login
A255576
Integers k such that Sum_{i=1..t-1} d(i)/d(i+1) is prime, where d(1), ..., d(t) are the divisors of k in ascending order.
1
16, 64, 729, 1024, 1536, 6250, 9375, 16384, 19683, 39366, 1179648, 4194304, 6770688, 9765625, 14348907, 29229255, 39062500, 67108864, 125000000, 128472708, 335544320, 1337982976, 10460353203
OFFSET
1,1
COMMENTS
The corresponding primes are 2, 3, 2, 5, 13, 5, 5, 7, 3, 11, 41, 11, 89, 2, 5, 37, 19, 13, 53, 37, ...
a(n) is a power of 2 for n = 1, 2, 4, 8, 12, 18, ... with the corresponding primes 2, 3, 5, 7, 11, 13, ...
a(n) is a perfect square for n = 1, 2, 3, 4, 8, 12, 14, 17, 18, ... with the corresponding primes 2, 3, 2, 5, 7, 11, 2, 19, 13, ...
EXAMPLE
64 is in the sequence because the divisors of 64 are {1, 2, 4, 8, 16, 32, 64} and 1/2 + 2/4 + 4/8 + 8/16 + 16/32 + 32/64 = 3 is prime.
MATHEMATICA
Do[s=0; Do[s=s+Divisors[n][[i]]/Divisors[n][[i+1]], {i, 1, Length[Divisors[n]]-1}]; If[PrimeQ[s]&&!PrimeQ[n], Print[n]], {n, 10^6}]
Select[Range[40000], PrimeQ[Total[#[[1]]/#[[2]]&/@Partition[ Divisors[ #], 2, 1]]]&] (* The program generates the first 10 terms of the sequence. To generate more, increase the Range constant. *) (* Harvey P. Dale, Feb 06 2022 *)
PROG
(Python)
from sympy import isprime, divisors
from fractions import Fraction
def ok(n):
divs = divisors(n)
f = sum(Fraction(dn, dd) for dn, dd in zip(divs[:-1], divs[1:]))
return f.denominator == 1 and isprime(f.numerator)
print([k for k in range(1, 40000) if ok(k)]) # Michael S. Branicky, Feb 06 2022
CROSSREFS
Subsequence of A227993.
Sequence in context: A175209 A141840 A203281 * A065404 A327496 A330824
KEYWORD
nonn,more
AUTHOR
Michel Lagneau, Feb 25 2015
EXTENSIONS
a(20) inserted and a(22)-a(23) from Michael S. Branicky, Feb 06 2022 using A227993
STATUS
approved