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

Author

J. M. Bergot and Robert Israel, Nov 25 2021

Keywords

• nonn
• hard

Examples

			a(3) = 66 is a term because the concatenation of its prime factors is 2311 and 2311 == 1 (mod 66).
		

Links

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

Crossrefs

Cf. A037276, A259047.

Programs

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 *)

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

Extensions

a(28)-a(32) from Martin Ehrenstein, Nov 27 2021