathematica

Select[Range[500]2+1, ! PrimeQ[#] && FindInstance[n*(4 + n*(s-2)-s)/2 == # && s >= 3 && n >= 3, {s, n}, Integers] == {} &] (* Giovanni Resta, Jul 13 2016 *)

Python

from sympy import isprime
A274967_list = []
for n in range(3,10**6,2):
    if not isprime(n):
        k = 3
        while k*(k+1) <= 2*n:
            if not (2*(k*(k-2)+n)) % (k*(k - 1)):
                break
            k += 1
        else:
            A274967_list.append(n) # Chai Wah Wu, Jul 28 2016

Sage

def is_a(n):
    if is_even(n): return False
    if is_prime(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..913) if is_a(n)]) # Peter Luschny, Jul 28 2016

Extensions

a(10)-a(52) from Giovanni Resta, Jul 13 2016