OFFSET
1,1
LINKS
Rick L. Shepherd and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 231 terms from Rick L. Shepherd)
Chai Wah Wu, Algorithms for complementary sequences, arXiv:2409.05844 [math.NT], 2024. See also Integers (2025) Vol. 25, Art. No. A95. See p. 24.
EXAMPLE
a(1) = 2*3*5*7*11*13*17*19*23*29 = 6469693230 = prime(10)# = A002110(10), the 10th primorial number.
MATHEMATICA
f[om_, lm_ : 0] := Block[{i, j, k, nn, w},
i = Abs[om]; j = 1;
If[lm == 0, nn = Times @@ Prime@ Range[i], nn = Abs[lm]];
w = ConstantArray[1, i];
Union@ Reap[ Do[
While[Set[k, Times @@ Map[Prime, Accumulate@ w]]; k <= nn,
Sow[k]; j = 1; w[[-j]]++];
If[j == i, Break[],
j++; w[[-j]]++; w = PadRight[w[[;; -j]], i, 1] ],
{n, Infinity}] ][[-1, 1]] ];
f[10, 10^11] (* Michael De Vlieger, Apr 19 2025 *)
PROG
(PARI) IsInA281222(n) = n > 0 && issquarefree(n) && bigomega(n) == 10
(PARI) list(lim, pr=10, maxp=lim\vecprod(primes(pr-1)))=if(pr==1, return(primes([2, min(lim\1, maxp)]))); my(v=List(), pr1=pr-1, mx=prod(i=1, pr1, prime(i))); forprime(p=prime(pr), min(lim\mx, maxp), my(u=list(lim\p, pr1, p-1)); for(i=1, #u, listput(v, p*u[i]))); Set(v) \\ Charles R Greathouse IV, Feb 03 2023; corrected by David A. Corneth, Jul 22 2025
(Python)
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A281222(n):
def g(x, a, b, c, m): yield from (((d, ) for d in enumerate(primerange(b+1, isqrt(x//c)+1), a+1)) if m==2 else (((a2, b2), )+d for a2, b2 in enumerate(primerange(b+1, integer_nthroot(x//c, m)[0]+1), a+1) for d in g(x, a2, b2, c*b2, m-1)))
def f(x): return int(n+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x, 0, 1, 1, 10)))
def bisection(f, kmin=0, kmax=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
return bisection(f) # Chai Wah Wu, Aug 29 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Rick L. Shepherd, Jan 17 2017
STATUS
approved
