OFFSET
1,1
COMMENTS
Positive integers that have exactly seven odd divisors.
Numbers k such that the symmetric representation of sigma(k) has 7 subparts. - Omar E. Pol, Dec 28 2016
Numbers that can be formed in exactly 6 ways by summing sequences of 2 or more consecutive positive integers. - Julie Jones, Aug 13 2018
Numbers of the form p^6 * 2^k where p is an odd prime. - David A. Corneth, Aug 14 2018
Every term a(n) in the sequence has the form 2^m * p^6, 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 7; 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 < 2^(m+1) * p^2 < p^3 <= A003056(a(n)) < 2^(m+1) * p^3 so that SRS(a(n)) consists of 7 parts of width 1 - these are also terms in A357775 and in A357581 (column 7). The width pattern (see A376829) of any term in this sequence has length 13. - Hartmut F. W. Hoft, May 25 2026
LINKS
David A. Corneth, Table of n, a(n) for n = 1..10000
FORMULA
A001227(a(n)) = 7.
Sum_{n>=1} 1/a(n) = 2 * P(6) - 1/32 = 0.00289017370127..., where P(6) is the value of the prime zeta function at 6 (A085966). - Amiram Eldar, Sep 16 2024
EXAMPLE
From Hartmut F. W. Hoft, May 25 2026: (Start)
a(1) = 729 = 3^6 with parts {365, 123, 45, 27, 45, 123, 365} where odd divisor 27 starts the center part of SRS(a(1)), while a(2) = 1458 = 2*3^6 satisfies 2 < 3 < 2^2 so that SRS(a(2)) consists of the single part {3279} with width pattern 1 2 1 2 1 2 1 2 1 2 1 2 1.
a(6) = 15625 = 5^6 has 7 parts where odd divisor 125 starts the center part of SRS(a(6)), while a(10) = 62500 = 2^2 * 5^6 satisfies 2^2 < 5 < 2^3 so that SRS(a(10)) consists of the single part {136717}, again with width pattern 1 2 1 2 1 2 1 2 1 2 1 2 1. (End)
MATHEMATICA
power6[n_, k_] := Module[{n6=Prime[n]^6 , k6=Prime[k]^6 }, k6 NestWhileList[2# &, 1, # k6 <=n6 &]]
a267697[n_] := Sort[Flatten[Map[power6[n, #]&, Range[2, n]]]]
a267697[8] (* all terms k less than or equal to twice the 6-th power of the 8-th prime *)
(* Hartmut F. W. Hoft, May 25 2026 *)
PROG
(PARI) isok(n) = sumdiv(n, d, (d%2)) == 7; \\ Michel Marcus, Apr 03 2016
(PARI) upto(n) = {my(res = List()); forprime(p = 3, sqrtnint(n, 6), listput(res, p^6)); q = #res; for(i = 1, q, odd = res[i]; for(j = 1, logint(n \ odd, 2), listput(res, odd <<= 1))); listsort(res); res} \\ David A. Corneth, Aug 14 2018
(Python)
from sympy import integer_log, primerange, integer_nthroot
def A267697(n):
def bisection(f, kmin=0, kmax=1):
while f(kmax) > kmax: kmax <<= 1
kmin = kmax >> 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax
def f(x): return int(n+x-sum(integer_log(x//p**6, 2)[0]+1 for p in primerange(3, integer_nthroot(x, 6)[0]+1)))
return bisection(f, n, n) # Chai Wah Wu, Feb 22 2025
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Omar E. Pol, Apr 03 2016
EXTENSIONS
More terms from Michel Marcus, Apr 03 2016
STATUS
approved