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A370494 Oblong numbers of the form (k-1)*k where k is the product of an odd number of distinct primes.

%I A370494 #11 Jan 28 2025 12:48:48
%S A370494 2,6,20,42,110,156,272,342,506,812,870,930,1332,1640,1722,1806,2162,
%T A370494 2756,3422,3660,4290,4422,4830,4970,5256,6006,6162,6806,7832,9312,
%U A370494 10100,10302,10506,10920,11342,11772,11990,12656,12882,16002,16770,17030,18632,18906,19182
%N A370494 Oblong numbers of the form (k-1)*k where k is the product of an odd number of distinct primes.
%H A370494 Amiram Eldar, <a href="/A370494/b370494.txt">Table of n, a(n) for n = 1..10000</a>
%F A370494 a(n) = A002378(A030059(n)-1).
%F A370494 Sum_{n>=1} 1/a(n) = (A368250 + A033150 - 1)/2 = 0.776922504035... .
%t A370494 Table[n*(n - 1), {n, Select[Range[150], MoebiusMu[#] == -1 &]}]
%o A370494 (PARI) lista(kmax) = forsquarefree(k=1, kmax, if(moebius(k) == -1, print1(k[1]*(k[1]-1), ", ")));
%o A370494 (Python)
%o A370494 from math import isqrt, prod
%o A370494 from sympy import primerange, integer_nthroot, primepi
%o A370494 def A370494(n):
%o A370494     def bisection(f,kmin=0,kmax=1):
%o A370494         while f(kmax) > kmax: kmax <<= 1
%o A370494         kmin = kmax >> 1
%o A370494         while kmax-kmin > 1:
%o A370494             kmid = kmax+kmin>>1
%o A370494             if f(kmid) <= kmid:
%o A370494                 kmax = kmid
%o A370494             else:
%o A370494                 kmin = kmid
%o A370494         return kmax
%o A370494     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 A370494     def f(x): return int(n+x-primepi(x)-sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,i)) for i in range(3,x.bit_length(),2)))
%o A370494     return (k:=bisection(f,n,n))*(k-1) # _Chai Wah Wu_, Jan 28 2025
%Y A370494 Complement of A370495 within A368249.
%Y A370494 Cf. A002378, A030059, A033150, A368250.
%K A370494 nonn,easy
%O A370494 1,1
%A A370494 _Amiram Eldar_, Feb 20 2024