A056169 -id:A056169 - cp's OEIS frontend

A056170 Number of non-unitary prime divisors of n.

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

Offset: 1

Labos Elemer, Jul 27 2000

A prime factor of n is unitary iff its exponent is 1 in the prime factorization of n. (Of course for any prime p, GCD(p, n/p) is either 1 or p. For a unitary prime factor it must be 1.)
Number of squared primes dividing n. - Reinhard Zumkeller, May 18 2002
a(A005117(n)) = 0; a(A013929(n)) > 0; a(A190641(n)) = 1. - Reinhard Zumkeller, Dec 29 2012
First differences of A013940. - Jason Kimberley, Feb 01 2017
Number of exponents larger than 1 in the prime factorization of n. - Antti Karttunen, Nov 28 2017

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

Cf. A000188, A001221, A003557, A013940, A034444, A046660, A048105, A056169, A085548, A124010, A162641, A212177, A275812, A295659, A295666.

Cf. A057427(a(n)) = 1 - A008966(n).

a056170 = length . filter (> 1) . a124010_row
-- Reinhard Zumkeller, Dec 29 2012

A056170:=func;
[A056170(n):n in[1..105]];
// Jason Kimberley, Jan 22 2017

A056170 := n -> nops(select(t -> (t[2]>1), ifactors(n)[2]));
seq(A056170(n),n=1..100); # Robert Israel, Jun 03 2014

a[n_] := Count[FactorInteger[n], {, k /; k > 1}]; Table[a[n], {n, 105}]  (* Jean-François Alcover, Mar 23 2011 *)
Table[Count[FactorInteger[n][[All,2]],?(#>1&)],{n,110}] (* _Harvey P. Dale, Jul 08 2019 *)

a(n)=my(f=factor(n)[,2]); sum(i=1,#f,f[i]>1) \\ Charles R Greathouse IV, May 18 2015

from sympy import factorint
def a(n):
    f = factorint(n)
    return sum([1 for i in f if f[i]!=1]) # Indranil Ghosh, Apr 24 2017

Additive with a(p^e) = 0 if e = 1, 1 otherwise.
G.f.: Sum_{k>=1} x^(prime(k)^2)/(1 - x^(prime(k)^2)). - Ilya Gutkovskiy, Jan 01 2017
a(n) = log_2(A000005(A071773(n))). - observed by Velin Yanev, Aug 20 2017, confirmed by Antti Karttunen, Nov 28 2017
From Antti Karttunen, Nov 28 2017: (Start)
a(n) = A001221(n) - A056169(n).
a(n) = omega(A000188(n)) = omega(A003557(n)) = omega(A057521(n)) = omega(A295666(n)), where omega = A001221.
For all n >= 1 it holds that:
a(A003557(n)) = A295659(n).
a(n) >= A162641(n).
(End)
Dirichlet g.f.: primezeta(2s)*zeta(s). - Benedict W. J. Irwin, Jul 11 2018
Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = Sum_{p prime} 1/p^2 = 0.452247... (A085548). - Amiram Eldar, Nov 01 2020
a(n) = A275812(n) - A046660(n). - Amiram Eldar, Jan 09 2024

Minor edits by Franklin T. Adams-Watters, Mar 23 2011

A055231 Powerfree part of n: product of primes that divide n only once.

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 1, 1, 10, 11, 3, 13, 14, 15, 1, 17, 2, 19, 5, 21, 22, 23, 3, 1, 26, 1, 7, 29, 30, 31, 1, 33, 34, 35, 1, 37, 38, 39, 5, 41, 42, 43, 11, 5, 46, 47, 3, 1, 2, 51, 13, 53, 2, 55, 7, 57, 58, 59, 15, 61, 62, 7, 1, 65, 66, 67, 17, 69, 70, 71, 1, 73, 74, 3, 19, 77, 78, 79, 5

Offset: 1

Labos Elemer, Jun 21 2000

The previous name was: Write n = K^2*F where F is squarefree and F = g*f where g = gcd(K,F) and f = F/g; then a(n) = f(n) = F(n)/g(n). Thus gcd(K^2,f) = 1.
Differs from A007913; they coincide if and only if g(n) = 1.
a(n) is the powerfree part of n; i.e., if n=Product(pi^ei) over all i (prime factorization) then a(n)=Product(pi^ei) over those i with ei=1; if n=b*c^2*d^3 then a(n) is minimum possible value of b. - Henry Bottomley, Sep 01 2000
Also denominator of n/rad(n)^2, where rad is the squarefree kernel of n (A007947), numerator: A062378. - Reinhard Zumkeller, Dec 10 2002
Largest unitary squarefree number dividing n (the unitary squarefree kernel of n). - Steven Finch, Mar 01 2004
From Bernard Schott, Dec 19 2022: (Start)
a(n) = 1 iff n is a squareful number (A001694).
1 < a(n) < n iff n is a nonsquarefree number that is not squareful (A332785).
a(n) = n iff n is a squarefree number (A005117). (End)

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / n^2 for n = 1..1000000

Positions of 1's: A001694.
Cf. A008833, A007913, A007947, A000188, A057521, A055773 (computed for n!), A056169 (number of prime divisors), A056671 (number of divisors), A092261 (sum of divisors of the n-th term), A197863, A332785.
Cf. A005117 (subsequence).

A055231 := proc(n)
    a := 1 ;
    if n > 1 then
        for f in ifactors(n)[2] do
            if op(2, f) = 1 then
                a := a*op(1, f) ;
            end if;
        end do:
    end if;
    a ;
end proc: # R. J. Mathar, Dec 23 2011

rad[n_] := Times @@ First /@ FactorInteger[n]; a[n_] := Denominator[n/rad[n]^2]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jun 20 2013, after Reinhard Zumkeller *)
f[p_, e_] := If[e==1, p, 1]; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 07 2020 *)

A055231(n)={
   local(a=1);
   f=factor(n) ;
   for(i=1,matsize(f)[1],
         if( f[i,2] ==1, a *=  f[i,1]
         )
   ) ;
   a ;
} /* R. J. Mathar, Mar 12 2012 */

a(n) = {my(f=factor(n)); for (k=1, #f~, if (f[k,2] > 1, f[k,2] = 0);); factorback(f);} \\ Michel Marcus, Aug 27 2017

from math import prod
from sympy import factorint
def A055231(n): return prod(p for p, e in factorint(n).items() if e == 1) # Chai Wah Wu, Nov 14 2022

;; With memoization-macro definec.
(definec (A055231 n) (if (= 1 n) 1 (* (if (= 1 (A067029 n)) (A020639 n) 1) (A055231 (A028234 n))))) ;; Antti Karttunen, Nov 28 2017

a(n) = A007913(n)/gcd(A008833(n), A007913(n)).
a(n) = n/A057521(n).
Multiplicative with a(p) = p and a(p^e) = 1 for e > 1. - Vladeta Jovovic, Nov 01 2001
Dirichlet g.f.: zeta(s)*Product_{primes p} (1 + p^(1-s) - p^(-s) - p^(1-2s) + p^(-2s)). - R. J. Mathar, Dec 21 2011
a(n) = A007947(n)/A071773(n). - observed by Velin Yanev, Aug 27 2017, confirmed by Antti Karttunen, Nov 28 2017
a(1) = 1; for n > 1, a(n) = A020639(n)^A063524(A067029(n)) * a(A028234(n)). - Antti Karttunen, Nov 28 2017
a(n*m) = a(n)*a(m)/(gcd(n,a(m))*gcd(m,a(n))) for all n and m > 0 (conjectured). - Velin Yanev, Feb 06 2019. [This follows easily from the comment of Vladeta Jovovic. - N. J. A. Sloane, Mar 14 2019]
From Vaclav Kotesovec, Dec 19 2019: (Start)
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{primes p} (1 - p^(1-3*s) + p^(2-3*s) - p^(2-2*s) + p^(-2*s) - p^(-s)).
Sum_{k=1..n} a(k) ~ c * Pi^2 * n^2 / 12, where c = Product_{primes p} (1 - 2/p^2 + 2/p^4 - 1/p^5) = 0.394913518073109872954607634745304266741971541072... (End)
a(n) = A197863(n)/n. - Amiram Eldar, Sep 01 2023

Name replaced with a simpler description (based on Henry Bottomley's comment) by Antti Karttunen, Nov 28 2017

Incorrect comments and example deleted by Peter Munn, Nov 30 2022

A080359 The smallest integer x > 0 such that the number of primes in (x/2, x] equals n.

Original entry on oeis.org

2, 3, 13, 19, 31, 43, 53, 61, 71, 73, 101, 103, 109, 113, 139, 157, 173, 181, 191, 193, 199, 239, 241, 251, 269, 271, 283, 293, 313, 349, 353, 373, 379, 409, 419, 421, 433, 439, 443, 463, 491, 499, 509, 523, 577, 593, 599, 601, 607, 613, 619, 647, 653, 659

Offset: 1

Labos Elemer, Feb 21 2003

nonn

a(n) is the same as: Smallest integer x > 0 such that the number of unitary-prime-divisors of x! equals n.
Let p_n be the n-th prime. If p_n>3 is in the sequence, then all integers (p_n-1)/2, (p_n-3)/2, ..., (p_(n-1)+1)/2 are composite numbers. - Vladimir Shevelev, Aug 12 2009
For n >= 3, denote by q(n) the prime which is the nearest from the left to a(n)/2. Then there exists a prime between 2q(n) and a(n). The converse, generally speaking, is not true; i.e., there exist primes that are outside the sequence, but possess such property (e.g., 131). - Vladimir Shevelev, Aug 14 2009
See sequence A164958 for a generalization. - Vladimir Shevelev, Sep 02 2009
a(n) is the n-th Labos prime.

			n=5: in 31! five unitary-prime-divisors appear (firstly): {17,19,23,29,31}, while other primes {2,3,5,7,11,13} are at least squared. Thus a(5)=31.
Consider a(9)=71. Then the nearest prime < 71/2 is q(9)=31, and between 2q(9) and a(9), i.e., between 62 and 71 there exists a prime (67). - _Vladimir Shevelev_, Aug 14 2009

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..4460 from Daniel Forgues)
N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, and J. Sondow, Generalized Ramanujan primes, arXiv:1108.0475 [math.NT], 2011.
N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, and J. Sondow, Generalized Ramanujan primes, Combinatorial and Additive Number Theory, Springer Proc. in Math. & Stat., CANT 2011 and 2012, Vol. 101 (2014), 1-13.
Ethan Berkove and Michael Brilleslyper, Subgraphs of Coprime Graphs on Sets of Consecutive Integers, Integers (2022) Vol. 22, #A47, see p. 8.
Vladimir Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT], 2009.
Vladimir Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes, arXiv:0909.0715 [math.NT], 2009-2011.
Vladimir Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4.
Jonathan Sondow, MathWorld: Ramanujan Prime
Jonathan Sondow, Ramanujan primes and Bertrand's postulate, arXiv:0907.5232 [math.NT], 2009-2010.
Jonathan Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly, 116 (2009), 630-635.

Cf. A056171, A056169, A000720, A000142.
Cf. A104272 (Ramanujan primes).
Cf. A060756, A080360 (largest integer x with n primes in (x/2,x]).
Cf. A164554, A164288, A164333, A164294, A164372, A164371, A212493, A212541.

nn=1000; t=Table[0, {nn+1}]; s=0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s<=nn && t[[s+1]]==0, t[[s+1]]=k], {k, Prime[3*nn]}]; Rest[t]
(* Second program: *)
a[1] = 2; a[n_] := a[n] = Module[{x = a[n-1]}, While[(PrimePi[x]-PrimePi[Quotient[x, 2]]) != n, x++ ]; x]; Array[a, 54] (* Jean-François Alcover, Sep 14 2018 *)

a(n) = {my(x = 1); while ((primepi(x) - primepi(x\2)) != n, x++;); x;} \\ Michel Marcus, Jan 15 2014

def A():
    i = 0; n = 1
    while True:
        p = prime_pi(i) - prime_pi(i//2)
        if p == n:
            yield i
            n += 1
        i += 1
A080359 = A()
[next(A080359) for n in range(54)] # Peter Luschny, Sep 03 2014

a(n) = Min{x; Pi[x]-Pi[x/2]=n} = Min{x; A056171(x)=n}=Min{x; A056169(n!)=n}; where Pi()=A000720().

a(n) <= A193507(n) (cf. A194186). - Vladimir Shevelev, Aug 18 2011

Definition corrected by Jonathan Sondow, Aug 10 2008

Shrunk title and moved part of title to comments by John W. Nicholson, Sep 18 2011

A283477 If 2n = 2^e1 + 2^e2 + ... + 2^ek [e1 .. ek distinct], then a(n) = A002110(e1) * A002110(e2) * ... * A002110(ek).

Original entry on oeis.org

1, 2, 6, 12, 30, 60, 180, 360, 210, 420, 1260, 2520, 6300, 12600, 37800, 75600, 2310, 4620, 13860, 27720, 69300, 138600, 415800, 831600, 485100, 970200, 2910600, 5821200, 14553000, 29106000, 87318000, 174636000, 30030, 60060, 180180, 360360, 900900, 1801800, 5405400, 10810800, 6306300, 12612600, 37837800, 75675600

Offset: 0

Antti Karttunen, Mar 16 2017

nonn

a(n) = Product of distinct primorials larger than one, obtained as Product_{i} A002110(1+i), where i ranges over the zero-based positions of the 1-bits present in the binary representation of n.
This sequence can be represented as a binary tree. Each child to the left is obtained as A283980(k), and each child to the right is obtained as 2*A283980(k), when their parent contains k:
                                      1
                                      |
                   ...................2....................
                  6                                       12
       30......../ \........60                 180......../ \......360
       / \                 / \                 / \                 / \
      /   \               /   \               /   \               /   \
     /     \             /     \             /     \             /     \
  210       420      1260       2520     6300       12600   37800       75600
etc.

Cf. A129912 (same terms, but sorted into ascending order).
Cf. A000120, A001221, A001222, A002110, A005187, A005361, A006068, A007814, A007913, A019565, A029931, A030308, A046660, A048675, A051903, A056169, A065120, A070939, A072411, A090880, A097248, A108951, A124757, A248663, A276075, A276085, A283475, A283483, A283980, A283984, A283985, A284001, A284002, A284003, A284005, A324287, A324289, A324341, A324342, A324343.
Cf. A005940, A052330, A322827, A323505 for other similar trees.
Cf. also A260443.

Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e == 1 :> {Times @@ Prime@ Range@ PrimePi@ p, e}] &[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[n, 2]], {n, 0, 43}] (* Michael De Vlieger, Mar 18 2017 *)

A283477(n) = prod(i=0,exponent(n),if(bittest(n,i),vecprod(primes(1+i)),1)) \\ Edited by M. F. Hasler, Nov 11 2019

from sympy import prime, primerange, factorint
from operator import mul
from functools import reduce
def P(n): return reduce(mul, [i for i in primerange(2, n + 1)])
def a108951(n):
    f = factorint(n)
    return 1 if n==1 else reduce(mul, [P(i)**f[i] for i in f])
def a019565(n): return reduce(mul, (prime(i+1) for i, v in enumerate(bin(n)[:1:-1]) if v == '1')) if n > 0 else 1 # after Chai Wah Wu
def a(n): return a108951(a019565(n))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 22 2017

from sympy import primorial
from math import prod
def A283477(n): return prod(primorial(i) for i, b in enumerate(bin(n)[:1:-1],1) if b =='1') # Chai Wah Wu, Dec 08 2022

(define (A283477 n) (A108951 (A019565 n)))
;; Recursive "binary tree" implementation, using memoization-macro definec:
(definec (A283477 n) (cond ((zero? n) 1) ((even? n) (A283980 (A283477 (/ n 2)))) (else (* 2 (A283980 (A283477 (/ (- n 1) 2)))))))

a(0) = 1; a(2n) = A283980(a(n)), a(2n+1) = 2*A283980(a(n)).
Other identities. For all n >= 0 (or for n >= 1):
a(2n+1) = 2*a(2n).
a(n) = A108951(A019565(n)).
A097248(a(n)) = A283475(n).
A007814(a(n)) = A051903(a(n)) = A000120(n).
A001221(a(n)) = A070939(n).
A001222(a(n)) = A029931(n).
A048675(a(n)) = A005187(n).
A248663(a(n)) = A006068(n).
A090880(a(n)) = A283483(n).
A276075(a(n)) = A283984(n).
A276085(a(n)) = A283985(n).
A046660(a(n)) = A124757(n).
A056169(a(n)) = A065120(n). [seems to be]
A005361(a(n)) = A284001(n).
A072411(a(n)) = A284002(n).
A007913(a(n)) = A284003(n).
A000005(a(n)) = A284005(n).
A324286(a(n)) = A324287(n).
A276086(a(n)) = A324289(n).
A267263(a(n)) = A324341(n).
A276150(a(n)) = A324342(n). [subsequences in the latter are converging towards this sequence]
G.f.: Product_{k>=0} (1 + prime(k + 1)# * x^(2^k)), where prime()# = A002110. - Ilya Gutkovskiy, Aug 19 2019

More formulas and the binary tree illustration added by Antti Karttunen, Mar 19 2017

Four more linking formulas added by Antti Karttunen, Feb 25 2019

A162642 Number of odd exponents in the canonical prime factorization of n.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Jul 08 2009

a(n) is also known as the squarefree rank of n. - Jason Kimberley, Jul 08 2017

The number of primes that are infinitary divisors of n. - Amiram Eldar, Oct 01 2023

Jason Kimberley, Table of n, a(n) for n = 1..20000
R. B. Eggleton, J. S. Kimberley and J. A. MacDougall, Square-free rank of integers, Journal of Combinatorial Mathematics and Combinatorial Computing, Vol. 106 (2018).
Index entries for sequences computed from exponents in factorization of n.

Cf. A000290 (positions of zeros), A001221, A001620, A002035, A007913, A056169, A162641, A295316, A295659, A295662, A295664.

A162642:=func;
[A162642(n):n in[1..105]]; // Jason Kimberley, Dec 30 2015

A162642 := proc(n) add ( op(2,f) mod 2 ,f=ifactors(n)[2]) ; end proc: # R. J. Mathar, Mar 30 2011

{0}~Join~Table[Count[Last /@ FactorInteger@ n, e_ /; OddQ@ e], {n, 2, 105}] (* Michael De Vlieger, Jan 06 2016 *)

a(n) = {my(f = factor(n)); sum(k=1, #f~, f[k,2] % 2);} \\ Michel Marcus, Jan 08 2016

;; With memoization-macro definec.
(definec (A162642 n) (if (= 1 n) 0 (+ (A000035 (A067029 n)) (A162642 (A028234 n))))) ;; Antti Karttunen, Nov 28 2017

a(n) = A001221(n) - A162641(n).
a(n) = A001221(A007913(n)). - Jason Kimberley, Jan 06 2016
a(A000290(n)) = 0, n > 0. - Michel Marcus, Jan 08 2016
G.f.: Sum_{i>=1} Sum_{j>=1} (-1)^j x^(prime(i)^j)/(x^(prime(i)^j) - 1). - Robert Israel, Jan 15 2016
From Antti Karttunen, Nov 28 2017: (Start)
Additive with a(p^e) = A000035(e).
a(n) = A056169(n) + A295662(n).
A056169(n) <= a(n) <= A056169(n) + A295659(n).
a(n) <= A295664(n).
(End)
Sum_{k=1..n} a(k) = n * log(log(n)) + c * n + O(n/log(n)), where c = gamma + Sum_{p prime} (log(1-1/p) + 1/(p+1)) = A077761 - A179119 = -0.0687327134... and gamma is Euler's constant (A001620). - Amiram Eldar, Dec 25 2021

A056171 a(n) = pi(n) - pi(floor(n/2)), where pi is A000720.

Original entry on oeis.org

0, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 3, 2, 2, 2, 3, 3, 4, 4, 4, 3, 4, 4, 4, 3, 3, 3, 4, 4, 5, 5, 5, 4, 4, 4, 5, 4, 4, 4, 5, 5, 6, 6, 6, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 6, 7, 7, 8, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 10, 9, 9, 9, 9, 9, 10, 10, 10, 9, 10, 10, 10, 9, 9, 9, 10, 10, 10, 10, 10, 9, 9, 9, 10, 10

Offset: 1

Labos Elemer, Jul 27 2000

Also, the number of unitary prime divisors of n!. A prime divisor of n is unitary iff its exponent is 1 in the prime power factorization of n. In general, gcd(p, n/p) = 1 or p. Here we count the cases when gcd(p, n/p) = 1.
A unitary prime divisor of n! is >= n/2, hence their number is pi(n) - pi(n/2). - Peter Luschny, Mar 13 2011
See also the references and links mentioned in A143227. - Jonathan Sondow, Aug 03 2008
From Robert G. Wilson v, Mar 20 2017: (Start)
First occurrence of k is at n = A080359(k).
The last occurrence of k is at n = A080360(k).
The number of times k appears is A080362(k). (End)
Lev Schnirelmann proved that for every n, a(n) > (1/log_2(n))*(n/3 - 4*sqrt(n)) - 1 - (3/2)*log_2(n). - Arkadiusz Wesolowski, Nov 03 2017

			10! = 2^8 * 3^2 * 5^2 * 7. The only unitary prime divisor is 7, so a(10) = 1.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 214.

Daniel Forgues, Table of n, a(n) for n=1..100000
Ethan Berkove and Michael Brilleslyper, Subgraphs of Coprime Graphs on Sets of Consecutive Integers, Integers, Vol. 22 (2022), #A47, see p. 8.

Cf. A000720, A001221, A034444, A048105, A048656, A048657, A056169, A056172.
Cf. A014085, A060715, A104272, A143223, A143224, A143225, A143226, A143227.
Cf. A080359, A080360, A080362.

A056171 := proc(x)
     numtheory[pi](x)-numtheory[pi](floor(x/2)) ;
end proc:
seq(A056171(n),n=1..130) ; # N. J. A. Sloane, Sep 01 2015
A056171 := n -> nops(select(isprime,[$iquo(n,2)+1..n])):
seq(A056171(i),i=1..98); # Peter Luschny, Mar 13 2011

s=0; Table[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; s, {k,100}]
Table[PrimePi[n]-PrimePi[Floor[n/2]],{n,100}] (* Harvey P. Dale, Sep 01 2015 *)

A056171=n->primepi(n)-primepi(n\2) \\ M. F. Hasler, Dec 31 2016

from sympy import primepi
[primepi(n) - primepi(n//2) for n in range(1,151)] # Indranil Ghosh, Mar 22 2017

[prime_pi(n)-prime_pi(floor(n/2)) for n in range(1,99)] # Stefano Spezia, Apr 22 2025

a(n) = A000720(n) - A056172(n). - Robert G. Wilson v, Apr 09 2017

a(n) = A056169(n!). - Amiram Eldar, Jul 24 2024

Definition simplified by N. J. A. Sloane, Sep 01 2015

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).

A277324 Odd bisection of A260443 (the even terms): a(n) = A260443((2*n)+1).

Original entry on oeis.org

2, 6, 18, 30, 90, 270, 450, 210, 630, 6750, 20250, 9450, 15750, 47250, 22050, 2310, 6930, 330750, 3543750, 1653750, 4961250, 53156250, 24806250, 727650, 1212750, 57881250, 173643750, 18191250, 8489250, 25467750, 2668050, 30030, 90090, 40020750, 1910081250, 891371250, 9550406250, 455814843750, 212713593750

Offset: 0

Antti Karttunen, Oct 10 2016

nonn

From David A. Corneth, Oct 22 2016: (Start)
The exponents of the prime factorization of a(n) are first nondecreasing, then nonincreasing.
The exponent of 2 in the prime factorization of a(n) is 1. (End)

			A method to find terms of this sequence, explained by an example to find a(7). To find k = a(7), we find k such that A048675(k) = 2*7+1 = 15. 7 has the binary partitions: {[7, 0, 0], [5, 1, 0], [3, 2, 0], [1, 3, 0], [3, 0, 1], [1, 1, 1]}. To each of those, we prepend a 1. This gives the binary partitions of 15 starting with a 1. For example, for the first we get [1, 7, 0, 0]. We see that only [1, 5, 1, 0], [1, 3, 2, 0] and [1, 1, 1, 1] start nondecreasing, then nonincreasing, so we only check those. These numbers will be the exponents in a prime factorization. [1, 5, 1, 0] corresponds to prime(1)^1 * prime(2)^5 * prime(3)^1 * prime(4)^0 = 2430. We find that [1, 1, 1, 1] gives k = 210 for which A048675(k) = 15 so a(7) = 210. - _David A. Corneth_, Oct 22 2016

Antti Karttunen, Table of n, a(n) for n = 0..1024

Cf. A005811, A005940, A007306, A007949, A156552, A260443, A277189, A277323, A283484, A283975, A284267, A284268, A284563, A284564, A284573.

Cf. A277200 (same sequence sorted into ascending order).

a[n_] := a[n] = Which[n < 2, n + 1, EvenQ@ n, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] &@ a[n/2], True, a[#] a[# + 1] &[(n - 1)/2]]; Table[a[2 n + 1], {n, 0, 38}] (* Michael De Vlieger, Apr 05 2017 *)

from sympy import factorint, prime, primepi
from operator import mul
def a003961(n):
    F=factorint(n)
    return 1 if n==1 else reduce(mul, [prime(primepi(i) + 1)**F[i] for i in F])
def a260443(n): return n + 1 if n<2 else a003961(a260443(n//2)) if n%2==0 else a260443((n - 1)//2)*a260443((n + 1)//2)
def a(n): return a260443(2*n + 1)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 21 2017

a(n) = A260443((2*n)+1).
a(0) = 2; for n >= 1, a(n) = A260443(n) * A260443(n+1).
Other identities. For all n >= 0:
A007949(a(n)) = A005811(n). [See comments in A125184.]
A156552(a(n)) = A277189(n), a(n) = A005940(1+A277189(n)).
A048675(a(n)) = 2n + 1. - David A. Corneth, Oct 22 2016
A001222(a(n)) = A007306(1+n).
A056169(a(n)) = A284267(n).
A275812(a(n)) = A284268(n).
A248663(a(n)) = A283975(n).
A000188(a(n)) = A283484(n).
A247503(a(n)) = A284563(n).
A248101(a(n)) = A284564(n).
A046523(a(n)) = A284573(n).
a(n) = A277198(n) * A284008(n).
a(n) = A284576(n) * A284578(n) = A284577(n) * A000290(A284578(n)).

More linking formulas added by Antti Karttunen, Apr 16 2017

A056671 1 + the number of unitary and squarefree divisors of n = number of divisors of reduced squarefree part of n.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Aug 10 2000

Note that 1 is regarded as free of squares of primes and is also a square number and a unitary divisor.

			n = 252 = 2*2*3*3*7 has 18 divisors, 8 unitary and 8 squarefree divisors of which 2 are unitary and squarefree, divisors {1,7};
n = 2520 = 2*2*2*3*3*5*7 has 48 divisors, 16 unitary and 16 squarefree divisors of which {1,5,7,35} are both, thus a(2520) = 4.
a(2520) = a(2^3*3^2*5*7) = a(2^3)*a(3^2)*a(5)*a(7) = 1*1*2*2 = 4.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
Index entries for sequences computed from exponents in factorization of n.

Cf. A000005, A007913, A034444, A055229, A055231, A056169, A056674.

Cf. A343443.

Array[DivisorSigma[0, #] &@ Denominator[#/Apply[Times, FactorInteger[#][[All, 1]]]^2] &, 105] (* or *)
Table[DivisorSum[n, 1 &, And[SquareFreeQ@ #, CoprimeQ[#, n/#]] &], {n, 105}] (* Michael De Vlieger, Jul 19 2017 *)
f[p_,e_] := If[e==1, 2, 1]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, May 14 2019 *)

A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); } \\ Charles R Greathouse IV, Aug 13 2013
A055231(n) = n/A057521(n);
A056671(n) = numdiv(A055231(n));
\\ Or:
A055229(n) = { my(c=core(n)); gcd(c, n/c); }; \\ This function from Charles R Greathouse IV, Nov 20 2012
A056671(n) = numdiv(core(n)/A055229(n)); \\ Antti Karttunen, Jul 19 2017

for(n=1, 100, print1(direuler(p=2, n, (1 + X - X^2)/(1-X))[n], ", ")) \\ Vaclav Kotesovec, Feb 11 2023

a(n) = vecprod(apply(x -> if(x == 1, 2, 1), factor(n)[, 2])); \\ Amiram Eldar, Apr 15 2025

from sympy import factorint, prod
def a(n): return 1 if n==1 else prod([2 if e==1 else 1 for p, e in factorint(n).items()])
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jul 19 2017

(define (A056671 n) (if (= 1 n) n (* (if (= 1 (A067029 n)) 2 1) (A056671 (A028234 n))))) ;; (After the given multiplicative formula) - Antti Karttunen, Jul 19 2017

a(n) = A000005(A055231(n)) = A000005(A007913(n)/A055229(n)).
Multiplicative with a(p) = 2 and a(p^e) = 1 for e > 1. a(n) = 2^A056169(n). - Vladeta Jovovic, Nov 01 2001
a(n) = A034444(n) - A056674(n). - Antti Karttunen, Jul 19 2017
From Vaclav Kotesovec, Feb 11 2023: (Start)
Dirichlet g.f.: zeta(s) * Product_{primes p} (1 + 1/p^s - 1/p^(2*s)).
Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 - 2/p^(2*s) + 1/p^(3*s)), (with a product that converges for s=1).
Let f(s) = Product_{primes p} (1 - 2/p^(2*s) + 1/p^(3*s)), then Sum_{k=1..n} a(k) ~ n * (f(1) * (log(n) + 2*gamma - 1) + f'(1)), where f(1) = Product_{primes p} (1 - 2/p^2 + 1/p^3) = A065464 = 0.42824950567709444021876..., f'(1) = f(1) * Sum_{primes p} (4*p-3) * log(p) / (p^3 - 2*p + 1) = 0.808661108949590913395... and gamma is the Euler-Mascheroni constant A001620. (End)
a(n) = Sum_{d|n, gcd(d,n/d)=1} mu(d)^2. - Wesley Ivan Hurt, May 25 2023
a(n) = Sum_{d|n} A343443(d)*mu(n/d). - Ridouane Oudra, Dec 18 2023

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.

cp's OEIS Frontend

A056170 Number of non-unitary prime divisors of n.

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A055231 Powerfree part of n: product of primes that divide n only once.

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A080359 The smallest integer x > 0 such that the number of primes in (x/2, x] equals n.

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A283477 If 2n = 2^e1 + 2^e2 + ... + 2^ek [e1 .. ek distinct], then a(n) = A002110(e1) * A002110(e2) * ... * A002110(ek).

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A162642 Number of odd exponents in the canonical prime factorization of n.

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A056171 a(n) = pi(n) - pi(floor(n/2)), where pi is A000720.

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