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A349705
Numbers k such that the concatenation in increasing order of their prime factors, with multiplicity, is congruent to 1 (mod k).
1
36, 39, 66, 1435, 5714, 6410, 13861, 22564, 27346, 33137, 45542, 79260, 171860, 268218, 442068, 486127, 675423, 2287527, 3710027, 9610766, 14318290, 26293568, 29361702, 49703324, 227358366, 433100023, 442960845, 479174118, 1221238938, 1243718114, 4053362596, 8620689655
OFFSET
1,1
LINKS
EXAMPLE
a(3) = 66 is a term because the concatenation of its prime factors is 2311 and 2311 == 1 (mod 66).
MAPLE
filter:= proc(n) local L, t;
lcat(map(t -> t[1]$t[2], sort( ifactors(n)[2], (a, b) -> a[1] < b[1]))) mod n = 1;
end proc:
select(filter, [$1..10^7]);
MATHEMATICA
upto=10^5; a={}; Do[If[Mod[FromDigits[Flatten[Map[IntegerDigits[ConstantArray[First[#], Last[#]]]&, FactorInteger[k]]]], k]==1, AppendTo[a, k]], {k, upto}]; a (* Paolo Xausa, Nov 26 2021 *)
PROG
(Python)
from sympy import factorint
def ok(k): return int("".join(map(str, factorint(k, multiple=True))))%k == 1
print([k for k in range(2, 10**5) if ok(k)]) # Michael S. Branicky, Nov 26 2021
(Python)
from itertools import count, islice
from sympy import factorint
def A349705_gen(startvalue=1): # generator of terms >= startvalue
for k in count(max(startvalue, 1)):
c = 0
for d in sorted(factorint(k, multiple=True)):
c = (c*10**len(str(d)) + d) % k
if c == 1:
yield k
A349705_list = list(islice(A349705_gen(), 10)) # Chai Wah Wu, Feb 28 2022
CROSSREFS
Sequence in context: A337861 A261373 A248372 * A129288 A083248 A360765
KEYWORD
nonn,hard
AUTHOR
J. M. Bergot and Robert Israel, Nov 25 2021
EXTENSIONS
a(28)-a(32) from Martin Ehrenstein, Nov 27 2021
STATUS
approved