A350352 Products of three or more distinct prime numbers.

30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 210, 222, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 330, 345, 354, 357, 366, 370, 374, 385, 390, 399, 402, 406, 410, 418, 426, 429, 430

Offset

1,1

Comments

First differs from A336568 in lacking 420.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Example

The terms and their prime indices begin:
     30: {1,2,3}     182: {1,4,6}      285: {2,3,8}
     42: {1,2,4}     186: {1,2,11}     286: {1,5,6}
     66: {1,2,5}     190: {1,3,8}      290: {1,3,10}
     70: {1,3,4}     195: {2,3,6}      310: {1,3,11}
     78: {1,2,6}     210: {1,2,3,4}    318: {1,2,16}
    102: {1,2,7}     222: {1,2,12}     322: {1,4,9}
    105: {2,3,4}     230: {1,3,9}      330: {1,2,3,5}
    110: {1,3,5}     231: {2,4,5}      345: {2,3,9}
    114: {1,2,8}     238: {1,4,7}      354: {1,2,17}
    130: {1,3,6}     246: {1,2,13}     357: {2,4,7}
    138: {1,2,9}     255: {2,3,7}      366: {1,2,18}
    154: {1,4,5}     258: {1,2,14}     370: {1,3,12}
    165: {2,3,5}     266: {1,4,8}      374: {1,5,7}
    170: {1,3,7}     273: {2,4,6}      385: {3,4,5}
    174: {1,2,10}    282: {1,2,15}     390: {1,2,3,6}

Mathematica

Select[Range[100], SquareFreeQ[#]&&PrimeOmega[#]>=3&]

Prog

(Python)
from sympy import factorint
def ok(n): f = factorint(n, multiple=True); return len(f) == len(set(f)) > 2
print([k for k in range(431) if ok(k)]) # Michael S. Branicky, Jan 14 2022
(Python)
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A350352(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(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())))
    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, n, n) # Chai Wah Wu, Sep 11 2024
(PARI) is(n, f=factor(n))=my(e=f[, 2]); #e>2 && vecmax(e)==1 \\ Charles R Greathouse IV, Jul 08 2022
(PARI) list(lim)=my(v=List()); forsquarefree(n=30, lim\1, if(#n[2][, 2]>2, listput(v, n[1]))); Vec(v) \\ Charles R Greathouse IV, Jul 08 2022

Crossrefs

This is the squarefree case of A033942.

Including squarefree semiprimes gives A120944.

The squarefree complement consists of 1 and A167171.

These are the Heinz numbers of the partitions counted by A347548.

A000040 lists prime numbers (exactly 1 prime factor).

A005117 lists squarefree numbers.

A006881 lists squarefree numbers with exactly 2 prime factors.

A007304 lists squarefree numbers with exactly 3 prime factors.

A046386 lists squarefree numbers with exactly 4 prime factors.

Cf. A000009, A000111, A001250, A002808, A027383, A349796.

Sequence in context: A238367 A225228 A336568 * A093599 A007304 A160350

Adjacent sequences: A350349 A350350 A350351 * A350353 A350354 A350355

Keyword

nonn

Author

Gus Wiseman, Jan 11 2022

Status

approved