Thank you for your comments William.
Yours is a good idea, I ran it and it is extremely fast in comparison! The first time I encountered this problem was for
$t=2019$, and since my implementation takes a lot of time, specially for the computation of
$k_2$, I could only get to
$t=20$. With your idea of bypassing the need to calculate
$k_1$ and
$k_2$, I could get to
$t=2019$ in under two minutes, thanks!
For the curious,
$k(2019)$ is
2305290672102308631501600234772689901495331656296010173782609457211747
7589012645449972855113002723152850266996581249955310937900637196208377
6015656037190992015601992708978972950218730384956136982230018199914037
1873790474857914308090439690528149039878522176523214171723899120931680
1407795795740287175481595074888464242169114272889833455443538979070710
7954615588986518937269460499977097780936892890288486500115267711198795
7752458754886945509455939738184415246125385661077598946496962529694021
3807498060952467101009725056900507200381670539199960267610955917970647
7348799285007520324333753433137317268595827312669270283710749289135338
3955488775416701421703591611585762587057816965055493289203079657757930
5552781639283256208264236157412839744713189755433693182114657440937393
5151735730964047333464200133046259636266617568495188850439992270539482
5070877950569111185484686258280683360985364614068620085244473016206039
9671853656249766670780202123346133265126530063266776108759382094754654
7716756458855458762042468590857137475911810094709371114227296091704480
6711606499107516880961144418896377536483208496898216007308760220924549
0972474888415801145088826737967372992873259567856816687494684460704424
3173061490180787964985981600770130719077612467850081453816286747633594
4284192328266852157788811299072458199422958623051583550247028984113634
4318289858861052813363039650908790965740078659153247276116386453200334
4836368400516024727032229892621807743635165529464487581545206230448918
1409645082500520804695354024823612306132917021276679585259953063129411
2545858077944461034768509061573193702273821820349478826128151042412868
5956793739278347138201495237865669448730673402603953205685107039200080
8952929651674242338774206988881423049199670391597860260553207517719017
9002932994455476535864508393906288302896457722437946004108041176775861
4329745301223902630302872947877380477704859782754757564959523661085897
7676261776516245911952437104048944340050558070801081399143586315076037
3235657211779132129451895845453642989758910041413588006513
