A369657 a(n) = A356253(n) - A003415(n).

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, 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, 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

Offset: 1

Author

Antti Karttunen, Feb 08 2024

Keywords

• nonn

Comments

From M. F. Hasler, Feb 14 2024: (Start)

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, ...

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.

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.

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)

Links

• Antti Karttunen, Table of n, a(n) for n = 1..30030

Crossrefs

Cf. A003415, A086801, A356253.

Programs

PARI

A356253(n) = vecmax(Vec(vecprod([(x + f[1])^f[2] | f<-factor(n)~])))
A003415(n) = if(n>1, vecsum([n/f[1]*f[2] | f<-factor(n)~]), 0)
A369657(n) = A356253(n)-A003415(n) \\ This and above edited by M. F. Hasler, Feb 14 2024

Formula

a(9*prime(n)) = 3*A086801(n) for n > 1. - Thomas Scheuerle, Feb 14 2024