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A176949 Composite numbers n for which A176948(n) = n. 5
4, 8, 14, 20, 26, 32, 38, 44, 50, 56, 62, 68, 74, 77, 80, 86, 98, 104, 110, 116, 119, 122, 128, 134, 140, 143, 146, 152, 158, 161, 164, 170, 182, 187, 188, 194, 200, 203, 206, 209, 212, 218, 221, 224, 230, 236, 242, 248, 254, 266, 272, 278, 284, 290, 296, 299, 302 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

If p >= 3 is prime, then A176948(p) = p. The sequence lists composite numbers with this property.

It is interesting that there is a large overlap with terms in A140164 (but there are exceptions, e.g. 77).

From Daniel Forgues, Jul 15 2016: (Start)

Composite numbers n which are not of form (k/2)*[(m-2)*k - (m-4)] for any m >= 3 and k >= 3, thus not m-gonal numbers for any order k >= 3.

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 k >= 3. (End)

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 order 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.

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

a(1) = 4 is the only square number: 4-gonal with order k = 2. (End)

An integer n which is congruent to k (mod t_{k-1}) with 3 <= t_{k-1} < n, i.e. n = j * t_{k-1} + k with k >= 3 and j >= 1, is an m-gonal number of order k, with m = j + 2, where t_{k-1} is a triangular number. If all the congruence tests fail, a composite n belongs to this sequence. - Daniel Forgues, Aug 02 2016

LINKS

Giovanni Resta, Table of n, a(n) for n = 1..10000

OEIS Wiki, Polygonal numbers

EXAMPLE

8 is in the sequence since it is composite and is an octagonal number, but not a heptagonal number, hexagonal number, pentagonal number, etc. 10 is not in the sequence because even though it is composite and a decagonal number, it is also a triangular number: 10 = 1 + 2 + 3 + 4. - Michael B. Porter, Jul 16 2016

MATHEMATICA

Select[Range[302], CompositeQ@ # && 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_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..302) if is_a(n)] # Peter Luschny, Jul 28 2016

(PARI) listc(nn) = {forcomposite(c=1, nn, sp = c; forstep(k=c, 3, -1, if (ispolygonal(c, k), sp=k); ); if (sp == c, print1(c, ", ")); ); } \\ Michel Marcus, Sep 06 2016

CROSSREFS

Cf. A175873, A176744, A176747, A176774, A176775, A176874, A176948, A274967, A274968.

Sequence in context: A189676 A067699 A066941 * A274968 A173522 A049420

Adjacent sequences:  A176946 A176947 A176948 * A176950 A176951 A176952

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, Apr 29 2010, Apr 30 2010

EXTENSIONS

Offset corrected and sequence extended by R. J. Mathar, May 03 2010

STATUS

approved

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Last modified August 17 15:10 EDT 2017. Contains 290635 sequences.