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

%I A281222 #42 Jul 23 2025 14:43:38
%S A281222 6469693230,6915878970,8254436190,8720021310,9146807670,9592993410,
%T A281222 10407767370,10485364890,10555815270,11125544430,11532931410,
%U A281222 11797675890,11823922110,12095513430,12328305990,12598876290,12929686770,13162479330,13220677470,13467764310
%N A281222 Products of 10 distinct primes (squarefree 10-almost primes).
%H A281222 Rick L. Shepherd and Giovanni Resta, <a href="/A281222/b281222.txt">Table of n, a(n) for n = 1..10000</a> (first 231 terms from Rick L. Shepherd)
%H A281222 Chai Wah Wu, <a href="https://arxiv.org/abs/2409.05844">Algorithms for complementary sequences</a>, arXiv:2409.05844 [math.NT], 2024.
%e A281222 a(1) = 2*3*5*7*11*13*17*19*23*29 = 6469693230 = prime(10)# = A002110(10), the 10th primorial number.
%t A281222 f[om_, lm_ : 0] := Block[{i, j, k, nn, w},
%t A281222   i = Abs[om]; j = 1;
%t A281222   If[lm == 0, nn = Times @@ Prime@ Range[i], nn = Abs[lm]];
%t A281222   w = ConstantArray[1, i];
%t A281222   Union@ Reap[ Do[
%t A281222     While[Set[k, Times @@ Map[Prime, Accumulate@ w]]; k <= nn,
%t A281222       Sow[k]; j = 1; w[[-j]]++];
%t A281222     If[j == i, Break[],
%t A281222       j++; w[[-j]]++; w = PadRight[w[[;; -j]], i, 1] ],
%t A281222     {n, Infinity}] ][[-1, 1]] ];
%t A281222 f[10, 10^11] (* _Michael De Vlieger_, Apr 19 2025 *)
%o A281222 (PARI) IsInA281222(n) = n > 0 && issquarefree(n) && bigomega(n) == 10
%o A281222 (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
%o A281222 (Python)
%o A281222 from math import isqrt, prod
%o A281222 from sympy import primerange, integer_nthroot, primepi
%o A281222 def A281222(n):
%o A281222     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)))
%o A281222     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)))
%o A281222     def bisection(f,kmin=0,kmax=1):
%o A281222         while f(kmax) > kmax: kmax <<= 1
%o A281222         while kmax-kmin > 1:
%o A281222             kmid = kmax+kmin>>1
%o A281222             if f(kmid) <= kmid:
%o A281222                 kmax = kmid
%o A281222             else:
%o A281222                 kmin = kmid
%o A281222         return kmax
%o A281222     return bisection(f) # _Chai Wah Wu_, Aug 29 2024
%Y A281222 Intersection of A005117 and A046314.
%K A281222 nonn
%O A281222 1,1
%A A281222 _Rick L. Shepherd_, Jan 17 2017