A277707 -id:A277707 - cp's OEIS frontend

A277330 a(0)=1, a(1)=2, a(2n) = A003961(a(n)), a(2n+1) = lcm(a(n),a(n+1))/gcd(a(n),a(n+1)).

1, 2, 3, 6, 5, 2, 15, 30, 7, 10, 3, 30, 35, 2, 105, 210, 11, 70, 21, 30, 5, 10, 105, 42, 77, 70, 3, 210, 385, 2, 1155, 2310, 13, 770, 231, 30, 55, 70, 105, 6, 7, 2, 21, 42, 385, 10, 165, 66, 143, 110, 231, 210, 5, 70, 1155, 66, 1001, 770, 3, 2310, 5005, 2, 15015, 30030, 17, 10010, 3003, 30, 715, 770, 105, 66, 91, 154, 231, 6, 385, 70, 15, 42, 11, 14, 3, 42, 55, 2

Offset: 0

Antti Karttunen, Oct 27 2016

nonn

Each term is a squarefree number, A005117.

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A001511, A003961, A005117, A007913, A019565, A048675, A055396, A260443, A264977, A277707.
Cf. A023758 (positions where coincides with A260443).
Cf. A277701, A277712, A277713 for the positions of 2's, 3's and 6's in this sequence, which are also the first three rows of array A277710.
Cf. also A255483.

a(0) = 1, a(1) = 2, a(2n) = A003961(a(n)), a(2n+1) = lcm(a(n),a(n+1))/gcd(a(n),a(n+1)).
Other identities. For all n >= 0:
a(n) = A007913(A260443(n)).
a(n) = A019565(A264977(n)), A048675(a(n)) = A264977(n).
A055396(a(n)) = A277707(A260443(n)) = A001511(n).

A277697 a(n) = index of the least unitary prime divisor of n or 0 if no such prime-divisor exists.

Original entry on oeis.org

0, 1, 2, 0, 3, 1, 4, 0, 0, 1, 5, 2, 6, 1, 2, 0, 7, 1, 8, 3, 2, 1, 9, 2, 0, 1, 0, 4, 10, 1, 11, 0, 2, 1, 3, 0, 12, 1, 2, 3, 13, 1, 14, 5, 3, 1, 15, 2, 0, 1, 2, 6, 16, 1, 3, 4, 2, 1, 17, 2, 18, 1, 4, 0, 3, 1, 19, 7, 2, 1, 20, 0, 21, 1, 2, 8, 4, 1, 22, 3, 0, 1, 23, 2, 3, 1, 2, 5, 24, 1, 4, 9, 2, 1, 3, 2, 25, 1, 5, 0, 26, 1, 27, 6, 2

Offset: 1

Antti Karttunen, Oct 28 2016

nonn

			For n = 8 = 2*2*2, none of the prime divisors are unitary, thus a(8) = 0.
For n = 20 = 2*2*5 = prime(1)^2 * prime(3), the prime divisor 2 is not unitary, but 5 (= prime(3)) is, thus a(20) = 3.
For n = 36 = 2*2*3*3, none of the prime divisors are unitary, thus a(36) = 0.

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

Cf. A028234, A055396, A067029.
Cf. A001694 (positions of zeros).
Cf. also A080368, A277698, A277707.

Table[If[Length@ # == 0, 0, PrimePi@ First@ #] &@ Select[FactorInteger[n][[All, 1]], GCD[#, n/#] == 1 &], {n, 105}] (* Michael De Vlieger, Oct 30 2016 *)

a(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 2] == 1, return(primepi(f[i, 1])))); 0;} \\ Amiram Eldar, Jul 28 2024

from sympy import factorint, primepi, isprime, primefactors
def a049084(n): return primepi(n)*(1*isprime(n))
def a055396(n): return 0 if n==1 else a049084(min(primefactors(n)))
def a028234(n):
    f = factorint(n)
    return 1 if n==1 else n/(min(f)**f[min(f)])
def a067029(n):
    f=factorint(n)
    return 0 if n==1 else f[min(f)]
def a(n): return 0 if n==1 else a055396(n) if a067029(n)==1 else a(a028234(n)) # Indranil Ghosh, May 15 2017

(definec (A277697 n) (cond ((= 1 n) 0) ((= 1 (A067029 n)) (A055396 n)) (else (A277697 (A028234 n)))))

a(1) = 0; for n > 1, if A067029(n) = 1, then a(n) = A055396(n), otherwise a(n) = a(A028234(n)).

A277708 a(n) = Least prime divisor of n with an odd exponent, or 1 if n is a perfect square.

Original entry on oeis.org

1, 2, 3, 1, 5, 2, 7, 2, 1, 2, 11, 3, 13, 2, 3, 1, 17, 2, 19, 5, 3, 2, 23, 2, 1, 2, 3, 7, 29, 2, 31, 2, 3, 2, 5, 1, 37, 2, 3, 2, 41, 2, 43, 11, 5, 2, 47, 3, 1, 2, 3, 13, 53, 2, 5, 2, 3, 2, 59, 3, 61, 2, 7, 1, 5, 2, 67, 17, 3, 2, 71, 2, 73, 2, 3, 19, 7, 2, 79, 5, 1, 2, 83, 3, 5, 2, 3, 2, 89, 2, 7, 23, 3, 2, 5, 2, 97, 2, 11, 1, 101, 2, 103, 2, 3

Offset: 1

Antti Karttunen, Oct 28 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A020639, A007913, A008578, A277707.
Cf. A000290 (after its initial zero-term gives the positions of ones in this sequence).
Cf. also A277698.

a(n) = my(f = factor(core(n))); if (!#f~, 1, vecmin(f[,1])); \\ Michel Marcus, Oct 30 2016

from sympy import primefactors
from sympy.ntheory.factor_ import core
def lpf(n): return 1 if n==1 else primefactors(n)[0]
def a(n): return lpf(core(n)) # Indranil Ghosh, May 17 2017

a(n) = A008578(1+A277707(n)).

a(n) = A020639(A007913(n)).

cp's OEIS Frontend

A277330 a(0)=1, a(1)=2, a(2n) = A003961(a(n)), a(2n+1) = lcm(a(n),a(n+1))/gcd(a(n),a(n+1)).

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A277697 a(n) = index of the least unitary prime divisor of n or 0 if no such prime-divisor exists.

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A277708 a(n) = Least prime divisor of n with an odd exponent, or 1 if n is a perfect square.

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PARI

Python

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