A333789 Numbers k for which A333790(k) < A073934(k).

119, 143, 187, 209, 221, 238, 239, 286, 319, 357, 374, 407, 418, 419, 429, 442, 443, 451, 476, 478, 479, 561, 572, 595, 627, 638, 663, 667, 671, 703, 713, 714, 715, 717, 748, 779, 803, 814, 833, 836, 838, 839, 851, 858, 859, 884, 886, 887, 902, 935, 943, 952, 953, 956, 957, 958, 979, 989, 1001, 1045, 1067, 1071, 1073, 1105, 1111, 1122

Offset: 1

Author

Antti Karttunen, Apr 12 2020

Keywords

• nonn

Comments

Numbers n for which the {smallest path sum when iterating from n to 1 with nondeterministic map k -> k - k/p, where p is any prime factor of k} cannot be obtained by always selecting the smallest prime factor of k (A020639). See the example in A333790 how that simple heuristic fails when starting from k=119.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..36576; all terms <= 2^17

Crossrefs

Cf. A020639, A060681, A073934, A333790.

Programs

Mathematica

Block[{a, b, nn = 1122}, a = Min@ Map[Total, #] & /@ Nest[Function[{a, n}, Append[a, Join @@ Table[Flatten@ Prepend[#, n] & /@ a[[n - n/p]], {p, FactorInteger[n][[All, 1]]}]]] @@ {#, Length@ # + 1} &, {{{1}}}, nn]; b = Array[If[# == 1, 1, Total@ NestWhileList[If[PrimeQ@ #, # - 1, # - #/FactorInteger[#][[1, 1]] ] &, #, # > 1 &]] &, nn]; Select[Range@ nn, a[[#]] < b[[#]] &]] (* Michael De Vlieger, Apr 15 2020 *)

PARI

search_up_to = 2^17;
A333790list(up_to) = { my(v=vector(up_to)); v[1] = 1; for(n=2, up_to, v[n] = n+vecmin(apply(p -> v[n-n/p], factor(n)[, 1]~))); (v); };
v333790 = A333790list(search_up_to);
A333790(n) = v333790[n];
A073934(n) = if(1==n,n,n + A073934(n-(n/vecmin(factor(n)[,1]))));
isA333789(n) = (A073934(n)!=A333790(n));