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

Author

Paolo P. Lava, Oct 03 2018

Keywords

• nonn

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

Examples

			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.
		

Crossrefs

Cf. A000225, A001065, A186103, A320021.

Programs

Maple

with(numtheory): P:=proc(q) local a,j,k,n; for n from 1 to q do
a:=0; k:=0; while a
				

Formula

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

Extensions

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