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

%I A274968 #75 Mar 13 2020 17:44:22
%S A274968 4,8,14,20,26,32,38,44,50,56,62,68,74,80,86,98,104,110,116,122,128,
%T A274968 134,140,146,152,158,164,170,182,188,194,200,206,212,218,224,230,236,
%U A274968 242,248,254,266,272,278,284,290,296,302
%N A274968 Even numbers n >= 4 which are not m-gonal number for 3 <= m < n.
%C A274968 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.
%C A274968 Except for a(1) = 4, all a(n) are congruent to 2 (mod 6), although from 8 to 302, the numbers
%C A274968      92:  5-gonal of order 8,
%C A274968     176:  5-gonal of order 11, 8-gonal of order 8,
%C A274968     260: 11-gonal of order 8,
%C A274968   are not in this sequence.
%C A274968 Even numbers n for which A176948(n) = n.
%C A274968 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
%C A274968     k = {(m-4) + sqrt[(m-4)^2 + 8*(m-2)*n]}/[2*(m-2)] with m = 3, thus
%C A274968     k <= (1/2)*{-1 + sqrt[1 + 8*n]}.
%C A274968 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 A274968     m = [2n + 2*k*(k-2)]/[k*(k-1)] with k = 3, thus m <= (n+3)/3.
%C A274968 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 A274968 a(1) = 4 is the only square number: 4-gonal with order k = 2.
%C A274968 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
%H A274968 Chai Wah Wu, <a href="/A274968/b274968.txt">Table of n, a(n) for n = 1..10000</a>
%H A274968 OEIS Wiki, <a href="/wiki/Polygonal_numbers">Polygonal numbers</a>
%e A274968 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]}.
%o A274968 (Sage)
%o A274968 def is_A274968(n):
%o A274968     if is_odd(n): return False
%o A274968     for m in (3..(n+3)//3):
%o A274968         if pari('ispolygonal')(n, m):
%o A274968             return False
%o A274968     return True
%o A274968 print([n for n in (3..302) if is_A274968(n)]) # _Peter Luschny_, Jul 28 2016
%o A274968 (Python)
%o A274968 A274968_list = []
%o A274968 for n in range(4,10**6,2):
%o A274968     k = 3
%o A274968     while k*(k+1) <= 2*n:
%o A274968         if not (2*(k*(k-2)+n)) % (k*(k - 1)):
%o A274968             break
%o A274968         k += 1
%o A274968     else:
%o A274968         A274968_list.append(n) # _Chai Wah Wu_, Jul 28 2016
%o A274968 (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
%Y A274968 Cf. A051872, A176948, A176949, A274967.
%K A274968 nonn
%O A274968 1,1
%A A274968 _Daniel Forgues_, Jul 12 2016