

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
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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)*[(m2)*k  (m4)] for any m >= 3 and k >= 3, thus not mgonal numbers for any order k >= 3.
An mgonal number, m >= 3, i.e., of the form n = (k/2)*[(m2)*k  (m4)], yields a nontrivial factorization of n if and only if k >= 3. (End)
Since we are looking for solutions of (m2)*k^2  (m4)*k  2*n = 0,
with m >= 3 and k >= 3, the largest order k we need to consider is
k = {(m4) + sqrt[(m4)^2 + 8*(m2)*n]}/[2*(m2)] with m = 3, thus
k <= (1/2)*{1 + sqrt[1 + 8*n]}.
Or, since we are looking for solutions of 2n = m*k*(k1)  2*k*(k2),
with m >= 3 and k >= 3, the largest m we need to consider is
m = [2n + 2*k*(k2)]/[k*(k1)] with order k = 3, thus m <= (n+3)/3.
Composite numbers n which are divisible by 3 are mgonal 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 mgonal 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: 4gonal with order k = 2. (End)
An integer n which is congruent to k (mod t_{k1}) with 3 <= t_{k1} < n, i.e. n = j * t_{k1} + k with k >= 3 and j >= 1, is an mgonal number of order k, with m = j + 2, where t_{k1} 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 ngonal numbers, and will thus have A176948(n)=n unless they have a nontrivial polygonal decomposition. Thus this is just the sequence of nonpolygonal 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 order3 numbers; numbers of the form 6x+4 are always order4 numbers; small odd composites not divisible by 3 are usually divisible by 5, and are thus order5 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 order3 or 6 numbers (those are divisible by 3); order4 numbers (2 is not a square (mod 6)); order5 numbers (all odd); or order7 numbers (all == 1 (mod 3)).
The first 6x+2 composite not in the list is order8 pentagonal number 92.
(End)


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*(s2)  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: A312683 A312684 A312685 * A274968 A317109 A173522
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



