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A097008
a(n) = index of first appearance of n in A096862.
2
1, 2, 5, 11, 19, 43, 53, 101, 1297, 883, 1009, 1037, 1051, 985, 2391, 12101, 13457, 21887, 42683, 69697, 50177, 115601, 113669, 88897, 156817, 184477, 247487, 245029, 187273, 287543, 211031, 287093, 1001447, 5398093, 9741229, 7757137
OFFSET
1,2
COMMENTS
a(n) = smallest k such that A096863(k) + A096993(k) = n.
a(n) = smallest k such that n equals the index of the term that completes the first cycle in the trajectory of k under iteration of f(x) = A062402(x) = sigma(phi(x)).
LINKS
EXAMPLE
The trajectory of 19 under iteration of f(x) is 19, 39, 60, 31, 72, 60, 31, 72, ...; the cycle (60, 31, 72) is completed at the fifth term and for j < 19 the first cycle in trajectory of j under iteration of f(x) is completed at the first, second, third or fourth term, hence a(5) = 19.
The trajectory of 247487 under iteration of f(x) is 247487, 787200, 507873, 1282842, 1395372, 1476096, 1572096, 1089403, 3669120, 2621120, 4464096, 3963960, 2946240, 2538280, 3265416, 2877420, 1965840, 2227680, 1310680, 1591200, 1277874, 1307124, 1110488, 2010960, 1488032, 1981496, 2239920, 1965840, ...; the cycle (1965840, 2227680,
..., 2239920) is completed at the 27th term and for j < 247487 the first cycle in trajectory of j under iteration of f(x) is completed at an earlier term, hence a(27) = 247487.
MATHEMATICA
sf[x_] :=DivisorSigma[1, EulerPhi[x]]; nsf[x_, ho_] :=NestList[sf, x, ho]; luf[x_, ho_] :=Length[Union[nsf[x, ho]]]; t=Table[0, {35}]; Do[s=luf[n, 100]; If[s<36&&t[[s]]==0, t[[s]]=n], {n, 1, 1500000}]; t
PROG
(PARI) {v=vector(40); for(n=1, 10000000, k=n; s=Set(k); until(setsearch(s, k=sigma(eulerphi(k))), s=setunion(s, Set(k))); a=#s; if(a<=m&&v[a]==0, v[a]=n)); v} /* Klaus Brockhaus, Jul 16 2007 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Labos Elemer, Jul 26 2004
EXTENSIONS
Edited, a(27) and a(33) corrected and a(34) through a(36) added by Klaus Brockhaus, Jul 16 2007
STATUS
approved