A347769 - OEIS

A347769

a(0) = 0; a(1) = 1; for n > 1, a(n) = A001065(a(n-1)) = sigma(a(n-1)) - a(n-1) (the sum of aliquot parts of a(n-1)) if this is not yet in the sequence; otherwise a(n) is the smallest number missing from the sequence.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 15, 13, 14, 17, 18, 21, 19, 20, 22, 23, 24, 36, 55, 25, 26, 27, 28, 29, 30, 42, 54, 66, 78, 90, 144, 259, 45, 33, 31, 32, 34, 35, 37, 38, 39, 40, 50, 43, 41, 44, 46, 47, 48, 76, 64, 63, 49, 51, 52, 53, 56, 57, 58, 59, 60, 108, 172

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

OFFSET

0,3

COMMENTS

This sequence is a permutation of the nonnegative integers iff Catalan's aliquot sequence conjecture (also called Catalan-Dickson conjecture) is true.

a(563) = 276 is the smallest number whose aliquot sequence has not yet been fully determined.

As long as the aliquot sequence of 276 is not known to be finite or eventually periodic, a(563+k) = A008892(k).

LINKS

Eric Chen, Table of n, a(n) for n = 0..2703 (all currently known terms, after the b-file of A008892)

Eric Weisstein's World of Mathematics, Aliquot sequence

Eric Weisstein's World of Mathematics, Catalan's Aliquot Sequence Conjecture

Wikipedia, Aliquot sequence

EXAMPLE

a(0) = 0, a(1) = 1;

since A001065(a(1)) = 0 has already appeared in this sequence, a(2) = 2;

since A001065(a(2)) = 1 has already appeared in this sequence, a(3) = 3;

...

a(11) = 11;

since A001065(a(11)) = 1 has already appeared in this sequence, a(12) = 12;

since A001065(a(12)) = 16 has not yet appeared in this sequence, a(13) = A001065(a(12)) = 16;

since A001065(a(13)) = 15 has not yet appeared in this sequence, a(14) = A001065(a(13)) = 15;

since A001065(a(14)) = 9 has already appeared in this sequence, a(15) = 13;

...

PROG

(PARI) A347769_list(N)=print1(0, ", "); if(N>0, print1(1, ", ")); v=[0, 1]; b=1; for(n=2, N, if(setsearch(Set(v), sigma(b)-b), k=1; while(k<=n, if(!setsearch(Set(v), k), b=k; k=n+1, k++)), b=sigma(b)-b); print1(b, ", "); v=concat(v, b))

CROSSREFS

Cf. A032451.

Cf. A001065 (sum of aliquot parts).

Cf. A003023, A044050, A098007, A098008: ("length" of aliquot sequences, four versions).

Cf. A007906.

Cf. A115060 (maximum term of aliquot sequences).

Cf. A115350 (termination of the aliquot sequences).

Cf. A098009, A098010 (records of "length" of aliquot sequences).

Cf. A290141, A290142 (records of maximum term of aliquot sequences).

Cf. A000396, A063990, A122726, A206708, A347770.

Cf. A080907, A063769, A121507, A121508, A131884, A126016.