OFFSET
1,1
COMMENTS
Positive integers that have exactly five odd divisors.
Numbers k such that the symmetric representation of sigma(k) has 5 subparts. - Omar E. Pol, Dec 28 2016
Also numbers that can be expressed as the sum of k > 1 consecutive positive integers in exactly 4 ways; e.g., 81 = 40+41 = 26+27+28 = 11+12+13+14+15+16 = 5+6+7+8+9+10+11+12+13. - Julie Jones, Aug 13 2018
Every term a(n) in the sequence has the form 2^m * p^4, m >= 0 and p an odd prime. If p < 2^(m+1) then all divisors of a(n) belong to the single 2-dense sublist (see A237270) of all divisors of a(n) so that the symmetric representation of sigma for a(n), SRS(a(n)), consists of a single part of maximum width between 2 and 5; the latter occurs when p^4 < 2^(m+1) and makes SRS(a(n)) unimodal. If 2^(m+1) < p then 1 < 2^(m+1) < p < 2^(m+1) * p < p^2 <= A003056(a(n)) < 2^(m+1) * p^2 so that SRS(a(n)) consists of 5 parts of width 1 - these are also terms in A320066 and in A357581 (column 5). The width pattern (see A376829) of any term in this sequence has length 9. - Hartmut F. W. Hoft, May 25 2026
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
A001227(a(n)) = 5.
Sum_{n>=1} 1/a(n) = 2 * P(4) - 1/8 = 0.00289017370127..., where P(4) is the value of the prime zeta function at 4 (A085964). - Amiram Eldar, Sep 16 2024
EXAMPLE
From Hartmut F. W. Hoft, May 25 2026: (Start)
a(1) = 81 = 3^4 with parts {41, 15, 9, 15, 41} where odd divisor 9 starts the center part of SRS(a(1)), while a(2) = 162 = 2*3^4 satisfies 3 < 2^2 so that SRS(a(2)) consists of the single part {363} with width pattern 1 2 1 2 1 2 1 2 1.
a(17) = 11^4 = 14641 has 5 parts where odd divisor 121 starts the center part of SRS(a(17)), while a(33) = 8 * 11^4 = 117128 satisfies 11 < 2^4 so that SRS(a(33)) consists of the single part {241575}, again with width pattern 1 2 1 2 1 2 1 2 1. (End)
MAPLE
N:= 10^6: # for terms <= N
R:= NULL: p:= 2:
do
p:= nextprime(p);
if p^4 > N then break fi;
R:= R, seq(2^i*p^4, i=0..floor(log[2](N/p^4)));
od:
sort([R]); # Robert Israel, May 25 2026
MATHEMATICA
power2[n_, k_] := Module[{n4=Prime[n]^4 , k4=Prime[k]^4 }, k4 NestWhileList[2# &, 1, # k4 <=n4 &]]
a267696[n_] := Sort[Flatten[Map[power2[n, #]&, Range[2, n]]]]
a267696[8] (* all terms k less than or equal to twice the 4-th power of the 8-th prime; 260642 *)
(* Hartmut F. W. Hoft, May 25 2026 *)
PROG
(PARI) isok(n) = sumdiv(n, d, (d%2)) == 5; \\ Michel Marcus, Apr 03 2016
(GAP) A:=List([1..700000], n->DivisorsInt(n));;
B:=List([1..Length(A)], i->Filtered(A[i], IsOddInt));;
a:=Filtered([1..Length(B)], i->Length(B[i])=5); # Muniru A Asiru, Aug 14 2018
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Omar E. Pol, Apr 03 2016
EXTENSIONS
More terms from Michel Marcus, Apr 03 2016
STATUS
approved