A176949 Composite numbers n for which A176948(n) = n.

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

Offset: 1

Author

Vladimir Shevelev, Apr 29 2010, Apr 30 2010

Keywords

• nonn

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 the form n = (k/2)*[(m-2)*k - (m-4)], yields a nontrivial factorization of n if and only if k >= 3.

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

From Jonathan Dushoff, Apr 05 2022: (Start)

All numbers n>2 are trivially n-gonal numbers, and will thus have A176948(n)=n unless they have a nontrivial polygonal decomposition. Thus this is just the sequence of non-polygonal composite numbers.

Note that the 2nd through 13th terms are in arithmetic progression.

Some reasons: many of the smaller odd numbers are prime (and thus don't appear); numbers of the form 6x (or 6x+3) are always order-3 numbers; numbers of the form 6x+4 are always order-4 numbers; small odd composites not divisible by 3 are usually divisible by 5, and are thus order-5 numbers.

In fact, the first number to break the arithmetic progression is the first product of distinct primes > 5.

Conversely, 6x+2 numbers cannot be order-3 or -6 numbers (those are divisible by 3); order-4 numbers (all == 4 (mod 6)); order-5 numbers (all odd); or order-7 numbers (all == 1 (mod 3)).

The first 6x+2 composite not in the list is order-8 pentagonal number 92. (End)

Examples

			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
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000
• OEIS Wiki, Polygonal numbers

Crossrefs

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

Programs

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 *)

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

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

Extensions

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