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A096529
Numbers whose divisors can be permuted so that all sums of triple adjacent divisors are primes.
2
4, 8, 9, 10, 12, 14, 15, 16, 20, 21, 24, 25, 26, 27, 28, 33, 34, 35, 36, 38, 39, 40, 44, 45, 52, 55, 56, 57, 58, 60, 63, 65, 68, 75, 76, 77, 81, 84, 85, 86, 88, 92, 93, 99, 100, 104, 105, 111, 115, 117, 119, 123, 124, 125, 129, 132, 135, 136, 140, 143, 145, 147
OFFSET
1,1
COMMENTS
Square of terms of A053182 are in this sequence. - Michel Marcus, May 08 2014
From Amiram Eldar, Nov 08 2024: (Start)
The possible values of the number of even divisors of even terms of this sequence is restricted by the number of odd divisors.
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.
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.
If m = 1, then d_odd(k) = 1 and e <= 4, so 16 = 2^4 is the largest power of 2 in this sequence.
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.
If m is composite, then d_odd(k) > 2 and e <= 2, and therefore k is not divisible by 8. (End)
LINKS
FORMULA
A096527(a(n)) > 0.
EXAMPLE
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.
PROG
(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; }
is2(n) = {my(d = divisors(n)); forperm(d, p, if(isok(p), return(1))); 0; }
is1(k) = {my(e = valuation(k, 2), o = k >> e); (e == 0) || (o == 1 && e <= 4) || (abs(e-2) * numdiv(o) <= 2); }
is(k) = is1(k) && is2(k); \\ Amiram Eldar, Nov 08 2024
CROSSREFS
Complement of A096530.
Sequence in context: A214489 A189207 A127162 * A351099 A193166 A155101
KEYWORD
nonn,changed
AUTHOR
Reinhard Zumkeller, Jun 23 2004
EXTENSIONS
a(30)-a(51) from Michel Marcus, May 03 2014
a(52) onwards from Amiram Eldar, Nov 08 2024
STATUS
approved