A056170 -id:A056170 - 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

A162641 Number of even exponents in canonical prime factorization of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Jul 08 2009

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A001221, A003557, A002035, A007913, A025487, A059841, A061742 (where records occur), A072587, A162642, A179119.

Cf. A268335 (positions of zeros), A295316.

Table[Count[FactorInteger[n][[All, -1]], ?EvenQ], {n, 105}] (* _Michael De Vlieger, Jul 23 2017 *)

A162641(n) = omega(n) - omega(core(n)); \\ Antti Karttunen, Jul 23 2017

(define (A162641 n) (if (= 1 n) 0 (+ (A059841 (A067029 n)) (A162641 (A028234 n))))) ;; Antti Karttunen, Jul 23 2017

a(n) = A001221(n) - A162642(n).
a(A002035(n)) = 0.
a(A072587(n)) > 0.
Additive with a(p^e) = A059841(e). - Antti Karttunen, Jul 23 2017
From Antti Karttunen, Nov 28 2017: (Start)
a(n) = A162642(A003557(n)).
a(n) <= A056170(n).
(End)
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} 1/(p*(p+1)) = 0.3302299262... (A179119). - Amiram Eldar, Dec 25 2021

A190641 Numbers having exactly one non-unitary prime factor.

Original entry on oeis.org

4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 68, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 104, 112, 116, 117, 120, 121, 124, 125, 126, 128, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164

Offset: 1

Reinhard Zumkeller, Dec 29 2012

nonn

Numbers k such that the powerful part of k, A057521(k), is a composite prime power (A246547). - Amiram Eldar, Aug 01 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio.
Carl Pomerance and Andrzej Schinzel, Multiplicative Properties of Sets of Residues, Moscow Journal of Combinatorics and Number Theory, Vol. 1, Iss. 1 (2011), pp. 52-66. See p. 61.

Subsequence of A013929 and of A327877.

Cf. A056170, A057521, A154945, A246547, A359466 (characteristic function).

a190641 n = a190641_list !! (n-1)
a190641_list = map (+ 1) $ elemIndices 1 a056170_list

Select[Range[164],Count[FactorInteger[#][[All, 2]], 1] == Length[FactorInteger[#]] - 1 &] (* Geoffrey Critzer, Feb 05 2015 *)

list(lim)=my(s=lim\4, v=List(), u=vectorsmall(s, i, 1), t, x); forprime(k=2, sqrtint(s), t=k^2; forstep(i=t, s, t, u[i]=0)); forprime(k=2, sqrtint(lim\1), for(e=2,logint(lim\1,k), t=k^e; for(i=1, #u, if(u[i] && gcd(k, i)==1, x=t*i; if(x>lim, break); listput(v, x))))); Set(v) \\ Charles R Greathouse IV, Aug 02 2016

isok(n) = my(f=factor(n)); #select(x->(x>1), f[,2]) == 1; \\ Michel Marcus, Jul 30 2017

A056170(a(n)) = 1.

a(n) ~ k*n, where k = Pi^2/(6*A154945) = 2.9816096.... - Charles R Greathouse IV, Aug 02 2016

A212172 Row n of table represents second signature of n: list of exponents >= 2 in canonical prime factorization of n, in nonincreasing order, or 0 if no such exponent exists.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 3, 2, 0, 3, 2, 0, 0, 0, 5, 0, 0, 0, 2, 2, 0, 0, 0, 3, 0, 0, 0, 2, 2, 0, 0, 4, 2, 2, 0, 2, 0, 3, 0, 3, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0

Offset: 1

Matthew Vandermast, Jun 03 2012

Length of row n equals A056170(n) if A056170(n) is positive, or 1 if A056170(n) = 0.
The multiset of exponents >=2 in the prime factorization of n completely determines a(n) for over 20 sequences in the database (see crossreferences). It also determines the fractions A034444(n)/A000005(n) and A037445(n)/A000005(n).
For squarefree numbers, this multiset is { } (the empty multiset). The use of 0 in the table to represent each n with no exponents >=2 in its prime factorization accords with the usual OEIS practice of using 0 to represent nonexistent elements when possible. In comments, the second signature of squarefree numbers will be represented as { }.
For each second signature {S}, there exist values of j and k such that, if the second signature of n is {S}, then A085082(n) is congruent to j modulo k.  These values are nontrivial unless {S} = { }. Analogous (but not necessarily identical) values of j and k also exist for each second signature with respect to A088873 and A181796.
Each sequence of integers with a given second signature {S} has a positive density, unlike the analogous sequences for prime signatures. The highest of these densities is 6/Pi^2 = 0.607927... for A005117 ({S} = { }).

			First rows of table read: 0; 0; 0; 2; 0; 0; 0; 3; 2; 0; 0; 2;...
12 = 2^2*3 has positive exponents 2 and 1 in its canonical prime factorization (1s are often left implicit as exponents). Since only exponents that are 2 or greater appear in a number's second signature, 12's second signature is {2}.
30 = 2*3*5 has no exponents greater than 1 in its prime factorization. The multiset of its exponents >= 2 is { } (the empty multiset), represented in the table with a 0.
72 = 2^3*3^2 has positive exponents 3 and 2 in its prime factorization, as does 108 = 2^2*3^3. Rows 72 and 108 both read {3,2}.

Jason Kimberley, Table of i, a(i) for i = 1..10575 (n = 1..10000)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Primefan, The First 2500 Integers Factored (1st of 5 pages)

A181800 gives first integer of each second signature. Also see A212171, A212173-A212181, A212642-A212644.
Functions determined by exponents >=2 in the prime factorization of n:
Multiplicative: A000688, A005361, A008966, A038538, A046951, A049419, A050361, A050377, A056624, A061704, A063775, A162510, A162511, A212181.
Additive: A046660, A056170.
Other: A007424, A051903 (for n > 1), A056626, A066301, A071325, A072411, A091050, A107078, A185102 (for n > 1), A212180.
Sequences that contain all integers of a specific second signature: A005117 (second signature { }), A060687 ({2}), A048109 ({3}).

&cat[IsEmpty(e)select [0]else Reverse(Sort(e))where e is[pe[2]:pe in Factorisation(n)|pe[2]gt 1]:n in[1..102]]; // Jason Kimberley, Jun 13 2012

row[n_] := Select[ FactorInteger[n][[All, 2]], # >= 2 &] /. {} -> 0 /. {k__} -> Sequence[k]; Table[row[n], {n, 1, 100}] (* Jean-François Alcover, Apr 16 2013 *)

For nonsquarefree n, row n is identical to row A057521(n) of table A212171.

A275734 Prime-factorization representations of "factorial base slope polynomials": a(0) = 1; for n >= 1, a(n) = A275732(n) * a(A257684(n)).

Original entry on oeis.org

1, 2, 3, 6, 2, 4, 5, 10, 15, 30, 10, 20, 3, 6, 9, 18, 6, 12, 2, 4, 6, 12, 4, 8, 7, 14, 21, 42, 14, 28, 35, 70, 105, 210, 70, 140, 21, 42, 63, 126, 42, 84, 14, 28, 42, 84, 28, 56, 5, 10, 15, 30, 10, 20, 25, 50, 75, 150, 50, 100, 15, 30, 45, 90, 30, 60, 10, 20, 30, 60, 20, 40, 3, 6, 9, 18, 6, 12, 15, 30, 45, 90, 30, 60, 9, 18, 27

Offset: 0

Antti Karttunen, Aug 08 2016

These are prime-factorization representations of single-variable polynomials where the coefficient of term x^(k-1) (encoded as the exponent of prime(k) in the factorization of n) is equal to the number of nonzero digits that occur on the slope (k-1) levels below the "maximal slope" in the factorial base representation of n. See A275811 for the definition of the "digit slopes" in this context.

			For n=23 ("321" in factorial base representation, A007623), all three nonzero digits are maximal for their positions (they all occur on "maximal slope"), thus a(23) = prime(1)^3 = 2^3 = 8.
For n=29 ("1021"), there are three nonzero digits, where both 2 and the rightmost 1 are on the "maximal slope", while the most significant 1 is on the "sub-sub-sub-maximal", thus a(29) = prime(1)^2 * prime(4)^1 = 2*7 = 28.
For n=37 ("1201"), there are three nonzero digits, where the rightmost 1 is on the maximal slope, 2 is on the sub-maximal, and the most significant 1 is on the "sub-sub-sub-maximal", thus a(37) = prime(1) * prime(2) * prime(4) = 2*3*7 = 42.
For n=55 ("2101"), the least significant 1 is on the maximal slope, and the digits "21" at the beginning are together on the sub-sub-maximal slope (as they are both two less than the maximal digit values 4 and 3 allowed in those positions), thus a(55) = prime(1)^1 * prime(3)^2 = 2*25 = 50.

Antti Karttunen, Table of n, a(n) for n = 0..40320
Index entries for sequences related to factorial base representation

Cf. A001221, A001222, A002110, A007489, A007814, A048675, A051903, A056169, A056170, A060130, A060502, A225901.
Cf. A257684, A275732.
Cf. A275811.
Cf. A260736, A275728, A275808, A275812, A275946, A275947, A275962.
Cf. A275804 (indices of squarefree terms), A275805 (of terms not squarefree).
Cf. also A275725, A275733, A275735, A276076 for other such prime factorization encodings of A060117/A060118-related polynomials.

from operator import mul
from sympy import prime, factorial as f
def a007623(n, p=2): return n if n0 else '0' for i in x)[::-1]
    return 0 if n==1 else sum(int(y[i])*f(i + 1) for i in range(len(y)))
def a(n): return 1 if n==0 else a275732(n)*a(a257684(n))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 19 2017

a(0) = 1; for n >= 1, a(n) = A275732(n) * a(A257684(n)).
Other identities and observations. For all n >= 0:
a(n) = A275735(A225901(n)).
a(A007489(n)) = A002110(n).
A001221(a(n)) = A060502(n).
A001222(a(n)) = A060130(n).
A007814(a(n)) = A260736(n).
A051903(a(n)) = A275811(n).
A048675(a(n)) = A275728(n).
A248663(a(n)) = A275808(n).
A056169(a(n)) = A275946(n).
A056170(a(n)) = A275947(n).
A275812(a(n)) = A275962(n).

A107078 Whether n has non-unitary prime divisors.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0

Offset: 1

Paul Barry, May 10 2005

Also the characteristic function of the numbers that are not squarefree: A013929. - Enrique Pérez Herrero, Jul 08 2012

The sequence of partial sums of this sequence is A057627. - Jason Kimberley, Feb 01 2017

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
Index entries for characteristic functions.

Cf. A087049. - R. J. Mathar, Aug 24 2008

Cf. A013929, A008683, A008966, A107078, A229099.

seq(1 - abs(numtheory:-mobius(n)), n = 1..101); # Peter Luschny, Jul 27 2023

Table[1-MoebiusMu[n]^2,{n,1,100}] (* Enrique Pérez Herrero, Jul 08 2012 *)

from sympy import mobius
def A107078(n): return int(not mobius(n)) # Chai Wah Wu, Dec 05 2024

a(n) = 1 if A056170(n)>0, 0 otherwise.
a(n) = A107079(n) - A013928(n+1).
a(n) = 1 - A008966(n). - Reinhard Zumkeller, Oct 03 2008
a(n) = Sum_{k=0..n-1} (mu(n-k-1) mod 2) - Sum_{k=0..n-1} (mu(n-k) mod 2).
a(n) = abs(mu(n) - (-1)^omega(n)) = (mu(n) - (-1)^omega(n))^2 = abs(A008683(n) - (-1)^A001221(n)). - Enrique Pérez Herrero, Apr 28 2012
a(n) = 1 - mu(n)^2. - Enrique Pérez Herrero, Jul 08 2012
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 - 6/Pi^2 (A229099). - Amiram Eldar, Jul 24 2022

A036785 Numbers divisible by the squares of two distinct primes.

Original entry on oeis.org

36, 72, 100, 108, 144, 180, 196, 200, 216, 225, 252, 288, 300, 324, 360, 392, 396, 400, 432, 441, 450, 468, 484, 500, 504, 540, 576, 588, 600, 612, 648, 675, 676, 684, 700, 720, 756, 784, 792, 800, 828, 864, 882, 900, 936, 968, 972, 980, 1000, 1008, 1044

Offset: 1

Alford Arnold

Not squarefree, not a nontrivial prime power and not in {squarefree} times {nontrivial prime powers}.

Numbers k such that A056170(k) > 1. The asymptotic density of this sequence is 1 - (6/Pi^2) * (1 + A154945) = 0.05668359058... - Amiram Eldar, Nov 01 2020

CRC Standard Mathematical Tables and Formulae, 30th ed., (1996) page 102-105.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A005117, A025475, A056170, A154945.
Equivalent sequence for 3 distinct primes: A318720.
Cf. A085986, A338539, A339245 (subsequences).
Subsequence of A038838.

Select[Range@ 1050, And[Length@ # > 1, Total@ Boole@ Map[# > 1 &, #[[All, -1]]] > 1] &@ FactorInteger@ # &] (* Michael De Vlieger, Apr 25 2017 *)
dstdpQ[n_]:=Length[Select[Sqrt[#]&/@Divisors[n],PrimeQ]]>1; Select[ Range[ 1100],dstdpQ] (* Harvey P. Dale, Jan 15 2020 *)

is(n)=my(f=vecsort(factor(n)[,2],,4));#f>1&&f[2]>1 \\ Charles R Greathouse IV, Nov 15 2012

More terms from Larry Reeves (larryr(AT)acm.org), Apr 03 2000

New name from Charles R Greathouse IV, Nov 15 2012

A275725 a(n) = A275723(A002110(1+A084558(n)), n); prime factorization encodings of cycle-polynomials computed for finite permutations listed in the order that is used in tables A060117 / A060118.

Original entry on oeis.org

2, 4, 18, 8, 12, 8, 150, 100, 54, 16, 24, 16, 90, 40, 54, 16, 36, 16, 60, 40, 36, 16, 24, 16, 1470, 980, 882, 392, 588, 392, 750, 500, 162, 32, 48, 32, 270, 80, 162, 32, 108, 32, 120, 80, 72, 32, 48, 32, 1050, 700, 378, 112, 168, 112, 750, 500, 162, 32, 48, 32, 450, 200, 162, 32, 72, 32, 300, 200, 108, 32, 48, 32, 630, 280, 378, 112, 252, 112, 450, 200

Offset: 0

Antti Karttunen, Aug 09 2016

nonn

In this context "cycle-polynomials" are single-variable polynomials where the coefficients (encoded with the exponents of prime factorization of n) are equal to the lengths of cycles in the permutation listed with index n in tables A060117 or A060118. See the examples.

			Consider the first eight permutations (indices 0-7) listed in A060117:
  1 [Only the first 1-cycle explicitly listed thus a(0) = 2^1 = 2]
  2,1 [One transposition (2-cycle) in beginning, thus a(1) = 2^2 = 4]
  1,3,2 [One fixed element in beginning, then transposition, thus a(2) = 2^1 * 3^2 = 18]
  3,1,2 [One 3-cycle, thus a(3) = 2^3 = 8]
  3,2,1 [One transposition jumping over a fixed element, a(4) = 2^2 * 3^1 = 12]
  2,3,1 [One 3-cycle, thus a(5) = 2^3 = 8]
  1,2,4,3 [Two 1-cycles, then a 2-cycle, thus a(6) = 2^1 * 3^1 * 5^2 = 150].
  2,1,4,3 [Two 2-cycles, not crossed, thus a(7) = 2^2 * 5^2 = 100]
and also the seventeenth one at n=16 [A007623(16)=220] where we have:
  3,4,1,2 [Two 2-cycles crossed, thus a(16) = 2^2 * 3^2 = 36].

Antti Karttunen, Table of n, a(n) for n = 0..40319
Indranil Ghosh, Python program for computing this sequence
Index entries for sequences related to factorial base representation

Cf. A000040, A001222, A001221, A002110, A007814, A046660, A048675, A051903, A056169, A056170, A084558, A243054, A257510, A275723, A275803, A275832.
Cf. A275807 (terms divided by 2).
Cf. also A060129, A060128, A060130, A060131, A072411, A275726, A275812, A275851.
Cf. also A275733, A275734, A275735 for other such prime factorization encodings of A060117/A060118-related polynomials.

(define (A275725 n) (A275723bi (A002110 (+ 1 (A084558 n))) n)) ;; Code for A275723bi given in A275723.

a(n) = A275723(A002110(1+A084558(n)), n).
Other identities:
A001221(a(n)) = 1+A257510(n) (for all n >= 1).
A001222(a(n)) = 1+A084558(n).
A007814(a(n)) = A275832(n).
A048675(a(n)) = A275726(n).
A051903(a(n)) = A275803(n).
A056169(a(n)) = A275851(n).
A046660(a(n)) = A060130(n).
A072411(a(n)) = A060131(n).
A056170(a(n)) = A060128(n).
A275812(a(n)) = A060129(n).
a(n!) = 2 * A243054(n) = A000040(n)*A002110(n) for all n >= 1.

A275812 Sum of exponents larger than one in the prime factorization of n: A001222(n) - A056169(n).

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 3, 2, 0, 3, 2, 0, 0, 0, 5, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, 2, 2, 0, 0, 4, 2, 2, 0, 2, 0, 3, 0, 3, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 0, 5, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0, 0, 3, 0, 2, 0, 2, 0, 0, 0, 5, 0, 2, 2, 4, 0, 0, 0, 3, 0, 0, 0, 5, 0, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0, 3

Offset: 1

Antti Karttunen, Aug 11 2016

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A001222, A001694, A005117, A028234, A046660, A056169, A056170, A057521, A067029, A247180.
Differs from A212172 for the first time at n=36, where a(36)=4, while A212172(36)=2.
Cf. A085548, A136141.

Table[Total@ Map[Last, Select[FactorInteger@ n, Last@ # > 1 &] /. {} -> {{0, 0}}], {n, 120}] (* Michael De Vlieger, Aug 11 2016 *)

a(n) = my(f = factor(n)); sum(k=1, #f~, if (f[k,2] > 1, f[k,2])); \\ Michel Marcus, Jul 19 2017

sub a275812 { vecsum( grep {$> 1} map {$->[1]} factor_exp(shift) ); } # Dana Jacobsen, Aug 15 2016

from sympy import factorint, primefactors
def a001222(n):
    return 0 if n==1 else a001222(n//primefactors(n)[0]) + 1
def a056169(n):
    f=factorint(n)
    return 0 if n==1 else sum(1 for i in f if f[i]==1)
def a(n):
    return a001222(n) - a056169(n)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 19 2017

a(1) = 0, and for n > 1, if A067029(n)=1 [when n is one of the terms of A247180], a(n) = a(A028234(n)), otherwise a(n) = A067029(n)+a(A028234(n)).
a(n) = A001222(n) - A056169(n).
a(n) = A001222(A057521(n)). - Antti Karttunen, Jul 19 2017
From Amiram Eldar, Sep 28 2023: (Start)
Additive with a(p) = 0, and a(p^e) = e for e >= 2.
a(n) >= 0, with equality if and only if n is squarefree (A005117).
a(n) <= A001222(n), with equality if and only if n is powerful (A001694).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} (1/p^2 + 1/(p*(p-1))) = A085548 + A136141 = 1.22540408909086062637... . (End)
a(n) = A046660(n) + A056170(n). - Amiram Eldar, Jan 09 2024

A056172 Number of non-unitary prime divisors of n!.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14

Offset: 1

Labos Elemer, Jul 27 2000

A non-unitary prime divisor for n! cannot exceed n/2.

			10! = 2^8 * 3^4 * 5^2 * 7. The non-unitary prime divisors are 2, 3, and 5 because their exponents exceed 1, so a(10) = 3.  The only unitary prime divisor of 10! is 7.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000720, A001221, A034444, A048105, A048656, A048657, A056170, A056171.

with(numtheory); A056172:=n->pi(floor(n/2)); seq(A056172(k),k=1..100); # Wesley Ivan Hurt, Sep 30 2013

Table[PrimePi[Floor[n/2]], {n,100}] (* Wesley Ivan Hurt, Sep 30 2013 *)

a(n) = pi(n/2).
A prime divisor of x is non-unitary iff its exponent is at least 2 in the prime power factorization of x. In general, GCD(p, x/p) = 1 or p. Cases are counted when GCD(p, n/p) > 1.
a(n) = A000720(n) - A056171(n). - Robert G. Wilson v, Apr 09 2017
a(n) = A056170(n!). - Amiram Eldar, Jul 24 2024

Example corrected by Jon E. Schoenfield, Sep 30 2013

cp's OEIS Frontend

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