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

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;
if a=n then print(n); fi; od; end: P(10^9);

Crossrefs

Cf. A000225, A001065, A186103, A320021.

Sequence in context: A303803 A304850 A306740 * A057463 A287124 A118258

Adjacent sequences: A320019 A320020 A320021 * A320023 A320024 A320025

Keyword

nonn

Author

Paolo P. Lava, Oct 03 2018

Extensions

a(38)-a(41) from Giovanni Resta, Oct 09 2018

Status

approved