A019268 Let Dedekind's psi(m) = product of (p+1)p^(e-1) for primes p, where p^e is a factor of m. Iterating psi(m) eventually results in a number of form 2^a*3^b. a(n) is the smallest number that requires n steps to reach such a number.

1, 5, 13, 37, 73, 673, 1993, 15013, 49681, 239233, 1065601, 8524807, 68198461, 545587687, 1704961513, 7811750017

Offset

0,2

Comments

There is a remarkable and unexplained agreement: if 5 is dropped from the list, 2, 673, 1993 and 239233 are replaced by 1, 1021, 29173 and 532801, the result is sequence A005113 (least prime of class n+, according to the Erdős-Selfridge classification of primes).

A019269(a(n)) = n and A019269(m) != n for m < a(n). [Reinhard Zumkeller, Apr 12 2012]

References

Peter Giblin, "Primes and Programming - an Introduction to Number Theory with Computation", page 118.

R. K. Guy, "Unsolved Problems in Number Theory", section B41.

Links

Table of n, a(n) for n=0..15.

Mathematica

psi[m_] := ({pp, ee} = FactorInteger[m] // Transpose; If[Max[pp] == 3, m, Times @@ (pp+1)*Times @@ (pp^(ee-1))]); a[0] = 1; a[1] = 5; a[n_] := a[n] = For[k = a[n - 1] (* assuming monotony *), True, k++, If[Length @ FixedPointList[psi, k] == n+2, Return[k]]]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 10}] (* Jean-François Alcover, Feb 19 2018 *)

Prog

(Haskell)
import Data.List (elemIndex)
import Data.Maybe (fromJust)
a019268 = (+ 1) . fromJust . (`elemIndex` a019269_list)
-- Reinhard Zumkeller, Apr 12 2012

Crossrefs

Cf. A005113, A082449.

Sequence in context: A089523 A058507 A111057 * A083413 A232879 A269803

Adjacent sequences: A019265 A019266 A019267 * A019269 A019270 A019271

Keyword

nonn,nice,more

Author

Jud McCranie

Extensions

More terms from Jud McCranie, Jan 15 1997

Initial element corrected by Reinhard Zumkeller, Apr 12 2012

Status

approved