A102466 -id:A102466 - cp's OEIS frontend

A086971 Number of semiprime divisors of n.

0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 2, 0, 3, 0, 1, 1, 1, 1, 3, 0, 1, 1, 2, 0, 3, 0, 2, 2, 1, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 1, 0, 4, 0, 1, 2, 1, 1, 3, 0, 2, 1, 3, 0, 3, 0, 1, 2, 2, 1, 3, 0, 2, 1, 1, 0, 4, 1, 1, 1, 2, 0, 4, 1, 2, 1, 1, 1, 2, 0, 2, 2, 3, 0, 3

Offset: 1

Reinhard Zumkeller, Sep 22 2003

nonn

Inverse Moebius transform of A064911. - Jonathan Vos Post, Dec 08 2004

G. H. Hardy and E. M. Wright, Section 17.10 in An Introduction to the Theory of Numbers, 5th ed., Oxford, England: Clarendon Press, 1979.

T. D. Noe, Table of n, a(n) for n = 1..10000
E. A. Bender and J. R. Goldman, On the Applications of Mobius Inversion in Combinatorial Analysis, Amer. Math. Monthly 82, (1975), 789-803.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric Weisstein's World of Mathematics, Semiprime.
Eric Weisstein's World of Mathematics, Divisor Function.
Eric Weisstein's World of Mathematics, Moebius Transform.

Cf. A001358, A064911, A001221, A000005, A000010, A004018, A007913, A056170, A079275, A001222, A220264 (least inverse).

a086971 = sum . map a064911 . a027750_row
-- Reinhard Zumkeller, Dec 14 2012

a:= proc(n) local l, m; l:=ifactors(n)[2]; m:=nops(l);
       m*(m-1)/2 +add(`if`(i[2]>1, 1, 0), i=l)
    end:
seq(a(n), n=1..120);  # Alois P. Heinz, Jul 18 2013

semiPrimeQ[n_] := PrimeOmega@ n == 2; f[n_] := Length@ Select[Divisors@ n, semiPrimeQ@# &]; Array[f, 105] (* Zak Seidov, Mar 31 2011 and modified by Robert G. Wilson v, Dec 08 2012 *)
a[n_] := Count[e = FactorInteger[n][[;; , 2]], ?(# > 1 &)] + (o = Length[e])*(o - 1)/2; Array[a, 100] (* _Amiram Eldar, Jun 30 2022 *)

/* The following definitions of a(n) are equivalent. */
a(n) = sumdiv(n,d,bigomega(d)==2)
a(n) = f=factor(n); j=matsize(f)[1]; sum(m=1,j,f[m,2]>=2) + binomial(j,2)
a(n) = f=factor(n); j=omega(n); sum(m=1,j,f[m,2]>=2) + binomial(j,2)
a(n) = omega(n/core(n)) + binomial(omega(n),2)
/* Rick L. Shepherd, Mar 06 2006 */

a(n) = A106404(n) + A106405(n). - Reinhard Zumkeller, May 02 2005
a(n) = omega(n/core(n)) + binomial(omega(n),2) = A001221(n/A007913(n)) + binomial(A001221(n),2) = A056170(n) + A079275(n). - Rick L. Shepherd, Mar 06 2006
From Reinhard Zumkeller, Dec 14 2012: (Start)
a(n) = Sum_{k=1..A000005(n)} A064911(A027750(n,k)).
a(A220264(n)) = n and a(m) <> n for m < A220264(n); a(A008578(n)) = 0; a(A002808(n)) > 0; for n > 1: a(A102466(n)) <= 1 and a(A102467(n)) > 1; A066247(n) = A057427(a(n)). (End)
G.f.: Sum_{k = p*q, p prime, q prime} x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 25 2017

Entry revised by N. J. A. Sloane, Mar 28 2006

A102467 Positive integers k such that d(k) <> Omega(k) + omega(k), where d = A000005, Omega = A001222 and omega = A001221.

Original entry on oeis.org

1, 12, 18, 20, 24, 28, 30, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 66, 68, 70, 72, 75, 76, 78, 80, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 124, 126, 130, 132, 135, 136, 138, 140, 144, 147, 148, 150, 152, 153, 154, 156

Offset: 1

Reinhard Zumkeller, Jan 09 2005

nonn

These are the numbers which are neither prime powers (>1) nor semiprimes. - M. F. Hasler, Jan 31 2008

For n > 1, positive integers k with a composite divisor, d < k, that is relatively prime to k/d. For example 12 is in the sequence since 4 (composite) is coprime to 12/4 = 3. - Wesley Ivan Hurt, Apr 25 2020

			10 is not in the sequence since d(10) = 4 is equal to Omega(10) + omega(10) = 2 + 2 = 4.
12 is in the sequence since d(12) = 6 is not equal to Omega(12) + omega(12) = 3 + 2 = 5. - _Wesley Ivan Hurt_, Apr 25 2020

T. D. Noe, Table of n, a(n) for n = 1..1000
Joerg Arndt, Matters Computational (The Fxtbook), section 40.4.1.3 "Testing for irreducibility without GCD computations", pp. 839-840.

Cf. A000005 (tau), A001221 (omega), A001222 (Omega).

Cf. A102466, A135767.

a102467 n = a102467_list !! (n-1)
a102467_list = [x | x <- [1..], a000005 x /= a001221 x + a001222 x]
-- Reinhard Zumkeller, Dec 14 2012

with(numtheory):
q:= n-> is(tau(n)<>bigomega(n)+nops(factorset(n))):
select(q, [$1..200])[];  # Alois P. Heinz, Jul 14 2023

Select[Range[200], DivisorSigma[0, #] != PrimeOmega[#] + PrimeNu[#]&] (* Jean-François Alcover, Jun 22 2018 *)

is(n)=my(f=factor(n)[,2]); #f!=1 && f!=[1,1]~ \\ Charles R Greathouse IV, Oct 19 2015

def is_A102467(n) :
    return sloane.A001221(n) != 1 and sloane.A001222(n) != 2
def A102467_list(n) :
    return [k for k in (1..n) if is_A102467(k)]
A102467_list(156)  # Peter Luschny, Feb 07 2012

Complement of A102466; A000005(a(n)) <> A001221(a(n)) + A001222(a(n)).

For n > 1, A086971(a(n)) > 1. - Reinhard Zumkeller, Dec 14 2012

Name changed by Wesley Ivan Hurt, Apr 25 2020

A110981 a(n) = the number of aperiodic subsets S of the n-th roots of 1 with zero sum (i.e., there is no r different from 1 such that r*S=S).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 24, 0, 6, 0, 0, 0, 236, 0, 0, 0, 18, 0, 3768, 0, 0, 0, 0, 0, 20384, 0, 0, 0, 7188, 0, 227784, 0, 186, 480, 0, 0, 1732448, 0, 237600, 0, 630, 0, 16028160, 0, 306684, 0, 0, 0, 341521732, 0, 0, 4896, 0, 0, 1417919208

Offset: 1

Max Alekseyev, Jan 20 2008

nonn

We count these subsets only modulo rotations (multiplication by a nontrivial root of unity).
A103314(n) = a(n)*n + 2^n - A001037(n)*n. Note that as soon as a(n)=0, we have simply A103314(n) = 2^n - A001037(n)*n. This makes it especially interesting to study those n for which a(n)=0. Quite surprisingly, it appears that the sequence of such n coincides with A102466.
From Max Alekseyev, Jan 31 2008: (Start)
Every subset of the set U(n) = { 1=r^0, r^1, ..., r^(n-1) } of the n-th power roots of 1 (where r is a fixed primitive root) defines a binary word w of the length n where the j-th bit is 1 iff the root r^j is included in the subset.
If d is the period of w with respect to cyclic rotations (thus d|n) then the periodic part of w uniquely defines some binary Lyndon word of the length d (see A001037). In turn, each binary Lyndon word of the length d, where dThe binary Lyndon words of the length n are different in this respect: only some of them correspond to n distinct zero-sum subsets of U(n) while the others do not correspond to such subsets at all. A110981(n) gives the number of binary Lyndon words of the length n that correspond to zero-sum subsets of U(n). (End)

Max Alekseyev and M. F. Hasler, Table of n, a(n) for n = 1..164

Cf. A001037, A103314, A107847, A164896.

a(n) = A001037(n) - A107847(n) ( = A001037(n) - (2^n - A103314(n))/n ). - M. F. Hasler, Jan 31 2008

Additional comments from M. F. Hasler, Jan 31 2008

A354144 Prime powers together with semiprimes.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 71, 73, 74, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 119, 121, 122, 123, 125, 127, 128, 129, 131, 133, 134, 137

Offset: 1

N. J. A. Sloane, May 20 2022

nonn

Union of A000961 and A001358.

Although this is 1 together with A102466, it has arisen sufficiently often that it deserves its own entry.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A000961, A001221, A001222, A001358, A102466.

isPPorSP := proc(n) if n=1 or nops(numtheory[factorset](n)) = 1 or numtheory[bigomega](n) = 2 then 'true' else 'false'; fi; end;

A135767 sigma_0(n)-omega(n)-Omega(n) (sigma_0 = A000005 = # divisors, omega = A001221 = # prime factors, Omega = A001222 = # prime factors with multiplicity).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 2, 0, 1, 1, 0, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 5, 0, 0, 1, 0, 0, 2, 0, 1, 0, 2, 0, 5, 0, 0, 1, 1, 0, 2, 0, 3, 0, 0, 0, 5, 0, 0, 0, 2, 0, 5, 0, 1, 0, 0, 0, 4, 0, 1, 1, 3, 0, 2, 0, 2, 2

Offset: 1

M. F. Hasler, Jan 14 2008

A102467 = { n | a(n)>0 } ; A102466 = { n | a(n)=0 } = { n | omega(n)=1 or Omega(n)=2 }: these are exactly the prime powers (>1) and semiprimes. For all other numbers a(n) > 0 since for each of the Omega(n) prime power divisors, other divisors are obtained by multiplying it with another prime factor, which gives more than omega(n) different additional divisors. a(n)>0 is also equivalent to A001037(n) > A107847(n), i.e. there are strictly fewer nonzero sums of non-periodic subsets of U_n (n-th roots of unity) than there are non-periodic binary words of length n. Otherwise stated, a(n)>0 if there is a non-periodic subset of U_n with zero sum. Non-periodic means having no rotational symmetry (except for identity).

M. F. Hasler, Table of n, a(n) for n = 1..10000

Cf. A102466, A102467 ; A001037, A107847.

a[n_] := DivisorSigma[0, n] - PrimeOmega[n] - PrimeNu[n];
Array[a, 105] (* Jean-François Alcover, Jun 21 2018 *)

A135767(n)=numdiv(n)-omega(n)-bigomega(n)

a(n)=0 <=> omega(n)=1 or Omega(n)=2 <=> n is semiprime or a prime power (>1) <=> A001037(n) = A107847(n) <=> all non-periodic subsets of U_n have nonzero sum

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A086971 Number of semiprime divisors of n.

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A102467 Positive integers k such that d(k) <> Omega(k) + omega(k), where d = A000005, Omega = A001222 and omega = A001221.

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A110981 a(n) = the number of aperiodic subsets S of the n-th roots of 1 with zero sum (i.e., there is no r different from 1 such that r*S=S).

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A354144 Prime powers together with semiprimes.

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A135767 sigma_0(n)-omega(n)-Omega(n) (sigma_0 = A000005 = # divisors, omega = A001221 = # prime factors, Omega = A001222 = # prime factors with multiplicity).

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