A162642 -id:A162642 - cp's OEIS frontend

A056169 Number of unitary prime divisors of n.

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

Offset: 1

Labos Elemer, Jul 27 2000

The zeros of this sequences are the powerful numbers (A001694). There are no arbitrarily long subsequences with a given upper bound; for example, every sequence of 4 values includes one divisible by 2 but not 4, so there are no more than 3 consecutive zeros. Similarly, there can be no more than 23 consecutive values with none divisible by both 2 and 3 but neither 4 nor 9 (so a(n) >= 2), etc. In general, this gives an upper bound that is a (relatively) small multiple of the k-th primorial number (prime(k)#). One suspects that the actual upper bounds for such subsequences are quite a bit lower; e.g., Erdős conjectured that there are no three consecutive powerful numbers. - Franklin T. Adams-Watters, Aug 08 2006
In particular, for every A048670(k)*A002110(k) consecutive terms, at least one is greater than or equal to k. - Charlie Neder, Jan 03 2019
Following Catalan's conjecture (which became Mihăilescu's theorem in 2002), the first case of two consecutive zeros in this sequence is for a(8) and a(9), because 8 = 2^3 and 9 = 3^2, and there are no other consecutive zeros for consecutive powers. However, there are other pairs of consecutive zeros at powerful numbers (A001694, A060355). The next example is a(288) = a(289) = 0, because 288 = 2^5 * 3^2 and 289 = 17^2, then also a(675) and a(676). - Bernard Schott, Jan 06 2019
a(2k-1) is the number of primes p such that p || x + y and p^2 || x^(2k-1) + y^(2k-1) for some positive integers x and y. For any positive integers x, y and k > 1, there is no prime p such that p || x + y and p^2 || x^(2k) + y^(2k). - Jinyuan Wang, Apr 08 2020

			9 = 3^2 so a(9) = 0; 10 = 2 * 5 so a(10) = 2; 11 = 11^1 so a(11) = 1.

T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Catalan's Conjecture.
Index entries for sequences computed from exponents in factorization of n.

Cf. A001221, A001694, A002110, A034444, A056170, A055231, A076445, A162642, A275812, A295659, A295662, A295664, A001694.
See also A060355.
Cf. A077761, A085548.

a056169 = length . filter (== 1) . a124010_row
-- Reinhard Zumkeller, Sep 10 2013

a:= n-> nops(select(i-> i[2]=1, ifactors(n)[2])):
seq(a(n), n=1..120);  # Alois P. Heinz, Mar 27 2017

Join[{0},Table[Count[Transpose[FactorInteger[n]][[2]],1],{n,2,110}]] (* Harvey P. Dale, Mar 15 2012 *)
Table[DivisorSum[n, 1 &, And[PrimeQ@ #, CoprimeQ[#, n/#]] &], {n, 105}] (* Michael De Vlieger, Nov 28 2017 *)

a(n)=my(f=factor(n)[,2]); sum(i=1,#f,f[i]==1) \\ Charles R Greathouse IV, Apr 29 2015

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

;; With memoization-macro definec.
(definec (A056169 n) (if (= 1 n) 0 (+ (if (= 1 (A067029 n)) 1 0) (A056169 (A028234 n))))) ;; Antti Karttunen, Nov 28 2017

A prime factor of n is unitary iff its exponent is 1 in prime factorization of n. In general, gcd(p, n/p) = 1 or = p.
Additive with a(p^e) = 1 if e = 1, 0 otherwise.
a(n) = #{k: A124010(n,k) = 1, k = 1..A001221}. - Reinhard Zumkeller, Sep 10 2013
From Antti Karttunen, Nov 28 2017: (Start)
a(1) = 0; for n > 1, a(n) = A063524(A067029(n)) + a(A028234(n)).
a(n) = A001221(A055231(n)) = A001222(A055231(n)).
a(n) = A001221(n) - A056170(n) = A001221(n) - A001221(A000188(n)).
a(n) = A001222(n) - A275812(n).
a(n) = A162642(n) - A295662(n).
a(n) <= A162642(n) <= a(n) + A295659(n).
a(n) <= A295664(n).
(End)
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B - C), where B is Mertens's constant (A077761) and C = Sum_{p prime} (1/p^2) = 0.452247... (A085548). - Amiram Eldar, Sep 28 2023

A316523 Number of odd multiplicities minus number of even multiplicities in the canonical prime factorization of n.

Original entry on oeis.org

0, 1, 1, -1, 1, 2, 1, 1, -1, 2, 1, 0, 1, 2, 2, -1, 1, 0, 1, 0, 2, 2, 1, 2, -1, 2, 1, 0, 1, 3, 1, 1, 2, 2, 2, -2, 1, 2, 2, 2, 1, 3, 1, 0, 0, 2, 1, 0, -1, 0, 2, 0, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 0, -1, 2, 3, 1, 0, 2, 3, 1, 0, 1, 2, 0, 0, 2, 3, 1, 0, -1, 2, 1, 1

Offset: 1

Gus Wiseman, Jul 05 2018

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A000607, A071321, A100118, A112798.
Cf. A187039 (where a(n)=0). - Michel Marcus, Jul 08 2018
Cf. A077761, A162641, A162642, A179119.

f:= proc(n) local F;
  F:= map(t -> t[2],ifactors(n)[2]);
  2*nops(select(type,F,odd))-nops(F);
end proc:
map(f, [$1..100]); # Robert Israel, Aug 27 2018

Table[Total[-(-1)^If[n==1,{},FactorInteger[n][[All,2]]]],{n,100}]

a(n) = my(f=factor(n)); -sum(k=1, #f~, (-1)^(f[k,2])); \\ Michel Marcus, Jul 08 2018; corrected Jun 13 2022

If i and j are coprime, a(i*j) = a(i)+a(j). - Robert Israel, Aug 27 2018
From Amiram Eldar, Oct 05 2023: (Start)
Additive with a(p^e) = (-1)^(e+1).
a(n) = A162642(n) - A162641(n).
Sum_{k=1..n} a(k) = n * log(log(n)) + c * n + O(n/log(n)), where c = A077761 - 2*A179119 = -0.398962... . (End)

A366528 Sum of odd prime indices of n.

Original entry on oeis.org

0, 1, 0, 2, 3, 1, 0, 3, 0, 4, 5, 2, 0, 1, 3, 4, 7, 1, 0, 5, 0, 6, 9, 3, 6, 1, 0, 2, 0, 4, 11, 5, 5, 8, 3, 2, 0, 1, 0, 6, 13, 1, 0, 7, 3, 10, 15, 4, 0, 7, 7, 2, 0, 1, 8, 3, 0, 1, 17, 5, 0, 12, 0, 6, 3, 6, 19, 9, 9, 4, 0, 3, 21, 1, 6, 2, 5, 1, 0, 7, 0, 14, 23, 2

Offset: 1

Gus Wiseman, Oct 22 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239(n).

			The prime indices of 198 are {1,2,2,5}, so a(198) = 1+5 = 6.

Zeros are A066207, counted by A035363.
The triangle for this rank statistic is A113685, without zeros A365067.
For count instead of sum we have A257991, even A257992.
Nonzeros are A366322, counted by A086543.
The even version is A366531, halved A366533, triangle A113686.
A000009 counts partitions into odd parts, ranks A066208.
A053253 = partitions with all odd parts and conjugate parts, ranks A352143.
A066967 adds up sums of odd parts over all partitions.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A162641 counts even prime exponents, odd A162642.
A352142 = odd indices with odd exponents, counted by A117958.
Cf. A000720, A055396, A130780, A171966, A174713, A325698, A325700, A366748.

Table[Total[Cases[FactorInteger[n], {p_?(OddQ@*PrimePi),k_}:>PrimePi[p]*k]],{n,100}]

a(n) = A056239(n) - A366531(n).

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

A293442 Multiplicative with a(p^e) = A019565(e).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 6, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 12, 3, 4, 6, 6, 2, 8, 2, 10, 4, 4, 4, 9, 2, 4, 4, 12, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 12, 4, 12, 4, 4, 2, 12, 2, 4, 6, 15, 4, 8, 2, 6, 4, 8, 2, 18, 2, 4, 6, 6, 4, 8, 2, 10, 5, 4, 2, 12, 4, 4, 4, 12, 2, 12, 4, 6, 4, 4, 4, 20, 2, 6, 6, 9, 2, 8, 2, 12, 8

Offset: 1

Antti Karttunen, Oct 31 2017

From Peter Munn, Apr 06 2021: (Start)
a(n) is determined by the prime signature of n.
Compare with the multiplicative, self-inverse A225546, which also maps 2^e to the squarefree number A019565(e). However, this sequence maps p^e to the same squarefree number for every prime p, whereas A225546 maps the e-th power of progressively larger primes to progressively greater powers of A019565(e).
Both sequences map powers of squarefree numbers to powers of squarefree numbers.
(End)

Sequences used in a definition of this sequence: A000188, A003961, A019565, A028234, A059895, A067029, A162642.
Sequences with related definitions: A225546, A293443, A293444.
Cf. also A293214.
Sequences used to express relationship between terms of this sequence: A006519, A007913, A008833, A064989, A334747.
Sequences related via this sequence: (A001222, A048675, A064547), (A007814, A162642), (A087207, A267116), (A248663, A268387).

f[n_] := If[n == 1, 1, Apply[Times, Prime@ Flatten@ Position[Reverse@ IntegerDigits[Last@ #, 2], 1]] * f[n/Apply[Power, #]] &@ FactorInteger[n][[1]]]; Array[f, 105] (* Michael De Vlieger, Oct 31 2017 *)

a(1) = 1; for n > 1, a(n) = A019565(A067029(n)) * a(A028234(n)).
Other identities. For all n >= 1:
a(a(n)) = A293444(n).
A048675(a(n)) = A001222(n).
A001222(a(n)) = A064547(n) = A048675(A293444(n)).
A007814(a(n)) = A162642(n).
A087207(a(n)) = A267116(n).
A248663(a(n)) = A268387(n).
From Peter Munn, Mar 14 2021: (Start)
Alternative definition: a(1) = 1; a(2) = 2; a(n^2) = A003961(a(n)); a(A003961(n)) = a(n); if A059895(n, k) = 1, a(n*k) = a(n) * a(k).
For n >= 3, a(n) < n.
a(2n) = A334747(a(A006519(n))) * a(n/A006519(n)), where A006519(n) is the largest power of 2 dividing n.
a(2n+1) = a(A064989(2n+1)).
a(n) = a(A007913(n)) * a(A008833(n)) = 2^A162642(n) * A003961(a(A000188(n))).
(End)

A229125 Numbers of the form p * m^2, where p is prime and m > 0: union of A228056 and A000040.

Original entry on oeis.org

2, 3, 5, 7, 8, 11, 12, 13, 17, 18, 19, 20, 23, 27, 28, 29, 31, 32, 37, 41, 43, 44, 45, 47, 48, 50, 52, 53, 59, 61, 63, 67, 68, 71, 72, 73, 75, 76, 79, 80, 83, 89, 92, 97, 98, 99, 101, 103, 107, 108, 109, 112, 113, 116, 117, 124, 125, 127, 128, 131, 137, 139, 147, 148, 149

Offset: 1

Chris Boyd, Sep 14 2013

nonn

No term is the product of two other terms.
Squares of terms and pairwise products of distinct terms form a subsequence of A028260.
Numbers n such that A162642(n) = 1. - Jason Kimberley, Oct 10 2016
Numbers k such that A007913(k) is a prime number. - Amiram Eldar, Jul 27 2020

Chris Boyd, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical notes, IX. On the set of integers representable as a product of a prime and square, Acta Arithmetica, Vol. 7 (1962), pp. 417-420.

Subsequence of A026424.
Cf. A028260, A162642, A229153.
Cf. A007913, A013661.
Subsequences: A000040, A030078, A050997, A054753, A092759, A179643, A179665, A246551.

With[{nn=70},Take[Union[Flatten[Table[p*m^2,{p,Prime[Range[nn]]},{m,nn}]]], nn]] (* Harvey P. Dale, Dec 02 2014 *)

test(n)=isprime(core(n))
for(n=1,200,if(test(n), print1(n",")))

from math import isqrt
from sympy import primepi
def A229125(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 f(x): return n+x-sum(primepi(x//y**2) for y in range(1,isqrt(x)+1))
    return bisection(f,n,n) # Chai Wah Wu, Jan 30 2025

The number of terms not exceeding x is (Pi^2/6) * x/log(x) + O(x/(log(x))^2) (Cohen, 1962). - Amiram Eldar, Jul 27 2020

A055076 Multiplicity of Max{gcd(d, n/d)} when d runs over divisors of n.

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 2, 1, 4, 2, 2, 2, 4, 4, 1, 2, 2, 2, 2, 4, 4, 2, 4, 1, 4, 2, 2, 2, 8, 2, 2, 4, 4, 4, 1, 2, 4, 4, 4, 2, 8, 2, 2, 2, 4, 2, 2, 1, 2, 4, 2, 2, 4, 4, 4, 4, 4, 2, 4, 2, 4, 2, 1, 4, 8, 2, 2, 4, 8, 2, 2, 2, 4, 2, 2, 4, 8, 2, 2, 1, 4, 2, 4, 4, 4, 4, 4, 2, 4, 4, 2, 4, 4, 4, 4, 2, 2, 2, 1, 2, 8, 2, 4, 8

Offset: 1

Labos Elemer, Jun 13 2000

Number of distinct values of gcd(d, n!/d) if d runs over divisors of n! seems to be A046951(n).
a(n) = 1 iff n is a square. - Bernard Schott, Oct 22 2019
a(n) is the number of the unitary divisors (cf. A077610) of n that are exponentially odd (A268335). - Amiram Eldar, Nov 11 2022
The number of infinitary divisors of n that are squarefree (A005117). - Amiram Eldar, Jan 09 2024

			n=120, the set of gcd(d, 120/d) values for the 16 divisors of 120 is {1,2,1,2,1,2,1,2,2,1,2,1,2,1,2,1}. The max is 2 and it occurs 8 times, so a(120)=8. This sequence seems to consist of powers of 2.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms).
Christian Krause, LODA, an assembly language, a computational model and a tool for mining integer sequences.
Index entries for sequences computed from exponents in factorization of n.

Cf. A000188, A005117, A007913, A034444, A077610, A162642, A268335, A295316, A350389.

with(numtheory):
a:= n->(p->coeff(p, x, degree(p)))(add(x^igcd(d, n/d), d=divisors(n))):
seq(a(n), n=1..105);  # Alois P. Heinz, Jul 21 2015

a[n_] := With[{g = GCD[#, n/#]& /@ Divisors[n]}, Count[g, Max[g]]];
Array[a, 105] (* Jean-François Alcover, Mar 28 2017 *)
f[p_, e_] := 2^Mod[e, 2]; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Nov 11 2022 *)

A055076(n) = if(1==n,n,my(es=factor(n)[,2]~); prod(i=1,#es,2^(es[i]%2))); \\ Antti Karttunen, Apr 05 2021

;; With memoization-macro definec.
(definec (A055076 n) (if (= 1 n) n (* (+ 1 (A000035 (A067029 n))) (A055076 (A028234 n))))) ;; Antti Karttunen, Dec 02 2017

Multiplicative with a(p^e) = 2^(e mod 2). - Vladeta Jovovic, Dec 13 2002
a(n) = 2^A162642(n). - Antti Karttunen, Dec 02 2017
a(n) = A034444(A007913(n)). [Found by LODA miner, see C. Krause link. Essentially the same formula as the above ones] - Antti Karttunen, Apr 05 2021
From Amiram Eldar, Sep 09 2023: (Start)
a(n) = A034444(A350389(n)).
Dirichlet g.f.: zeta(2*s) * Product_{p prime} (1 + 2/p^s). (End)
From Vaclav Kotesovec, Sep 09 2023: (Start)
Let f(s) = Product_{p prime} (1 - 3/p^(2*s) + 2/p^(3*s)).
Dirichlet g.f.: zeta(s)^2 * zeta(2*s) * f(s).
Sum_{k=1..n} a(k) ~ (Pi^2 * f(1) * n / 6) * (log(n) + 2*gamma - 1 + 12*zeta'(2)/Pi^2 + f'(1)/f(1)), where
f(1) = A065473 = Product_{primes p} (1 - 3/p^2 + 2/p^3) = 0.286747428434478734107892712789838446434331844097056995641477859336652243...,
f'(1) = f(1) * Sum_{primes p} 6*log(p) / (p^2 + p - 2) = f(1) * 2.798014228561519243358371276385174449737670294137200281334256087932048625...
and gamma is the Euler-Mascheroni constant A001620. (End)

A187039 Numbers that have equal counts of even and odd exponents of primes in their factorization.

Original entry on oeis.org

1, 12, 18, 20, 28, 44, 45, 48, 50, 52, 63, 68, 72, 75, 76, 80, 92, 98, 99, 108, 112, 116, 117, 124, 147, 148, 153, 162, 164, 171, 172, 175, 176, 188, 192, 200, 207, 208, 212, 236, 242, 244, 245, 261, 268, 272, 275, 279, 284, 288, 292, 304, 316, 320, 325, 332

Offset: 1

Vladimir Shevelev, Mar 02 2011

nonn

Numbers k such that A162641(k) = A162642(k). - Amiram Eldar, Sep 27 2021

			108 = 2^2*3^3 has one even and one odd exponent in its factorization and therefore qualifies.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Zak Seidov)

Cf. A162641, A162642.

IsA187039:=func< n | #[ a: a in P | IsEven(a) ] eq #[ a: a in P | IsOdd(a) ] where P is [ g[2]: g in F ] where F is Factorization(n) >; [ n: n in [1..500] | IsA187039(n) ]; // Klaus Brockhaus, Mar 04 2011

Reap[Do[fi=FactorInteger[n];la=Mod[Last/@fi,2];If[Count[la,1]==Count[la,0],Sow[n]] ,{n,1,10000}]][[2,1]] (* Zak Seidov, Mar 04 2011 *)
eoeQ[n_]:=Module[{f=FactorInteger[n][[All,2]]},Count[ f,?OddQ]== Length[ f]/2]; Join[{1},Select[Range[400],eoeQ]] (* _Harvey P. Dale, Sep 23 2016 *)

A295664 Exponent of the highest power of 2 dividing number of divisors of n: a(n) = A007814(A000005(n)); 2-adic valuation of tau(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 28 2017

In the prime factorization of n = p1^e1 * ... pk^ek, add together the number of trailing 1-bits in each exponent e when they are written in binary.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A007814, A028234, A056169, A067029, A077761, A162642, A295663.

Cf. A000290 (positions of zeros).

Table[IntegerExponent[DivisorSigma[0, n], 2], {n, 120}] (* Michael De Vlieger, Nov 28 2017 *)

a(n) = valuation(numdiv(n), 2); \\ Michel Marcus, Nov 30 2017

from sympy import divisor_count
def A295664(n): return (~(m:=int(divisor_count(n))) & m-1).bit_length() # Chai Wah Wu, Jul 05 2022

Additive with a(p^e) = A007814(1+e).
a(1) = 0; for n > 1, a(n) = A007814(1+A067029(n)) + a(A028234(n)).
a(n) = A007814(A000005(n)).
a(n) >= A162642(n) >= A056169(n).
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) =-0.223720656976344505701..., where f(x) = -x + (1-x) * Sum_{k>=1} x^(2^k-1)/(1-x^(2^k)). - Amiram Eldar, Sep 28 2023

A295659 Number of exponents larger than 2 in the prime factorization of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 28 2017

nonn

			For n = 120 = 2^3 * 3^1 * 5^1 there is only one exponent larger than 2, thus a(120) = 1.
For n = 216 = 2^3 * 3^3 there are two exponents larger than 2, thus a(216) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A001221, A003557, A053150, A056169, A056170, A085541, A162642, A295657, A295662, A295883, A295884.

Cf. A004709 (positions of zeros), A046099 (of nonzeros), A212793.

Array[Count[FactorInteger[#][[All, -1]], ?(# > 2 &)] &, 105] (* _Michael De Vlieger, Nov 28 2017 *)

a(n) = { my(v = factor(n)[, 2], i=0); for(x=1, length(v), if(v[x]>2, i++)); i; } \\ Iain Fox, Nov 29 2017

Additive with a(p^e) = 1 if e>2, 0 otherwise.
a(n) = 0 iff A212793(n) = 1.
a(n) = A001221(A053150(n)).
a(n) = A056170(A003557(n)).
a(n) >= A295662(n) = A162642(n) - A056169(n).
a(n) = A295883(n) + A295884(n).
Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = Sum_{p prime} 1/p^3 = 0.174762... (A085541). - Amiram Eldar, Nov 01 2020

cp's OEIS Frontend

A056169 Number of unitary prime divisors of n.

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A316523 Number of odd multiplicities minus number of even multiplicities in the canonical prime factorization of n.

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A366528 Sum of odd prime indices of n.

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

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A293442 Multiplicative with a(p^e) = A019565(e).

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A229125 Numbers of the form p * m^2, where p is prime and m > 0: union of A228056 and A000040.

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A055076 Multiplicity of Max{gcd(d, n/d)} when d runs over divisors of n.

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