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A230078
Complement of A138929: positive integers not of the form 2*p^k, k >= 0, p a prime (also 2).
5
1, 3, 5, 7, 9, 11, 12, 13, 15, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 30, 31, 33, 35, 36, 37, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 61, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 83, 84, 85, 87, 88, 89, 90, 91, 92, 93, 95, 96, 97, 99, 100
OFFSET
1,2
COMMENTS
The complement relative to the positive integers of the present sequence is A138929.
The sequence includes all positive integers of the forms (i) odd, (ii) 2^k*p, p an odd prime and k>=2, and (iii) 2^e0*p1^e1*p2^e2 *** pk^ek, k >= 2, with odd primes p1, ..., pk, and each exponent from {e0, ..., ek} is >= 1.
For a(n) > 1 a regular a(n)-gon, with length ratio (smallest diagonal)/side rho(a(n)) = 2*cos(Pi/a(n)), the inverse of rho(a(n)), which is an element of the algebraic number field Q(rho(a(n))), is in fact a Q(rho(a(n)))-integer. For a(1)=1 rho(1) = -2, and the inverse is not a Q-integer.
EXAMPLE
Even members a(n) of the form (ii) 2^k*p, p an odd prime and k>=2 are: 12, 20, 24, 28, 36, 40, 44, 48, 52, 56, 68, 72, 76, 80, 88, 92, 96, 100,...
Even members a(n) of the form (iii), given above, include 30, 42, 60, 66, 70, 78, 84, 90, ...
For the regular 5-gon (pentagon) rho(5) = tau = (1 + sqrt(5))/ 2 (the golden section). The number field is Q(rho(5)), and for the inverse one has 1/rho(5) = -1*1 + 1* rho(5) (in the power basis <1, rho(5)>, in which Q(rho(5))-integers have integer coefficients).
For the regular 7-gon rho(7) = 2*cos(Pi/7), (approximately 1.801937736) is of degree 3, and 1/rho(7) = 2*1 + 1*rho(7) - 1*rho(7)^2, (approximately 0.5549581320), hence a Q(rho(7))- integer.
For Gauss' regular 17-gon rho(17) = 2*cos(Pi/17) (approximately 1.965946199) is of degree 8 and 1/rho(17) = -4*1+ 10*rho(17)^1 + 10*rho(17)^2 - 15*rho(17)^3 -6*rho(17)^4 + 7*rho(17)^5 + 1*rho(17)^6 -1*rho(17)^7, (approximately 0.5086610), hence this is a Q(rho(17))- integer.
MATHEMATICA
With[{upto = 100}, Complement[Range[upto], 2*Join[{1}, Select[Range[upto/2], PrimePowerQ]]]] (* Paolo Xausa, Aug 30 2024 *)
PROG
(Python)
from sympy import primepi, integer_nthroot
def A230078(n):
if n == 1: return 1
def f(x): return int(n+1+sum(primepi(integer_nthroot(x>>1, k)[0]) for k in range(1, (x>>1).bit_length())))
kmin, kmax = 0, 1
while f(kmax) > kmax:
kmax <<= 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax # Chai Wah Wu, Aug 29 2024
CROSSREFS
Cf. A138929 (complement), 2*A020513, A230079 (1/rho(a(n))).
Sequence in context: A162495 A107315 A340855 * A275669 A191275 A260392
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Nov 02 2013
STATUS
approved