A277698 -id:A277698 - cp's OEIS frontend

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

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)).

A080368 a(n) is the least unitary prime divisor of n, or 0 if no such prime divisor exists.

Original entry on oeis.org

0, 2, 3, 0, 5, 2, 7, 0, 0, 2, 11, 3, 13, 2, 3, 0, 17, 2, 19, 5, 3, 2, 23, 3, 0, 2, 0, 7, 29, 2, 31, 0, 3, 2, 5, 0, 37, 2, 3, 5, 41, 2, 43, 11, 5, 2, 47, 3, 0, 2, 3, 13, 53, 2, 5, 7, 3, 2, 59, 3, 61, 2, 7, 0, 5, 2, 67, 17, 3, 2, 71, 0, 73, 2, 3, 19, 7, 2, 79, 5, 0, 2, 83, 3, 5, 2, 3, 11, 89, 2, 7, 23, 3, 2

Offset: 1

Labos Elemer, Feb 21 2003

			For n = 252100 = 2*2*3*5*5*7*11*11, the unitary prime divisors are {3,7}, the smallest is 3, so a(252100) = 3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A034444, A056169, A080367, A277697.
Cf. A020639, A027748, A055231, A124010.
Cf. A001694 (positions of zeros).
Cf. A277698 for a variant which gives 1's instead of 0's for numbers with no unitary prime divisors (A001694).

a080368 n = if null us then 0 else fst $ head us
  where us = filter ((== 1) . snd) $ zip (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Jul 23 2014

ffi[x_] := Flatten[FactorInteger[x]]; lf[x_] := Length[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; gb[x_] := GCD[ba[x], x/ba[x]]; fpg[x_] := Flatten[Position[gb[x], 1]]; upd[x_] := Part[ba[x], fpg[x]]; mxu[x_] := Max[upd[x]]; miu[x_] := Min[upd[x]]; Do[If[Equal[upd[n], {}], Print[0]]; If[ !Equal[upd[n], {}], Print[miu[n]]], {n, 2, 256}]
Table[If[Or[n == 1, Length@ # == 0], 0, First@ #] &@ Select[FactorInteger[n][[All, 1]], GCD[#, n/#] == 1 &], {n, 94}] (* Michael De Vlieger, Oct 30 2016 *)
a[n_] := If[(p = Select[FactorInteger[n], Last[#] == 1 &][[;; , 1]]) == {}, 0, Min[p]]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Aug 17 2024 *)

a(n) = {my(f = factor(n), pmin = 0); for(i = 1, #f~, if(f[i, 2] == 1, if(pmin == 0, pmin = f[i, 1], if(f[i, 1] < pmin, pmin = f[i, 1])))); pmin;} \\ Amiram Eldar, Aug 17 2024

from sympy import factorint, prime, 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 a277697(n): return 0 if n==1 else a055396(n) if a067029(n)==1 else a277697(a028234(n))
def a(n): return 0 if a277697(n)==0 else prime(a277697(n)) # Indranil Ghosh, May 16 2017

(define (A080368 n) (if (zero? (A277697 n)) 0 (A000040 (A277697 n)))) ;; Antti Karttunen, Oct 28 2016

If A277697(n) = 0, then a(n) = 0, otherwise a(n) = A000040(A277697(n)). - Antti Karttunen, Oct 28 2016
from Amiram Eldar, Aug 17 2024: (Start)
a(n) = 0 if and only of n is powerful (A001694).
a(n) = A020639(A055231(n)) if n is not powerful. (End)

a(1)=0 inserted by Reinhard Zumkeller, Jul 23 2014

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)).

A371601 Nonsquarefree numbers whose largest nonunitary prime divisor is smaller than their smallest unitary prime divisor, if it exists.

Original entry on oeis.org

4, 8, 9, 12, 16, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 52, 56, 60, 63, 64, 68, 72, 76, 80, 81, 84, 88, 92, 96, 99, 100, 104, 108, 112, 116, 117, 120, 121, 124, 125, 128, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 164, 168, 169, 171, 172, 175, 176

Offset: 1

Amiram Eldar, Mar 29 2024

Subsequence of A283050 and first differs from it at n = 100: A283050(100) = 300 = 2^2 * 3 * 5^2 is not a term of this sequence.
Powerful numbers and nonpowerful numbers k such that 1 < A249740(k) < A277698(k), or equivalently, 1 < A006530(A057521(k)) < A020639(A055231(k)).
The asymptotic density of this sequence is (6/Pi^2) * Sum_{p prime} f(p)/(p^2-p+1) = 0.32131800923..., where f(p) = Product_{primes q <= p} (q^2-q+1)/(q^2-1).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A013929 and A283050.

Cf. A001694, A006530, A020639, A052485, A055231, A057521, A059956, A249740, A277698.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Max[e] > 1 && (Min[e] > 1 || Max[e[[FirstPosition[e, 1][[1]] ;; -1]]] == 1)]; Select[Range[200], q]

is(n) = {my(e = apply(x->if(x > 1, 2, 1), factor(n)[,2])); n > 1 && vecmax(e) > 1 && vecsort(e, , 4) == e;}

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Scheme

Formula

A080368 a(n) is the least unitary prime divisor of n, or 0 if no such prime divisor exists.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Scheme

Formula

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Python

Formula

A371601 Nonsquarefree numbers whose largest nonunitary prime divisor is smaller than their smallest unitary prime divisor, if it exists.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI