A328897 - OEIS

%I #18 Dec 06 2021 03:13:51

%S 27,45,75,105,117,135,147,165,187,189,231,243,245,275,285,297,315,333,

%T 345,357,375,387,403,405,423,425,427,429,435,437,459,473,495,507,525,

%U 555,567,583,585,605,621,627,637,663,665,675,693,729,731,735,741,763,765,775,777,795

%N Odd numbers k > 1 such that A005179(k-1) > A005179(k) < A005179(k+1).

%C As only square numbers have an odd number of divisors, for odd k, A005179(k) is usually larger than either A005179(k-1) or A005179(k+1) (or both). This sequence lists the exceptions. There are 71 terms below 10^3, 963 terms below 10^4, 11179 terms below 10^5. It seems that the density of this sequence over all the odd numbers is > 0.2.

%C Is there any odd k such that A005179(k) is smaller than A005179(k-3), A005179(k-1), A005179(k+1) and A005179(k+3)? There is no such k < 10^5.

%H Jianing Song, <a href="/A328897/b328897.txt">Table of n, a(n) for n = 1..11179</a> (all terms below 10^5)

%e 27 is a term because the smallest number with 27 divisors is 900, which is smaller than both A005179(26) = 12288 and A005179(28) = 960, so 27 is a term.

%e 45 is a term because the smallest number with 45 divisors is 3600, which is smaller than both A005179(44) = 15360 and A005179(46) = 12582912, so 45 is a term.

%p A := [seq(A005179(n), n=1..800)];

%p isA := k -> k::odd and A[k] < A[k-1] and A[k] < A[k+1]:

%p select(isA, [$3..799]); # _Peter Luschny_, Oct 30 2019

%o (PARI) isA328897(k) = (k%2&&k>1) && A005179(k)<A005179(k-1) && A005179(k)<A005179(k+1) \\ Corrected by _Jianing Song_, Dec 05 2021

%Y Cf. A005179, A339863.

%K nonn

%O 1,1

%A _Jianing Song_, Oct 30 2019