A028234 -id:A028234 - cp's OEIS frontend

A059895 Table a(i,j) = product prime[k]^(Ei[k] AND Ej[k]) where Ei and Ej are the vectors of exponents in the prime factorizations of i and j; AND is the bitwise operation on binary representation of the exponents.

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 2, 1, 1, 5, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 6, 1, 4, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 7, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 1, 5, 1, 3, 1, 1, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1

Offset: 1

Marc LeBrun, Feb 06 2001

Analogous to GCD, with AND replacing MIN.

			The top left 18 X 18 corner of the array:
1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1
1,  2,  1,  1,  1,  2,  1,  2,  1,  2,  1,  1,  1,  2,  1,  1,  1,  2
1,  1,  3,  1,  1,  3,  1,  1,  1,  1,  1,  3,  1,  1,  3,  1,  1,  1
1,  1,  1,  4,  1,  1,  1,  4,  1,  1,  1,  4,  1,  1,  1,  1,  1,  1
1,  1,  1,  1,  5,  1,  1,  1,  1,  5,  1,  1,  1,  1,  5,  1,  1,  1
1,  2,  3,  1,  1,  6,  1,  2,  1,  2,  1,  3,  1,  2,  3,  1,  1,  2
1,  1,  1,  1,  1,  1,  7,  1,  1,  1,  1,  1,  1,  7,  1,  1,  1,  1
1,  2,  1,  4,  1,  2,  1,  8,  1,  2,  1,  4,  1,  2,  1,  1,  1,  2
1,  1,  1,  1,  1,  1,  1,  1,  9,  1,  1,  1,  1,  1,  1,  1,  1,  9
1,  2,  1,  1,  5,  2,  1,  2,  1, 10,  1,  1,  1,  2,  5,  1,  1,  2
1,  1,  1,  1,  1,  1,  1,  1,  1,  1, 11,  1,  1,  1,  1,  1,  1,  1
1,  1,  3,  4,  1,  3,  1,  4,  1,  1,  1, 12,  1,  1,  3,  1,  1,  1
1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, 13,  1,  1,  1,  1,  1
1,  2,  1,  1,  1,  2,  7,  2,  1,  2,  1,  1,  1, 14,  1,  1,  1,  2
1,  1,  3,  1,  5,  3,  1,  1,  1,  5,  1,  3,  1,  1, 15,  1,  1,  1
1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, 16,  1,  1
1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, 17,  1
1,  2,  1,  1,  1,  2,  1,  2,  9,  2,  1,  1,  1,  2,  1,  1,  1, 18
A(864,1944) = A(2^5*3^3,2^3*3^5) = 2^(5 AND 3)* 3^(3 AND 5) = 2^1*3^1 = 6.

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of the array

Cf. A003985 (A004198), A003989, A028234, A059896, A059897, A067029, A267115, A284578.

a[i_, i_] := i;
a[i_, j_] := Module[{f1 = FactorInteger[i], f2 = FactorInteger[j], e1, e2}, Scan[(e1[#[[1]]] = #[[2]])&, f1]; Scan[(e2[#[[1]]] = #[[2]])&, f2]; Times @@ (#^BitAnd[e1[#], e2[#]]& /@ Intersection[f1[[All, 1]], f2[[All, 1]]]) ];
Table[a[i - j + 1, j], {i, 1, 15}, {j, 1, i}] // Flatten (* Jean-François Alcover, Jun 19 2018 *)

(define (A059895 n) (A059895bi (A002260 n) (A004736 n)))
(define (A059895bi a b) (let loop ((a a) (b b) (m 1)) (cond ((= 1 a) m) ((= 1 b) m) ((equal? (A020639 a) (A020639 b)) (loop (A028234 a) (A028234 b) (* m (expt (A020639 a) (A004198bi (A067029 a) (A067029 b)))))) ((< (A020639 a) (A020639 b)) (loop (A028234 a) b m)) (else (loop a (A028234 b) m)))))
;; Antti Karttunen, Apr 11 2017

From Antti Karttunen, Apr 11 2017: (Start)
A(x,y) = A059896(x,y) / A059897(x,y).
A(x,y) * A059896(x,y) = A(x,y)^2 * A059897(x,y) = x*y.
(End)

Data section extended to 120 terms by Antti Karttunen, Apr 11 2017

A278908 Multiplicative with a(p^e) = 2^omega(e), where omega = A001221.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Nov 30 2016

The number of exponential unitary (or e-unitary) divisors of n and the number of exponential squarefree exponential divisors (or e-squarefree e-divisors) of n. These are divisors of n = Product p(i)^a(i) of the form Product p(i)^b(i) where each b(i) is a unitary divisor of a(i) in the first case, or each b(i) is a squarefree divisor of a(i) in the second case. - Amiram Eldar, Dec 29 2018

Antti Karttunen, Table of n, a(n) for n = 1..10000
Xiaodong Cao and Wenguang Zahi, Some arithmetic functions involving exponential divisors, Journal of Integer Sequences, Vol. 13 (2010), Article 10.3.7, eq (20).
Nicuşor Minculete and László Tóth, Exponential unitary divisors, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 35 (2011), pp. 205-216.
László Tóth, On certain arithmetic functions involving exponential divisors, II, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 27 (2007), pp. 155-166; arXiv preprint, arXiv:0708.3557 [math.NT], 2007-2009.
Xiangzhen Zhao, Min Liu, and Yu Huang, Mean value for the function t^(e)(n) over square-full numbers, Scientia Magna, Vol. 8, No. 3 (2012), pp. 110-114.
Index entries for sequences computed from exponents in factorization of n

Cf. A001221.

A278908 := proc(n)
    local a,p,e;
    a := 1;
    if n =1 then
        ;
    else
        for p in ifactors(n)[2] do
            e := op(2,p) ;
            a := a*2^A001221(e) ;
        end do:
    end if;
    a ;
end proc:

Table[Times @@ Apply[Times, FactorInteger[n] /. {p_, e_} /; p > 1 :> 2^PrimeNu[e]], {n, 105}] (* Michael De Vlieger, Jul 29 2017 *)

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = 2^omega(f[k,2]); f[k,2] = 1); factorback(f); \\ Michel Marcus, Jul 28 2017

(define (A278908 n) (if (= 1 n) n (* (A000079 (A001221 (A067029 n))) (A278908 (A028234 n))))) ;; Antti Karttunen, Jul 27 2017

Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = Product_{p prime} (1 + Sum_{k>=2} (2*omega(k) - 2^omega(k-1))/p^k) = 1.5431653193... (Tóth, 2007). - Amiram Eldar, Nov 08 2020

A349056 Number of weakly alternating permutations of the multiset of prime factors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 4, 1, 1, 2, 2, 2, 4, 1, 2, 2, 4, 1, 4, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 6, 1, 2, 3, 1, 2, 4, 1, 3, 2, 4, 1, 6, 1, 2, 3, 3, 2, 4, 1, 5, 1, 2, 1, 6, 2, 2, 2

Offset: 1

Gus Wiseman, Dec 02 2021

nonn

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. Then a sequence is alternating in the sense of A025047 iff it is a weakly alternating anti-run.

A prime index of n is a number m such that prime(m) divides n. For n > 1, the multiset of prime factors of n is row n of A027746. The prime indices A112798 can also be used.

			The following are the weakly alternating permutations for selected n:
n = 2   6    12    24     48      60     90     120     180
   ----------------------------------------------------------
    2   23   223   2223   22223   2253   2335   22253   22335
        32   232   2232   22232   2325   2533   22325   22533
             322   2322   22322   2523   3253   22523   23253
                   3222   23222   3252   3325   23252   23352
                          32222   3522   3352   25232   25233
                                  5232   3523   32225   25332
                                         5233   32522   32325
                                         5332   35222   32523
                                                52223   33252
                                                52322   33522
                                                        35232
                                                        52323
                                                        53322

Counting all permutations of prime factors gives A008480.
The variation counting anti-run permutations is A335452.
The strong case is A345164, with twins A344606.
Compositions of this type are counted by A349052, also A129852 and A129853.
Compositions not of this type are counted by A349053, ranked by A349057.
The version for patterns is A349058, strong A345194.
The version for ordered factorizations is A349059, strong A348610.
Partitions of this type are counted by A349060, complement A349061.
The complement is counted by A349797.
The non-alternating case is A349798.
A001250 counts alternating permutations, complement A348615.
A003242 counts Carlitz (anti-run) compositions.
A025047 counts alternating or wiggly compositions, ranked by A345167.
A056239 adds up prime indices, row sums of A112798, row lengths A001222.
A071321 gives the alternating sum of prime factors, reverse A071322.
A344616 gives the alternating sum of prime indices, reverse A316524.
A345165 counts partitions w/o an alternating permutation, ranked by A345171.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A348379 counts factorizations with an alternating permutation.
A349800 counts weakly but not strongly alternating compositions.
Cf. A028234, A051119, A096441, A335433, A335448, A344614, A344652, A344653, A345173, A345192.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[Permutations[primeMS[n]],whkQ[#]||whkQ[-#]&]],{n,100}]

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

A268387 Bitwise-XOR of the exponents of primes in the prime factorization of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 0, 1, 3, 2, 0, 1, 3, 1, 0, 0, 4, 1, 3, 1, 3, 0, 0, 1, 2, 2, 0, 3, 3, 1, 1, 1, 5, 0, 0, 0, 0, 1, 0, 0, 2, 1, 1, 1, 3, 3, 0, 1, 5, 2, 3, 0, 3, 1, 2, 0, 2, 0, 0, 1, 2, 1, 0, 3, 6, 0, 1, 1, 3, 0, 1, 1, 1, 1, 0, 3, 3, 0, 1, 1, 5, 4, 0, 1, 2, 0, 0, 0, 2, 1, 2, 0, 3, 0, 0, 0, 4, 1, 3, 3, 0, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 0, 5, 1, 1, 0, 3, 3, 0, 0, 3

Offset: 1

Antti Karttunen, Feb 05 2016

The sums of the first 10^k terms, for k = 1, 2, ..., are 11, 139, 1427, 14207, 141970, 1418563, 14183505, 141834204, 1418330298, 14183245181, ... . Apparently, the asymptotic mean of this sequence is limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1.4183... . - Amiram Eldar, Sep 10 2022

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

A003987, A028234, A059897 and A067029 are used to express relationships between sequence terms.
Cf. A268390 (indices of zeros).
Sequences with similar definitions: A267115, A267116.
Differs from A136566 for the first time at n=24, where a(24) = 2, while A136566(24) = 4.
Cf. A019565, A052330.

Table[BitXor @@ Map[Last, FactorInteger@ n], {n, 120}] (* Michael De Vlieger, Feb 12 2016 *)

a(n) = {my(f = factor(n)); my(b = 0); for (k=1, #f~, b = bitxor(b, f[k,2]);); b;} \\ Michel Marcus, Feb 06 2016

from functools import reduce
from operator import xor
from sympy import factorint
def A268387(n): return reduce(xor,factorint(n).values(),0) # Chai Wah Wu, Aug 31 2022

a(1) = 0; for n > 1: a(n) = A067029(n) XOR a(A028234(n)). [Here XOR stands for bitwise exclusive-or, A003987.]
Other identities and observations. For all n >= 1:
a(n) <= A267116(n) <= A001222(n).
From Peter Munn, Dec 02 2019 with XOR used as above: (Start)
Defined by: a(p^k) = k, for prime p; a(A059897(n,k)) = a(n) XOR a(k).
a(A052330(n XOR k)) = a(A052330(n)) XOR a(A052330(k)).
a(A019565(n XOR k)) = a(A019565(n)) XOR a(A019565(k)).
(End)

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

A290106 a(1) = 1; for n > 1, if n = Product prime(k)^e(k), then a(n) = Product (k)^(e(k)-1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 2, 3, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Antti Karttunen, Aug 13 2017

			For n = 21 = 3*7 = prime(2)^1 * prime(4)^1, a(n) = 2^0 * 4^0 = 1*1 = 1.
For n = 360 = 2^3 * 3^2 * 5^1 = prime(1)^3 * prime(2)^2 * prime(3)^1, a(n) = 1^2 * 2^1 * 3^0 = 1*2*1 = 2.

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

Cf. A003557, A003963, A007947, A156061, A290105.

Differs from A290104 for the first time at n=21.

Table[If[n == 1, 1, Apply[Times, Map[PrimePi[#1]^#2 & @@ # &, #]] / Apply[Times, PrimePi[#[[All, 1]] ]]] &@ FactorInteger@ n, {n, 120}] (* Michael De Vlieger, Aug 14 2017 *)

(define (A290106 n) (/ (A003963 n) (A156061 n)))
(define (A290106 n) (if (= 1 n) 1 (* (expt (A055396 n) (- (A067029 n) 1)) (A290106 (A028234 n)))))

Multiplicative with a(prime(k)^e) = k^(e-1).
a(n) = A003963(n) / A156061(n).
a(n) = A003963(A003557(n)) = A003963(n/A007947(n)).

A106490 Total number of bases and exponents in Quetian Superfactorization of n, excluding the unity-exponents at the tips of branches.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 09 2005 based on Leroy Quet's message ('Super-Factoring' An Integer) posted to SeqFan-mailing list on Dec 06 2003

nonn

Quetian Superfactorization proceeds by factoring a natural number to its unique prime-exponent factorization (p1^e1 * p2^e2 * ... pj^ej) and then factoring recursively each of the (nonzero) exponents in similar manner, until unity-exponents are finally encountered.

			a(64) = 3, as 64 = 2^6 = 2^(2^1*3^1) and there are three non-1 nodes in that superfactorization. Similarly, for 360 = 2^(3^1) * 3^(2^1) * 5^1 we get a(360) = 5. a(65536) = a(2^(2^(2^(2^1)))) = 4.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
A. Karttunen, Scheme-program for computing this sequence.
Index entries for sequences computed from exponents in factorization of n

Cf. A028234, A064372, A067029, A106444, A106491, A106492, A106493.
Cf. A276230 (gives first k such that a(k) = n, i.e., this sequence is a left inverse of A276230).
After n=1 differs from A038548 for the first time at n=24, where A038548(24)=4, while a(24)=3.

a:= proc(n) option remember; `if`(n=1, 0,
      add(1+a(i[2]), i=ifactors(n)[2]))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 07 2014

a[n_] := a[n] = If[n == 1, 0, Sum[1 + a[i[[2]]], {i,FactorInteger[n]}]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

A067029(n) = if(n<2, 0, factor(n)[1,2]);
A028234(n) = my(f = factor(n)); if (#f~, f[1, 1] = 1); factorback(f); /* after Michel Marcus */
a(n) = if(n<2, 0, 1 + a(A067029(n)) + a(A028234(n)));
for(n=1, 150, print1(a(n),", ")) \\ Indranil Ghosh, Mar 23 2017, after formula by Antti Karttunen

Additive with a(p^e) = 1 + a(e).
a(1) = 0; for n > 1, a(n) = 1 + a(A067029(n)) + a(A028234(n)). - Antti Karttunen, Mar 23 2017
Other identities. For all n >= 1:
a(A276230(n)) = n.
a(n) = A106493(A106444(n)).
a(n) = A106491(n) - A064372(n).

A124859 Multiplicative with p^e -> primorial(e), p prime and e > 0.

Original entry on oeis.org

1, 2, 2, 6, 2, 4, 2, 30, 6, 4, 2, 12, 2, 4, 4, 210, 2, 12, 2, 12, 4, 4, 2, 60, 6, 4, 30, 12, 2, 8, 2, 2310, 4, 4, 4, 36, 2, 4, 4, 60, 2, 8, 2, 12, 12, 4, 2, 420, 6, 12, 4, 12, 2, 60, 4, 60, 4, 4, 2, 24, 2, 4, 12, 30030, 4, 8, 2, 12, 4, 8, 2, 180, 2, 4, 12, 12, 4, 8, 2, 420, 210, 4, 2, 24, 4, 4

Offset: 1

Reinhard Zumkeller, Nov 10 2006

			From _Michael De Vlieger_, Mar 06 2017: (Start)
a(2) = 2 since 2 = 2^1, thus primorial p_1# = 2.
a(4) = 6 since 4 = 2^2, thus primorial p_2# = 2*3 = 6.
a(6) = 4 because 6 is squarefree with omega(6)=2, thus 2^2 = 4.
a(8) = 30 since 8 = 2^3, thus primorial p_3# = 2*3*5 = 30.
a(10) = 4 since 10 is squarefree with omega(10)=2, thus 2^2 = 4.
a(12) = 12 since 12 = 2^1 * 3^2, thus primorials p_1# * p_2# = 2*6 = 12.
(End)

Indranil Ghosh, Table of n, a(n) for n = 1..5000 (first 1000 terms from R. Zumkeller)
Eric Weisstein's World of Mathematics, Prime Factorization
Eric Weisstein's World of Mathematics, Primorial
Index entries for sequences computed from exponents in factorization of n

Cf. A000040, A000079, A001221, A001222, A002110, A005117, A028234, A046523, A067029, A072048, A108951, A156552, A181819, A181821, A238745, A278159, A278222.

A124859 := proc(n)
    local a,pf;
    a := 1;
    for pf in ifactors(n)[2] do
        a := a*A002110(pf[2]) ;
    end do:
    a ;
end proc:
seq(A124859(n),n=1..80) ; # R. J. Mathar, Oct 06 2017

Table[Which[n == 1, 1, SquareFreeQ@ n, 2^PrimeNu@ n, True, Times @@ Map[Times @@ Prime@ Range@ # &, #[[All, -1]]]] &@ FactorInteger@ n, {n, 86}] (* Michael De Vlieger, Mar 06 2017 *)

a(n) = {my(f = factor(n)); for (k=1, #f~, f[k,1] = prod(j=1, f[k,2], prime(j)); f[k,2] = 1;); factorback(f);} \\ Michel Marcus, Nov 16 2015

from sympy.ntheory.factor_ import core
from sympy import factorint, primorial, primefactors
from operator import mul
def omega(n): return 0 if n==1 else len(primefactors(n))
def a(n):
    f=factorint(n)
    return n if n<3 else 2**omega(n) if core(n) == n else reduce(mul, [primorial(f[i]) for i in f]) # Indranil Ghosh, May 13 2017

(define (A124859 n) (cond ((= 1 n) 1) (else (* (A002110 (A067029 n)) (A124859 (A028234 n)))))) ;; Antti Karttunen, Mar 06 2017

a(A000040(x)^n) = A002110(n); a(A002110(n)) = A000079(n);
a(A005117(n)) = 2^A001221(A005117(n)) = A072048(n);
A001221(a(n)) = A051903(n); A001222(a(n)) = A001222(n).
From Antti Karttunen, Mar 06 2017: (Start)
a(1) = 1, for n > 1, a(n) = A002110(A067029(n)) * a(A028234(n)).
a(n) = A278159(A156552(n)).
a(A278159(n)) = A278222(n).
a(a(n)) = A046523(n). [after Matthew Vandermast's May 19 2012 formula for the latter sequence]
A181819(a(n)) = A238745(n). [after Matthew Vandermast's formula for the latter sequence]
(End)
a(n) = A108951(A181819(n)). [Primorial inflation of the prime shadow of n] - Antti Karttunen, Sep 15 2023

A276079 Numbers n such that prime(k)^(k+1) divides n for some k.

Original entry on oeis.org

4, 8, 12, 16, 20, 24, 27, 28, 32, 36, 40, 44, 48, 52, 54, 56, 60, 64, 68, 72, 76, 80, 81, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 135, 136, 140, 144, 148, 152, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188, 189, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232, 236, 240, 243, 244, 248, 252, 256, 260, 264, 268, 270, 272

Offset: 1

Antti Karttunen, Aug 18 2016

nonn

The asymptotic density of this sequence is 1 - Product_{i>=1} 1-prime(i)^(-1-i) = 0.2789766... - Amiram Eldar, Oct 21 2020

			625 = 5*5*5*5 = prime(3)^4 so it is divisible by prime(3)^(3+1), and thus 625 is included in the sequence.

Antti Karttunen, Table of n, a(n) for n = 1..5000

Positions of nonzeros in A276077.
Complement: A276078.
Cf. A000040, A000720, A008586 (a subsequence).
Differs from its subsequence A100716 for the first time at n=175, where a(175) = 625, while that value is missing from A100716.

from sympy import primepi, isprime, primefactors, factorint
def a028234(n):
    f=factorint(n)
    minf = min(f)
    return 1 if n==1 else n//(minf**f[minf])
def a067029(n):
    f=factorint(n)
    return 0 if n==1 else f[min(f)]
def a049084(n): return primepi(n) if isprime(n) else 0
def a055396(n): return 0 if n==1 else a049084(min(primefactors(n)))
def a(n): return 0 if n==1 else a(a028234(n)) + (1 if a067029(n) > a055396(n) else 0)
print([n for n in range(1, 301) if a(n)!=0]) # Indranil Ghosh, Jun 21 2017

cp's OEIS Frontend

A059895 Table a(i,j) = product prime[k]^(Ei[k] AND Ej[k]) where Ei and Ej are the vectors of exponents in the prime factorizations of i and j; AND is the bitwise operation on binary representation of the exponents.

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A278908 Multiplicative with a(p^e) = 2^omega(e), where omega = A001221.

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A349056 Number of weakly alternating permutations of the multiset of prime factors of n.

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

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A268387 Bitwise-XOR of the exponents of primes in the prime factorization of n.

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

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A290106 a(1) = 1; for n > 1, if n = Product prime(k)^e(k), then a(n) = Product (k)^(e(k)-1).

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