A115060 Maximum peak of aliquot sequence starting at n.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 13, 14, 15, 16, 17, 21, 19, 22, 21, 22, 23, 55, 25, 26, 27, 28, 29, 259, 31, 32, 33, 34, 35, 55, 37, 38, 39, 50, 41, 259, 43, 50, 45, 46, 47, 76, 49, 50, 51, 52, 53, 259, 55, 64, 57, 58, 59, 172, 61, 62, 63, 64, 65, 259

Offset: 1

Author

Sergio Pimentel, Mar 06 2006

Keywords

• nonn

Comments

According to Catalan's conjecture all aliquot sequences end in a prime followed by 1, a perfect number, a friendly pair or an aliquot cycle. Some sequences seem to be open ended and keep growing forever i.e. 276. Most sequences only go down (i.e. 10 - 8 - 7 - 1), so for most cases in this sequence, a(n) = n. The first number to achieve a significantly high peak is 138

Examples

			a(24)=55 because the aliquot sequence starting at 24 is: 24 - 36 - 55 - 17 - 1 so the maximum peak of this sequence is 55.
		

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..275
• W. Creyaufmueller, Aliquot Sequences.
• Paul Zimmerman, Aliquot Sequences.

Crossrefs

Cf. A003023, A005114, A007906, A037020, A044050, A063769, A098007, A098008, A098009, A098010.

Programs

Python

from sympy import divisor_sigma as sigma
def aliquot(n):
    alst = []; seen = set(); i = n
    while i and i not in seen: alst.append(i); seen.add(i); i = sigma(i) - i
    return alst
def aupton(terms): return [max(aliquot(n)) for n in range(1, terms+1)]
print(aupton(66)) # Michael S. Branicky, Jul 11 2021

Extensions

More terms from Jinyuan Wang, Jul 11 2021