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A329464
Poulet numbers (Fermat pseudoprimes to base 2) k such that sopfr(k) is also a Poulet number, where sopfr(k) is the sum of the primes dividing k with repetition (A001414).
1
261523801, 407131165, 762278161, 1144998841, 1267600105, 1538242161, 1618206745, 1632785701, 1984685185, 2265650401, 2607116865, 2769136833, 2830242961, 3121418161, 3521441665, 4202755165, 4320404641, 4786205041, 5013061801, 5154023161, 6647529601, 6850760365
OFFSET
1,1
COMMENTS
The corresponding values of sopfr(a(n)) are 1105, 13741, 4371, 1105, 4033, 4681, 1729, 4371, 4681, 1105, 1729, 8481, 8321, 1105, 4681, 2701, 1729, 1105, 1105, 4033, 1105, 2821, ...
The least term with 3 prime factors is 762278161 = 337 * 673 * 3361.
The least term with 5 prime factors is 261523801 = 7 * 11 * 17 * 241 * 829.
The least term with 7 prime factors is 1672711724593 = 7 * 13 * 17 * 37 * 73 * 97 * 4127.
The least term with 9 prime factors is 402664105330201 = 11 * 13 * 17 * 31 * 37 * 41 * 43 * 101 * 811.
The nonsquarefree terms are 69727862019625993, 2052198214756489801, 2615108159340724321, ...
Since most Poulet numbers are squarefree, the sequence of Poulet numbers k such that sopf(k) is also a Poulet number (sopf(k) is the sum of the distinct primes dividing k, A008472) is similar, with the nonsquarefree terms being 1784634771193, 2464196735286469, 3732511553381521, 5715437749541521, 35442651934429741, 58035415075638001, ...
LINKS
EXAMPLE
261523801 = 7 * 11 * 17 * 241 * 829 is a term since it is a Poulet number, and 7 + 11 + 17 + 241 + 829 = 1105 is also a Poulet number.
MATHEMATICA
pouletQ[n_] := CompositeQ[n] && PowerMod[2, n - 1, n] == 1; sopf[n_] := Total[FactorInteger[n][[;; , 1]]]; s={}; Do[If[pouletQ[n] && pouletQ[sopf[n]], AppendTo[s, n]], {n, 2, 3*10^9}]; s
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Nov 13 2019
STATUS
approved