

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
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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

Amiram Eldar, Table of n, a(n) for n = 1..10000


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

Cf. A001414, A001567, A008472.
Sequence in context: A231202 A226448 A250433 * A332314 A125576 A233501
Adjacent sequences: A329461 A329462 A329463 * A329465 A329466 A329467


KEYWORD

nonn


AUTHOR

Amiram Eldar, Nov 13 2019


STATUS

approved



