OFFSET
1,1
COMMENTS
It can be shown that if tau(k) + ... + tau(k+7) = 40, the octuple (tau(k), tau(k+1), tau(k+2), tau(k+3), tau(k+4), tau(k+5), tau(k+6), tau(k+7)) must be one of the following, each of which might plausibly occur infinitely often:
(2, 4, 4, 6, 4, 8, 4, 8), which first occurs at k = 7321991041, 7391371681, 22487905441, ...;
(2, 4, 4, 8, 4, 8, 4, 6), which first occurs at k = 884811061, 1457032501, 11434075381, ...;
(6, 4, 8, 4, 8, 4, 4, 2), which first occurs at k = 3573315892, 8557865812, 28044279892, ...;
(8, 4, 8, 4, 6, 4, 4, 2), which first occurs at k = 10780552, 62198632, 139738178152, ...;
or one of the following, each of which occurs only once:
(4, 4, 8, 2, 8, 2, 6, 6), which occurs only at k = 38;
(4, 8, 2, 8, 2, 6, 6, 4), which occurs only at k = 39;
(2, 8, 2, 6, 6, 4, 2, 10), which occurs only at k = 41;
(4, 6, 2, 8, 4, 8, 4, 4), which occurs only at k = 51;
(4, 8, 4, 4, 2, 12, 2, 4), which occurs only at k = 55;
(2, 6, 4, 8, 2, 12, 2, 4), which occurs only at k = 67;
(4, 2, 12, 4, 4, 4, 8, 2), which occurs only at k = 82.
The terms of the repeatedly occurring patterns form sequence A071370.
Tau(k) + ... + tau(k+7) >= 40 for all sufficiently large k; the only numbers k for which tau(k) + ... + tau(k+7) < 40 are 1..34, 36, 37, 40, 43, 46, 52, and 61.
FORMULA
{ k : Sum_{j=0..7} tau(k+j) = 40 }.
EXAMPLE
The table below lists each term k that is the smallest one having a pattern (tau(k), ..., tau(k+7)) that appears repeatedly for large k. Each such pattern corresponds to one of the 4 possible orders in which the multipliers m=1..8 can appear among 8 consecutive integers of the form m*prime, and thus to a single residue of k modulo 2520; e.g., k=884811061 begins a run of 8 consecutive integers having the form (p, 2*q, 3*r, 8*s, 5*t, 6*u, 7*v, 4*w), where p, q, r, s, t, u, v, and w are distinct primes > 8, and all such runs satisfy k == 1261 (mod 2520).
.
. # divisors of factorization of k+j as
k+j for j = m*(prime > 8) for j =
n a(n)=k 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 k mod 2520
- ---------- - - - - - - - - -- -- -- -- -- -- -- -- ----------
8 10780552 8 4 8 4 6 4 4 2 8p 7q 6r 5s 4t 3u 2v w 2512
10 884811061 2 4 4 8 4 8 4 6 p 2q 3r 8s 5t 6u 7v 4w 1261
12 3573315892 6 4 8 4 8 4 4 2 4p 7q 6r 5s 8t 3u 2v w 1252
13 7321991041 2 4 4 6 4 8 4 8 p 2q 3r 4s 5t 6u 7v 8w 1
MATHEMATICA
Position[Plus @@@ Partition[Array[DivisorSigma[0, #] &, 100], 8, 1], 40] // Flatten (* Amiram Eldar, Jan 19 2022 *)
PROG
(Python) from sympy import divisor_count as tau
taulist = [1, 2, 2, 3, 2, 4, 2, 4]
for k in range(1, 10000000):
if sum(taulist) == 40: print(k, end=", ")
taulist.append(tau(k+8))
del taulist[0] # Karl-Heinz Hofmann, Jan 21 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Jon E. Schoenfield, Jan 19 2022
STATUS
approved