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A286324 a(n) is the number of bi-unitary divisors of n. 14
1, 2, 2, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 8, 2, 6, 4, 4, 4, 4, 2, 4, 4, 8, 2, 8, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 8, 4, 8, 4, 4, 2, 8, 2, 4, 4, 6, 4, 8, 2, 4, 4, 8, 2, 8, 2, 4, 4, 4, 4, 8, 2, 8, 4, 4, 2, 8, 4, 4, 4, 8, 2, 8 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) is the number of terms of the n-th row of A222266.

LINKS

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

D. Suryanarayana, The number of bi-unitary divisors of an integer, in The Theory of Arithmetic Functions pp 273-282, Lecture Notes in Mathematics book series (LNM, volume 251).

Eric Weisstein's World of Mathematics, Biunitary Divisor

FORMULA

Multiplicative with a(p^e) = e + (e mod 2). - Andrew Howroyd, Aug 05 2018

EXAMPLE

From Michael De Vlieger, May 07 2017: (Start)

a(1) = 1 since 1 is the empty product; all divisors of 1 (i.e., 1) have a greatest common unitary divisor that is 1. 1 is a unitary divisor of all numbers n.

a(p) = 2 since 1 and p have greatest common unitary divisor 1.

a(6) = 4 since the divisor pairs {1, 6} and {2, 3} have greatest common unitary divisor 1.

a(24) = 8 since {1, 24}, {2, 12}, {3, 8}, {4, 6} have greatest unitary divisors {1, {1, 3, 8, 24}}, {{1, 2}, {1, 3, 4, 12}}, {{1, 3}, {1, 8}}, {1, 4}, {1, 2, 3, 6}}: 1 is the greatest common unitary divisor among all 4 pairs.

(End)

MATHEMATICA

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; Table[DivisorSum[n, 1 &, Last@ Intersection[f@ #, f[n/#]] == 1 &], {n, 90}] (* Michael De Vlieger, May 07 2017 *)

f[p_, e_] := If[OddQ[e], e + 1, e]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 120] (* Amiram Eldar, Dec 19 2018 *)

PROG

(PARI) udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }

gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));

biudivs(n) = select(x->(gcud(x, n/x)==1), divisors(n));

a(n) = #biudivs(n);

(PARI) a(n)={my(f=factor(n)[, 2]); prod(i=1, #f, my(e=f[i]); e + e % 2)} \\ Andrew Howroyd, Aug 05 2018

CROSSREFS

Cf. A222266, A188999.

Sequence in context: A037445 A318307 A331109 * A318472 A186643 A342087

Adjacent sequences:  A286321 A286322 A286323 * A286325 A286326 A286327

KEYWORD

nonn,mult

AUTHOR

Michel Marcus, May 07 2017

STATUS

approved

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Last modified April 23 05:42 EDT 2021. Contains 343199 sequences. (Running on oeis4.)