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A274967 Odd composite numbers n which are not m-gonal number for 3 <= m < n. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..10000

OEIS Wiki, Polygonal numbers

EXAMPLE

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]}.

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 *)

PROG

(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

(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

CROSSREFS

Cf. A176774, A176948, A176949, A274968.

Sequence in context: A193570 A154534 A235867 * A229826 A330103 A176278

Adjacent sequences:  A274964 A274965 A274966 * A274968 A274969 A274970

KEYWORD

nonn,easy

AUTHOR

Daniel Forgues, Jul 12 2016

EXTENSIONS

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

STATUS

approved

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Last modified July 9 16:50 EDT 2020. Contains 335545 sequences. (Running on oeis4.)