A369657 - OEIS

%I #18 Feb 14 2024 14:26:31

%S 1,1,2,0,4,1,6,0,3,3,10,0,12,5,7,0,16,0,18,0,11,9,22,0,15,11,0,0,28,0,

%T 30,0,19,15,23,0,36,17,23,0,40,1,42,0,6,21,46,0,35,5,31,0,52,0,39,0,

%U 35,27,58,0,60,29,12,48,47,5,66,0,43,11,70,0,72,35,20,0,59,7,78,0,0,39,82,0,63,41,55,0,88,0,71

%N a(n) = A356253(n) - A003415(n).

%C From _M. F. Hasler_, Feb 14 2024: (Start)

%C a(n) = 0 for most n divisible by 4, except n = 64, 96, 128, 144, 160, 192, 216, 224, 240, 256, ... These exceptions include all proper multiples of 32 but also some other multiples of 4: 9*16, 27*8, 15*16, 21*16, ...

%C a(n) = 0 also for some n not a multiple of 4, namely 18*(6k + 1) for all k >= 0 except 2604, 18229, 33854, ... and 27*(4k + 1) for k >= 0 different from 101, 182, 236, ..., and others.

%C a(n) = 48 for all numbers of the form 32*p where p is prime, and for n = 171. (Are there any others?) This is by far the most frequent nonzero value: it can be seen as a horizontal line in the graph of the sequence.

%C a(n) = 11 for n = 5*709, 2*2833, 2*37*83, 2*29*107, 2*23*137, 2*17*191, 2*11*317, 2*7*569, 2*5*947, 2*3*2837, 2*3*53*67, ... This appears to be the second most frequent nonzero value. (End)

%H Antti Karttunen, <a href="/A369657/b369657.txt">Table of n, a(n) for n = 1..30030</a>

%F a(9*prime(n)) = 3*A086801(n) for n > 1. - _Thomas Scheuerle_, Feb 14 2024

%o (PARI)

%o A356253(n) = vecmax(Vec(vecprod([(x + f[1])^f[2] | f<-factor(n)~])))

%o A003415(n) = if(n>1, vecsum([n/f[1]*f[2] | f<-factor(n)~]), 0)

%o A369657(n) = A356253(n)-A003415(n) \\ This and above edited by _M. F. Hasler_, Feb 14 2024

%Y Cf. A003415, A086801, A356253.

%K nonn

%O 1,3

%A _Antti Karttunen_, Feb 08 2024