A370495 Oblong numbers of the form (k-1)*k where k is the product of an even number of distinct primes.

0, 30, 90, 182, 210, 420, 462, 650, 1056, 1122, 1190, 1406, 1482, 2070, 2550, 2970, 3192, 3306, 3782, 4160, 4692, 5402, 5852, 6642, 7140, 7310, 7482, 8190, 8556, 8742, 8930, 11130, 12210, 13110, 13806, 14042, 14762, 15006, 16512, 17556, 17822, 19740, 20022, 20306

Offset

1,2

Links

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

Formula

a(n) = A002378(A030229(n)-1).

Sum_{n>=2} 1/a(n) = (A368250 - A033150 + 1)/2 = 0.071711363929... .

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 A370495(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-1+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(2, x.bit_length(), 2)))
    return (k:=bisection(f, n, n))*(k-1) # Chai Wah Wu, Jan 28 2025

Crossrefs

Complement of A370494 within A368249.

Cf. A002378, A030059, A033150, A368250.

Sequence in context: A081807 A008445 A039462 * A138018 A380104 A004995

Adjacent sequences: A370492 A370493 A370494 * A370496 A370497 A370498

Keyword

nonn,easy

Author

Amiram Eldar, Feb 20 2024

Status

approved