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A306454 a(n) = A261327(n)/A013946(n). 1
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 25, 1, 1, 25, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 169, 1, 1, 1, 1, 1, 1, 25, 1, 1, 25, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 25, 1, 1, 25, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 289, 1, 1, 1, 1, 1, 841, 1, 1, 1, 25, 1, 1, 25, 1, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,11

COMMENTS

Are all terms odd squares?

b(n) = A013946(n)*A261327(n) = 25, 4, 169, 25, 841, 100, 2809, 289, 7225, 676, 625, ... . Are all terms squares?

a(n) = A008833(n^2+4) if n is odd and A008833((n^2+4)/4) if n is even, so a(n) is always an odd square. - Jianing Song, Feb 27 2019

Are the square roots only primes?

The sequence of period 4: repeat [25, 1, 1, 25] appears apparently every 25 terms.

From Robert Israel, Mar 20 2019: (Start)

The first term whose square root is not 1 or a prime is a(261) = 25^2.

a(11+25*k) is divisible by 25. The first term where a(11+25*k) > 25 is a(261)=a(11+25*10)=625.

The first term where a(12+25*k) > 1 is a(1212)=a(12+25*48)=169.

The first term where a(13+25*k) > 1 is a(213)=a(13+25*8)=289.

a(14+25*k) is divisible by 25. The first term where a(14+25*k) > 25 is a(364)=a(14+25*14)=625.

All prime factors of members of the sequence are in A002144.  For any p in A002144, there is k with 1 <= k < p^2/2 such that p^2 | a(n) if and only if n == k or -k (mod p^2).

- Robert Israel, Mar 20 2019

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

EXAMPLE

A261327(n) = 5, 2, 13, 5, 29, 10, 53, 17, 85, 26, 125, 37, 173, 50, ... .

A013946(n) = 5, 2, 13, 5, 29, 10, 53, 17, 85, 26,   5, 37, 173,  2, ... .

MAPLE

core:= proc(n) local t; mul(t[1], t=select(s -> s[2]::odd, ifactors(n)[2])) end proc:

map(n -> numer((4+n^2)/4)/core(n^2+4), [$1..100]); # Robert Israel, Mar 20 2019

MATHEMATICA

core[n_] := Times @@ Select[FactorInteger[n], OddQ[#[[2]]]&][[All, 1]];

a[n_] := Numerator[(n^2+4)/4]/core[n^2+4];

Array[a, 100] (* Jean-Fran├žois Alcover, Jan 04 2022 *)

PROG

(PARI)

A013946(n) = core(n^2+4); \\ From A013946

A261327(n) = if(n%2, n^2+4, (n/2)^2+1); \\ From A261327

A306454(n) = (A261327(n)/A013946(n)); \\ Antti Karttunen, Feb 28 2019

(PARI) A306454(n) = { my(k=((n^2)+4)/if(n%2, 1, 4)); k/core(k); }; \\ Antti Karttunen, Feb 28 2019, after Jianing Song's formula

CROSSREFS

Cf. A013946, A261327.

Cf. A002144, A008833.

Sequence in context: A040627 A040626 A040625 * A203550 A022188 A040636

Adjacent sequences:  A306451 A306452 A306453 * A306455 A306456 A306457

KEYWORD

nonn

AUTHOR

Paul Curtz, Feb 16 2019

STATUS

approved

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Last modified July 2 16:36 EDT 2022. Contains 355029 sequences. (Running on oeis4.)