A281222 Products of 10 distinct primes (squarefree 10-almost primes).

6469693230, 6915878970, 8254436190, 8720021310, 9146807670, 9592993410, 10407767370, 10485364890, 10555815270, 11125544430, 11532931410, 11797675890, 11823922110, 12095513430, 12328305990, 12598876290, 12929686770, 13162479330, 13220677470, 13467764310

Offset: 1

Author

Rick L. Shepherd, Jan 17 2017

Keywords

• nonn

Examples

			a(1) = 2*3*5*7*11*13*17*19*23*29 = 6469693230 = prime(10)# = A002110(10), the 10th primorial number.
		

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.

Crossrefs

Intersection of A005117 and A046314.

Programs

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 *)

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