A136566 -id:A136566 - cp's OEIS frontend

A136565 a(n) = sum of the distinct values making up the exponents in the prime-factorization of n.

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

Offset: 1

Leroy Quet, Jan 07 2008

nonn

The sums of the first 10^k terms, for k = 1, 2, ..., are 13, 192, 2089, 21405, 215730, 2162136, 21636277, 216410510, 2164253043, 21642998932, ... . Apparently, the asymptotic mean of this sequence is 2.164... . - Amiram Eldar, Jun 30 2025

			120 = 2^3 * 3^1 * 5^1. The exponents of the prime factorization are therefore 3,1,1. The distinct values which equal these exponents are 1 and 3. So a(120) = 1+3 = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537 (terms 1..1000 from Diana Mecum)
Index entries for sequences computed from exponents in factorization of n.

Cf. A066328, A071625, A088529, A136566, A136568, A181591, A181819.

Join[{0},Table[Total[Union[Transpose[FactorInteger[n]][[2]]]],{n,2,110}]] (* Harvey P. Dale, Jun 23 2013 *)

A136565(n) = vecsum(apply(primepi,factor(factorback(apply(e->prime(e),(factor(n)[,2]))))[,1])); \\ Antti Karttunen, Sep 06 2018

a(n) = A088529(n) = A181591(n) for n: 2 <= n < 24. - Reinhard Zumkeller, Nov 01 2010

a(n) = A066328(A181819(n)). - Antti Karttunen, Sep 06 2018

More terms from Diana L. Mecum, Jul 17 2008

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)

A136567 a(n) is the number of exponents occurring only once each in the prime factorization of n.

Original entry on oeis.org

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

Offset: 1

Leroy Quet, Jan 07 2008

nonn

Records are in A006939: 1, 2, 12, 360, 75600, ..., . - Robert G. Wilson v, Jan 20 2008

			4200 = 2^3 * 3^1 * 5^2 * 7^1. The exponents of the prime factorization are therefore 3,1,2,1. The exponents occurring exactly once are 2 and 3. So a(4200) = 2.

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

Cf. A056169, A071625, A133924, A136566, A181819.

For a(n)=0 see A130092 plus the term 1; for a(n)=1 see A000961.

f[n_] := Block[{fi = Sort[Last /@ FactorInteger@n]}, Count[ Count[fi, # ] & /@ Union@fi, 1]]; f[1] = 0; Array[f, 105] (* Robert G. Wilson v, Jan 20 2008 *)
Table[Boole[n != 1] Count[Split@ Sort[FactorInteger[n][[All, -1]]], ?(Length@ # == 1 &)], {n, 105}] (* _Michael De Vlieger, Jul 24 2017 *)

A136567(n) = { my(exps=(factor(n)[, 2]), m=prod(i=1, length(exps), prime(exps[i])), f=factor(m)[, 2]); sum(i=1, #f, f[i]==1); }; \\ Antti Karttunen, Jul 24 2017

(define (A136567 n) (A056169 (A181819 n))) ;; Antti Karttunen, Jul 24 2017

a(n) = A056169(A181819(n)). - Antti Karttunen, Jul 24 2017

More terms from Robert G. Wilson v, Jan 20 2008

cp's OEIS Frontend

A136565 a(n) = sum of the distinct values making up the exponents in the prime-factorization of n.

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

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A136567 a(n) is the number of exponents occurring only once each in the prime factorization of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Scheme

Formula

Extensions