%I #21 Feb 07 2022 02:39:40
%S 16,64,729,1024,1536,6250,9375,16384,19683,39366,1179648,4194304,
%T 6770688,9765625,14348907,29229255,39062500,67108864,125000000,
%U 128472708,335544320,1337982976,10460353203
%N 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.
%C The corresponding primes are 2, 3, 2, 5, 13, 5, 5, 7, 3, 11, 41, 11, 89, 2, 5, 37, 19, 13, 53, 37, ...
%C 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, ...
%C 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, ...
%e 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.
%t 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}]
%t 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 *)
%o (Python)
%o from sympy import isprime, divisors
%o from fractions import Fraction
%o def ok(n):
%o divs = divisors(n)
%o f = sum(Fraction(dn, dd) for dn, dd in zip(divs[:-1], divs[1:]))
%o return f.denominator == 1 and isprime(f.numerator)
%o print([k for k in range(1, 40000) if ok(k)]) # _Michael S. Branicky_, Feb 06 2022
%Y Subsequence of A227993.
%K nonn,more
%O 1,1
%A _Michel Lagneau_, Feb 25 2015
%E a(20) inserted and a(22)-a(23) from _Michael S. Branicky_, Feb 06 2022 using A227993