cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Antti Karttunen, Oct 28 2016

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    Table[If[Length@ # == 0, 0, PrimePi@ First@ #] &@ Select[FactorInteger[n][[All, 1]], GCD[#, n/#] == 1 &], {n, 105}] (* Michael De Vlieger, Oct 30 2016 *)
  • PARI
    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
  • Python
    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
  • Scheme
    (definec (A277697 n) (cond ((= 1 n) 0) ((= 1 (A067029 n)) (A055396 n)) (else (A277697 (A028234 n)))))

Formula

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

A277698 a(n) = least unitary prime divisor of n or 1 if no such prime-divisor exists.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Oct 28 2016

Keywords

Links

Crossrefs

Cf. A008578, A020639, A055231, A277697.
Cf. A001694 (positions of ones).
Cf. A080368 for a variant which gives 0's instead of 1's for numbers with no unitary prime divisors and also A277708 (the least prime factor with an odd exponent).
Differs from A134194 for the first time at n=18, where a(18) = 2, while A134194(18) = 3.

Programs

  • Mathematica
    Table[If[Length@ # == 0, 1, First@ #] &@ Select[FactorInteger[n][[All, 1]], GCD[#, n/#] == 1 &], {n, 105}] (* Michael De Vlieger, Oct 30 2016 *)
  • PARI
    a(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 2] == 1, return(f[i, 1]))); 1;} \\ Amiram Eldar, Jul 28 2024
  • Python
    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 a008578(n): return 1 if n==1 else prime(n - 1)
def a(n): return a008578(1 + a277697(n)) # Indranil Ghosh, May 16 2017
  • Scheme
    (define (A277698 n) (A008578 (+ 1 (A277697 n))))

Formula

a(n) = A008578(1+A277697(n)).
a(n) = A020639(A055231(n)). - Amiram Eldar, Jul 28 2024

A080367 Largest unitary prime divisor of n or a(n) = 0 if no such prime divisor exists.

Original entry on oeis.org

0, 2, 3, 0, 5, 3, 7, 0, 0, 5, 11, 3, 13, 7, 5, 0, 17, 2, 19, 5, 7, 11, 23, 3, 0, 13, 0, 7, 29, 5, 31, 0, 11, 17, 7, 0, 37, 19, 13, 5, 41, 7, 43, 11, 5, 23, 47, 3, 0, 2, 17, 13, 53, 2, 11, 7, 19, 29, 59, 5, 61, 31, 7, 0, 13, 11, 67, 17, 23, 7, 71, 0, 73, 37, 3, 19, 11, 13, 79, 5, 0, 41, 83, 7
Offset: 1

Views

Author

Labos Elemer, Feb 21 2003

Keywords

Comments

See [Grah, Section 5] for growth rate of the partial sums. - R. J. Mathar, Mar 03 2009

Examples

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

Links

Crossrefs

Cf. A034444, A056169, A080368.
Cf. A001694, A006530, A027748, A055231, A124010.

Programs

  • Haskell
    a080367 n = if null us then 0 else fst $ last us
  where us = filter ((== 1) . snd) $ zip (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Jul 23 2014
  • Mathematica
    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[mxu[n]]], {n, 2, 256}]
a[n_] := Max[Join[Select[FactorInteger[n], Last[#] == 1 &][[;; , 1]], {0}]]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Aug 17 2024 *)
  • PARI
    a(n) = {my(f = factor(n), pmax = 0); for(i = 1, #f~, if(f[i, 2] == 1 && f[i, 1] > pmax, pmax = f[i, 1])); pmax;} \\ Amiram Eldar, Aug 17 2024

Formula

from Amiram Eldar, Aug 17 2024: (Start)
a(n) = 0 if and only of n is powerful (A001694).
a(n) = A006530(A055231(n)) if n is not powerful. (End)
Showing 1-3 of 3 results.

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