Voozh

A320022

Numbers equal to the sum of the aliquot parts of the following k numbers, for some k.

1, 3, 7, 9, 15, 31, 33, 56, 63, 127, 135, 168, 255, 511, 1023, 2047, 2401, 4095, 5328, 8191, 16383, 17360, 21003, 32767, 41163, 54721, 65535, 131071, 262143, 524287, 557280, 1048575, 1060801, 2097151, 4194303, 5026561, 8388607, 10800111, 11108163, 14366401, 16777215

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Any number of the form 2^j-1, with j > 0, is part of the sequence (with k=1).

So far 1 <= k <= 3 (k = 2 for 9, 33, 135, 168, 2401, 5328, 21003, 41163, 54721, 1060801, 5026561, ...; k = 3 for 56, 17360, ...). Are there terms with k = 4, 5, 6, ...? No k=4 up to 10^9.

If we were looking at numbers equal to the sum of the aliquot parts of the previous k numbers and of the following k, for some k, the first terms would be 2263024 and 128508838576, as confirmed by Giovanni Resta.

Up to n = 6*10^12 there are no terms with k>3. - Giovanni Resta, Oct 11 2018

LINKS

Table of n, a(n) for n=1..41.

FORMULA

a(n) = Sum_{i = 1..k} A001065(a(n)+i), for some k.

EXAMPLE

1 is in the sequence because aliquot part of 2 is 1.

9 is in the sequence because aliquot parts of 10 are 1, 2, 5 and of 11 is 1: 1 + 2 + 5 + 1 = 9.

56 is in the sequence because aliquot parts of 57 are 1, 3, 19, of 58 are 1, 2, 29, of 59 is 1: 1 + 3 + 19 + 1 + 2 + 29 + 1 = 56.

MAPLE

with(numtheory): P:=proc(q) local a, j, k, n; for n from 1 to q do

a:=0; k:=0; while a<n do k:=k+1; a:=a+sigma(n+k)-n-k; od;