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A350593
Numbers k such that tau(k) + tau(k+1) = 6, where tau is the number of divisors function A000005.
7
5, 6, 7, 10, 13, 22, 37, 46, 58, 61, 73, 82, 106, 157, 166, 178, 193, 226, 262, 277, 313, 346, 358, 382, 397, 421, 457, 466, 478, 502, 541, 562, 586, 613, 661, 673, 718, 733, 757, 838, 862, 877, 886, 982, 997, 1018, 1093, 1153, 1186, 1201, 1213, 1237, 1282
OFFSET
1,1
COMMENTS
Since tau(k) + tau(k+1) = 6, (tau(k), tau(k+1)) must be (1,5), (2,4), (3,3), (4,2), or (5,1); of these, (1,5) and (5,1) are impossible (tau(m) = 1 only for m=1, but then neither m+1 nor m-1 would have 5 divisors), and (3,3) is also impossible (both k and k+1 would have to be squares of primes), so (tau(k), tau(k+1)) must be either (2,4) or (4,2).
For every prime p, tau(p) = 2. For every semiprime s, tau(s) = 4, with the exception of the squares of primes; for p prime, tau(p^2) = 3, since the divisors of p^2 are 1, p, and p^2.
The only numbers that have exactly 4 divisors but are not semiprimes are the cubes of primes; for prime p, the divisors of p^3 are 1, p, p^2, and p^3.
As a result, this sequence consists of:
(1) the primes p such that (p+1)/2 is prime (A005383), with the exception of p=3 (since p+1 = 4 has 3 divisors, not 4),
(2) semiprimes of the form prime - 1 (A077065), with the exception of the semiprime 4 (since it does not have 4 divisors), and
(3) the special case k = 7, since it is the unique prime p such that p+1 has 4 divisors but is not a semiprime.
For all k > 4, tau(k) + tau(k+1) >= 6; for k = 1..4, tau(k) + tau(k+1) = 3, 4, 5, 5.
LINKS
FORMULA
{ k : tau(k) + tau(k+1) = 6 }.
UNION(A005383 \ {3}, A077065 \ {4}, {7}).
a(n) = A164977(n+1) for n>=4. - Hugo Pfoertner, Jan 08 2022
EXAMPLE
k tau(k) tau(k+1) tau(k) + tau(k+1)
-- ------ -------- -----------------
1 1 2 1 + 2 = 3
2 2 2 2 + 2 = 4
3 2 3 2 + 3 = 5
4 3 2 3 + 2 = 5
5 2 4 2 + 4 = 6 so 5 = a(1)
6 4 2 4 + 2 = 6 so 6 = a(2)
7 2 4 2 + 4 = 6 so 7 = a(3)
8 4 3 4 + 3 = 7
9 3 4 3 + 4 = 7
10 4 2 4 + 2 = 6 so 10 = a(4)
11 2 6 2 + 6 = 8
12 6 2 6 + 2 = 8
13 2 4 2 + 4 = 6 so 13 = a(5)
MATHEMATICA
Select[Range[1300], Plus @@ DivisorSigma[0, # + {0, 1}] == 6 &] (* Amiram Eldar, Jan 08 2022 *)
Position[Total/@Partition[DivisorSigma[0, Range[1300]], 2, 1], 6]//Flatten (* Harvey P. Dale, Sep 03 2022 *)
PROG
(PARI) isok(k) = numdiv(k) + numdiv(k+1) == 6; \\ Michel Marcus, Jan 08 2022
(Python)
from itertools import count, islice
from sympy import divisor_count
def A350093_gen(): # generator of terms
a, b = divisor_count(1), divisor_count(2)
for k in count(1):
if a + b == 6:
yield k
a, b = b, divisor_count(k+2)
A350093_list = list(islice(A350093_gen(), 12)) # Chai Wah Wu, Jan 11 2022
CROSSREFS
Numbers k such that Sum_{j=0..N-1} tau(k+j) = 2*Sum_{k=1..N} tau(k): A000040 (N=1), (this sequence) (N=2), A350675 (N=3), A350686 (N=4), A350699 (N=5), A350769 (N=6), A350773 (N=7), A350854 (N=8).
Sequence in context: A206415 A162317 A162318 * A108910 A308193 A179274
KEYWORD
nonn
AUTHOR
Jon E. Schoenfield, Jan 08 2022
STATUS
approved