A139118 -id:A139118 - cp's OEIS frontend

A141241 a(n) = number of divisors of n-th positive integer with a nonprime number of divisors. a(n) = the number of divisors of A139118(n).

1, 4, 4, 4, 6, 4, 4, 6, 6, 4, 4, 8, 4, 4, 6, 8, 6, 4, 4, 4, 9, 4, 4, 8, 8, 6, 6, 4, 10, 6, 4, 6, 8, 4, 8, 4, 4, 12, 4, 6, 4, 8, 6, 4, 8, 12, 4, 6, 6, 4, 8, 10, 4, 12, 4, 4, 4, 8, 12, 4, 6, 4, 4, 4, 12, 6, 6, 9, 8, 8, 8, 4, 12, 8, 4, 10, 8, 4, 6, 6, 4, 4, 16, 4, 4, 6, 4, 12, 8, 4, 8, 12, 4, 4, 8, 8, 8, 12, 4

Offset: 1

Leroy Quet, Jun 16 2008

nonn

a(1) = 1 and all other terms are composite, of course.

John Tyler Rascoe, Table of n, a(n) for n = 1..10000

Cf. A000005, A139118, A141242.

Select[DivisorSigma[0,Range[200]],!PrimeQ[#]&] (* Harvey P. Dale, Mar 20 2015 *)

for(i=1,200,if(!isprime(numdiv(i)),print1(numdiv(i)","))) \\ Franklin T. Adams-Watters, Apr 09 2009

from sympy import primepi, integer_nthroot, primerange, divisor_count
def A141241(n):
    def f(x): return int(n+sum(primepi(integer_nthroot(x,k-1)[0]) for k in primerange(x.bit_length()+1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return divisor_count(m) # Chai Wah Wu, Feb 22 2025

a(n) = A000005(A139118(n)). - Michel Marcus, Feb 26 2025

More terms from Franklin T. Adams-Watters, Apr 09 2009

A268388 "Fermi-Dirac composites": numbers k for which A064547(k) > 1.

Original entry on oeis.org

6, 8, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120

Offset: 1

Antti Karttunen, Feb 09 2016, after Vladimir Shevelev's Apr 2010 comment in A176699

nonn

			6 = 2^1 * 3^1 is present, as there are altogether two 1-bits in the exponents (1 and 1 also in binary), which is more than one.
64 = 2^6 is present, as the binary representation of 6 is "110", which contains more than one 1-bit. This is also the first term not present in A139118.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Subsequence of A002808.
Cf. A050376 (complement without 1).
Cf. A064547.
Cf. A176699 (subsequence), A000379 (also subsequence, without the initial 1).
Different from A139118.

Select[Range[120], Plus @@ DigitCount[Last /@ FactorInteger[#], 2, 1] > 1 &] (* Amiram Eldar, Nov 27 2020 *)

isok(n) = my(f = factor(n)[,2]); sum(k=1, #f, hammingweight(f[k])) > 1; \\ Michel Marcus, Feb 10 2016

from sympy import primepi, integer_nthroot
def A268388(n):
    def f(x): return int(n+1+sum(primepi(integer_nthroot(x,1<Chai Wah Wu, Feb 22 2025

A192690 Nonprime numbers with a nonprime number of nonprime divisors.

Original entry on oeis.org

1, 12, 16, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 124, 126, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164, 171, 172, 175, 176, 180, 184, 188

Offset: 1

Juri-Stepan Gerasimov, Oct 15 2011

nonn

			For example, 12 is composite and it has 6 divisors: 1, 2, 3, 4, 6, 12. Of these, 4 are not prime: 1, 4, 6, 12. Since 4 is not prime either, 12 is in the sequence.

Cf. A139118, A141468, A167175.

Cf. A018252. - Omar E. Pol, Oct 20 2011

NonPrimeDivisors[n_] := Length[Select[Divisors[n], ! PrimeQ[#] &]]; Select[Range[200], ! PrimeQ[#] && ! PrimeQ[NonPrimeDivisors[#]] &] (* T. D. Noe, Oct 20 2011 *)

def npd(n: int) -> int:
    return len([d for d in divisors(n) if not is_prime(d)])
def isA192690(n: int) -> bool:
    return not (is_prime(n) or is_prime(npd(n)))
A192690List = lambda b: [n for n in range(1, b) if isA192690(n)]
print(A192690List(189))  # Peter Luschny, Apr 22 2025

cp's OEIS Frontend

A141241 a(n) = number of divisors of n-th positive integer with a nonprime number of divisors. a(n) = the number of divisors of A139118(n).

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A268388 "Fermi-Dirac composites": numbers k for which A064547(k) > 1.

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A192690 Nonprime numbers with a nonprime number of nonprime divisors.

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