A274967 - OEIS

%I #65 Mar 13 2020 17:44:17

%S 77,119,143,161,187,203,209,221,299,319,323,329,371,377,391,407,413,

%T 437,473,493,497,517,527,533,539,551,581,583,589,611,623,629,649,667,

%U 689,707,713,731,737,749,767,779,791,799,803,817,851,869,893,899,901,913

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

%C 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.

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

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

%C    77 =   7·11, 119 =  7·17, 143 = 11·13, 161 =  7·23,

%C   187 =  11·17, 203 =  7·29, 209 = 11·19, 221 = 13·17,

%C   299 =  13·23, 319 = 11·29, 323 = 17·19, 329 =  7·47,

%C   371 =   7·53, 377 = 13·29, 391 = 17·23, 407 = 11·37,

%C   413 =   7·59, 437 = 19·23, 473 = 11·43, 493 = 17·29,

%C   497 =   7·71, 517 = 11·47, 527 = 17·31, 533 = 13·41,

%C   539 = 7·7·11, 551 = 19·29, 581 =  7·83, 583 = 11·53,

%C   589 =  19·31, 611 = 13·47, 623 =  7·89, 629 = 17·37,

%C   649 =  11·59, 667 = 23·29, 689 = 13·53, 707 = 7·101,

%C   713 =  23·31, 731 = 17·43, 737 = 11·67, 749 = 7·107,

%C   767 =  13·59, 779 = 19·41, 791 = 7·113, 799 = 17·47,

%C   803 =  11·73, 817 = 19·43, 851 = 23·37, 869 = 11·79,

%C   893 =  19·47, 899 = 29·31, 901 = 17·53, 913 = 11·83.

%C 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.

%C 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.

%C 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

%C     k = {(m-4) + sqrt[(m-4)^2 + 8*(m-2)*n]}/[2*(m-2)] with m = 3, thus

%C     k <= (1/2)*{-1 + sqrt[1 + 8*n]}.

%C 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

%C     m = [2n + 2*k*(k-2)]/[k*(k-1)] with k = 3, thus m <= (n+3)/3.

%H Chai Wah Wu, <a href="/A274967/b274967.txt">Table of n, a(n) for n = 1..10000</a>

%H OEIS Wiki, <a href="/wiki/Polygonal_numbers">Polygonal numbers</a>

%e 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]}.

%t 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 *)

%o (Sage)

%o def is_a(n):

%o     if is_even(n): return False

%o     if is_prime(n): return False

%o     for m in (3..(n+3)//3):

%o         if pari('ispolygonal')(n, m):

%o             return False

%o     return True

%o print([n for n in (3..913) if is_a(n)]) # _Peter Luschny_, Jul 28 2016

%o (Python)

%o from sympy import isprime

%o A274967_list = []

%o for n in range(3,10**6,2):

%o     if not isprime(n):

%o         k = 3

%o         while k*(k+1) <= 2*n:

%o             if not (2*(k*(k-2)+n)) % (k*(k - 1)):

%o                 break

%o             k += 1

%o         else:

%o             A274967_list.append(n) # _Chai Wah Wu_, Jul 28 2016

%Y Cf. A176774, A176948, A176949, A274968.

%K nonn,easy

%O 1,1

%A _Daniel Forgues_, Jul 12 2016

%E a(10)-a(52) from _Giovanni Resta_, Jul 13 2016