A234842 Primes that are reached by an ever increasing aliquot sequence.

463, 523, 983, 1153, 2851, 2969, 4339, 4507, 6121, 8263, 8893, 10093, 12451, 17911, 18427, 18913, 22807, 22811, 25033, 27961, 33223, 36781, 41849, 42643, 48571, 60091, 64237, 71503, 73303, 74131, 90217, 90481, 103813, 108263, 123601, 124447, 125863, 140443

Offset: 1

Author

Michel Marcus, Dec 31 2013

Keywords

• nonn

Comments

Note that the starting point of these aliquot sequences are not in increasing order, since for instance we have: 392->463->1 and 324->523->1, that is, with 392>324 while 463<523.

One can observe that the "ever increasing aliquot" part in the definition is not really necessary. A prime is in the sequence if there is an abundant number whose sum of proper divisors results into this prime. So sequence could also be defined as: Primes resulting from summing up the proper divisors of an abundant number. - Michel Marcus, Jan 05 2014

If we try to build the revert sequence listing the starting points of the aliquot sequences, we would get the following terms in increasing order 324, 392, 784, 800, 2304, 2450, 2704, 3600, 3872. But then for n=5352, we'd hit a sequence that begins 5352->8088->12192->20064 and keeps rising to a point where the factors of the last known term are not known. Then later, there are several other such aliquot sequences like 9336->14064->22392 or 10344->15576->27624 that have the same behavior. Thus the only sure terms of the revert sequence would be the terms listed earlier. - Michel Marcus, Jan 11 2014

Examples

			The aliquot sequence that begins with 10712 is always increasing before reaching prime 12451: 10712->11128->11552->12451->1, hence 12451 is in the sequence.
20422951 also belongs here with the aliquot sequence that starts at 14952, so a 13-tuple abundant (see factordb link).
People at the Aliquot Sequences project have found longer sequences that reach higher primes.
		

Links

• Michel Marcus, Table of n, a(n) for n = 1..100
• Factordb, Sequence starting with 14952
• Michel Marcus, Aliquot sequences leading to these primes
• Help needed for a sequence Thread on Aliquot Sequences Forum

Crossrefs

Cf. A001065, A005101, A080907, A081705, A098007, A098008, A115350.

Programs

PARI

prev(n) = {for (i=1, n, if ((sigma(i) - i) == n, return (i));); return (0);}
lista(nn) = {forprime(p=2, nn, if (prev(p), print1(p, ", ");););} \\ simplified by Michel Marcus, Jan 11 2014