The On-Line Encyclopedia of Integer Sequences (OEIS)

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A368980

The number of exponential divisors of n that are squares (A000290).
(history; published version)

#9 by OEIS Server at Fri Jan 12 22:52:12 EST 2024

LINKS

Amiram Eldar, <a href="/A368980/b368980_1.txt">Table of n, a(n) for n = 1..10000</a>

#8 by N. J. A. Sloane at Fri Jan 12 22:52:12 EST 2024

STATUS

proposed

approved

Discussion

Fri Jan 12

22:52

OEIS Server: Installed first b-file as b368980.txt.

#7 by Amiram Eldar at Thu Jan 11 01:23:31 EST 2024

STATUS

editing

proposed

#6 by Amiram Eldar at Thu Jan 11 01:22:43 EST 2024

FORMULA

Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = 1.602317... (A327837 * sqrt(n).

#5 by Amiram Eldar at Thu Jan 11 01:21:40 EST 2024

FORMULA

a(n) <= A049419(n), with equality if and only if n = 1.

#4 by Amiram Eldar at Thu Jan 11 01:16:50 EST 2024

FORMULA

a(n) >= 0, with equality if and only if n is not a square number (A000037).

#3 by Amiram Eldar at Thu Jan 11 00:33:28 EST 2024

LINKS

Amiram Eldar, <a href="/A368980/b368980_1.txt">Table of n, a(n) for n = 1..10000</a>

#2 by Amiram Eldar at Thu Jan 11 00:29:58 EST 2024

NAME

allocated for Amiram EldarThe number of exponential divisors of n that are squares (A000290).

DATA

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

OFFSET

1,16

FORMULA

a(n) = A049419(n^2).

Multiplicative with a(p^e) = A183063(e), or equivalently, a(p^e) = 0 if e is odd, and A000005(e/2) if e is even.

a(n) >= 0, with equality if and only if n is not square (A000037).

Sum_{k=1..n} a(k) ~ A327837 * sqrt(n).

MATHEMATICA

f[p_, e_] := If[OddQ[e], 0, DivisorSigma[0, e/2]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PROG

(PARI) a(n) = vecprod(apply(x -> if(x%2, 0, numdiv(x/2)), factor(n)[, 2]));

CROSSREFS

Cf. A000005, A000037, A000290, A049419, A183063, A322791, A327837, A368979.

Similar sequences: A046951, A056624, A358260, A368978.

KEYWORD

allocated

nonn,easy,mult

AUTHOR

Amiram Eldar, Jan 11 2024

STATUS

approved

editing

#1 by Amiram Eldar at Thu Jan 11 00:19:45 EST 2024

NAME

allocated for Amiram Eldar

KEYWORD

allocated

STATUS

approved