A096529 - OEIS

%I #27 Nov 08 2024 08:37:20

%S 4,8,9,10,12,14,15,16,20,21,24,25,26,27,28,33,34,35,36,38,39,40,44,45,

%T 52,55,56,57,58,60,63,65,68,75,76,77,81,84,85,86,88,92,93,99,100,104,

%U 105,111,115,117,119,123,124,125,129,132,135,136,140,143,145,147

%N Numbers whose divisors can be permuted so that all sums of triple adjacent divisors are primes.

%C Square of terms of A053182 are in this sequence. - _Michel Marcus_, May 08 2014

%C From _Amiram Eldar_, Nov 08 2024: (Start)

%C The possible values of the number of even divisors of even terms of this sequence is restricted by the number of odd divisors.

%C Let k be a term and d_odd(k) = A001227(k) and d_even(k) = A183063 be its number of odd divisors and number of even divisors, respectively. When k is even, in a valid permutation of its divisors there must be two even divisors between two odd divisors, at most 2 before the first odd divisor, and at most 2 after the last odd divisor.

%C Therefore, d_even(k) - 2*(d_odd(k) - 1) <= 4. Let d(k) = A000005(k) = d_odd(k) + d_even(k), and let e = A007814(k) and m = A000265(k). Then, k = 2^e * m, d(k) = (e+1) * d(m) = (e+1) * d_odd(k), so d_even(k) = e * d_odd(k), and |e-2| * d_odd(k) <= 2.

%C If m = 1, then d_odd(k) = 1 and e <= 4, so 16 = 2^4 is the largest power of 2 in this sequence.

%C If m = p is a prime, then d_odd(k) = 2 and e <= 3, and therefore only terms of the form 2*p, 4*p or 8*p are possible. 2*p is a term if and only if p is a term of A106067.

%C If m is composite, then d_odd(k) > 2 and e <= 2, and therefore k is not divisible by 8. (End)

%H Amiram Eldar, <a href="/A096529/b096529.txt">Table of n, a(n) for n = 1..74</a>

%F A096527(a(n)) > 0.

%e Divisors of 24 are {1,2,3,4,6,8,12,24}: [2,8,3,12,4,1,24,6] -> (2+8+3,8+3+12,3+12+4,12+4+1,4+1+24,1+24+6) = (13,23,19,17,29,31): therefore 24 is a term.

%o (PARI) isok(p) = {my(n = #p); if(n < 3, return(0)); for(k = 1, n-2, if(!isprime(p[k]+p[k+1]+p[k+2]), return(0))); 1;}

%o is2(n) = {my(d = divisors(n)); forperm(d, p, if(isok(p), return(1))); 0;}

%o is1(k) = {my(e = valuation(k,2), o = k >> e); (e == 0) || (o == 1 && e <= 4) || (abs(e-2) * numdiv(o) <= 2);}

%o is(k) = is1(k) && is2(k); \\ _Amiram Eldar_, Nov 08 2024

%Y Complement of A096530.

%Y Cf. A053182, A096527.

%Y Cf. A000005, A000265, A001227, A007814, A106067, A183063.

%K nonn

%O 1,1

%A _Reinhard Zumkeller_, Jun 23 2004

%E a(30)-a(51) from _Michel Marcus_, May 03 2014

%E a(52) onwards from _Amiram Eldar_, Nov 08 2024