A349705 Numbers k such that the concatenation in increasing order of their prime factors, with multiplicity, is congruent to 1 (mod k).

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

Martin Ehrenstein, Table of n, a(n) for n = 1..39

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

Cf. A037276, A259047.

Sequence in context: A337861 A261373 A248372 * A129288 A083248 A360765

Adjacent sequences: A349702 A349703 A349704 * A349706 A349707 A349708

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