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

2, 6, 20, 42, 110, 156, 272, 342, 506, 812, 870, 930, 1332, 1640, 1722, 1806, 2162, 2756, 3422, 3660, 4290, 4422, 4830, 4970, 5256, 6006, 6162, 6806, 7832, 9312, 10100, 10302, 10506, 10920, 11342, 11772, 11990, 12656, 12882, 16002, 16770, 17030, 18632, 18906, 19182

Offset

1,1

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = A002378(A030059(n)-1).

Sum_{n>=1} 1/a(n) = (A368250 + A033150 - 1)/2 = 0.776922504035... .

Mathematica

Table[n*(n - 1), {n, Select[Range[150], MoebiusMu[#] == -1 &]}]

Prog

(PARI) lista(kmax) = forsquarefree(k=1, kmax, if(moebius(k) == -1, print1(k[1]*(k[1]-1), ", ")));
(Python)
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A370494(n):
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    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-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)))
    return (k:=bisection(f, n, n))*(k-1) # Chai Wah Wu, Jan 28 2025

Crossrefs

Complement of A370495 within A368249.

Cf. A002378, A030059, A033150, A368250.

Sequence in context: A087134 A378027 A383745 * A036689 A355390 A226326

Adjacent sequences: A370491 A370492 A370493 * A370495 A370496 A370497

Keyword

nonn,easy

Author

Amiram Eldar, Feb 20 2024

Status

approved