A275584 Primes p such that S_e(p-1)/S_o(p-1) is an integer, where S_e(x) is the sum of the even numbers and S_o(x) is the sum of the odd numbers in the Collatz iteration of x.

2, 3, 5, 17, 257, 59393, 65537, 331777, 534529, 1299457

Offset

1,1

Comments

Primes p such that A213909(p-1)/A213916(p-1) is an integer.

Primes of the form A274796 + 1.

Fermat primes (A019434) are terms. Also supersequence of A092506 (primes of the form 2^n+1).

Corresponding values of S_e/o(a(n)-1): 0, 2, 6, 30, 510, 1567, 131070, ...

Links

Table of n, a(n) for n=1..10.

Eric Weisstein's World of Mathematics, Collatz Problem

Wikipedia, Collatz conjecture

Formula

S_e/o(A092506(n)-1) = A033493(A092506(n))-1.

Example

Prime 59393 is a term because S_e/o(59392) = A213909(59392)/A213916(59392) = 119092/76 = 1567.

Mathematica

Select[Prime@ Range[10^5], IntegerQ[Divide @@ Map[Total, TakeDrop[#, LengthWhile[#, EvenQ]]]] &@ SortBy[#, OddQ] &@ NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, # - 1, # > 1 &] &] (* Michael De Vlieger, Oct 15 2018 *)

Prog

(Magma) [n+1: n in [A274796(m)] | IsPrime(n+1)]
(Magma) e:= [&+[not IsOdd(h) select h else 0: h in [k eq 1 select n else IsOdd(Self(k-1)) and not IsOne(Self(k-1)) select 3*Self(k-1)+1 else Self(k-1) div 2: k in [1..5*n]]]: n in [1..1000]]; o:= [&+[IsOdd(h) select h else 0: h in [k eq 1 select n else IsOdd(Self(k-1)) and not IsOne(Self(k-1)) select 3*Self(k-1)+1 else Self(k-1) div 2: k in [1..5*n]]]: n in [1..1000]]; [n+1: n in [1..1000] | IsPrime(n+1) and e[n] mod o[n] eq 0]

Crossrefs

Cf. A033493, A213909, A213916, A274796.

Sequence in context: A265426 A099936 A237251 * A092506 A275159 A127063

Adjacent sequences: A275581 A275582 A275583 * A275585 A275586 A275587

Keyword

nonn,more

Author

Jaroslav Krizek, Aug 04 2016

Extensions

More terms from Michael De Vlieger, Oct 15 2018.

Status

approved