1,1

n is f-triperfect iff sum_{k divides n} f(k) = 3*f(n). f-triperfect numbers appear to be rare for many f. For the usual f(n) = n, there are at least 6 f-triperfect numbers known < 3.2 x 10^10, the first being 120. However, there are no f-triperfect numbers < 10^6 for f(n) = n+1 and f(n) = n-1 (if one ignores the trivial n = 1).

Wells, D. Curious and Interesting Numbers, Revised Edition. Penguin Books, 1997. (See the entry on "120".)

Divisors of 4680 = {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 18, 20, 24, 26, 30, 36, 39, 40, 45, 52, 60, 65, 72, 78, 90, 104, 117, 120, 130, 156, 180, 195, 234, 260, 312, 360, 390, 468, 520, 585, 780, 936, 1170, 1560, 2340, 4680}; f applied to these yield {1, 3, 4, 7, 6, 12, 15, 13, 18, 28, 14, 24, 39, 42, 60, 42, 72, 91, 56, 90, 78, 98, 168, 84, 195, 168, 234, 210, 182, 360, 252, 392, 546, 336, 546, 588, 840, 1170, 1008, 1274, 1260, 1092, 2352, 2730, 3276, 5040, 7644, 16380}, which sum to 49140 = 3 * 16380 = 3 * f(4680). Hence 4680 is a term of the sequence.

f[x_] := DivisorSigma[1, x]; Do[If[ Apply[ Plus, Map[ f, Divisors[ n ] ] ] == 3*f[n], Print[n]], {n, 1, 10^7}]

Cf. A066218.

Sequence in context: A020422 A048131 A062626 this_sequence A022244 A114542 A114568

Adjacent sequences: A066728 A066729 A066730 this_sequence A066732 A066733 A066734

nonn

Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Jan 15 2002