A351975 Numbers k such that A037276(k) == -1 (mod k).

1, 6, 14, 18, 48, 124, 134, 284, 3135, 4221, 9594, 16468, 34825, 557096, 711676, 746464, 1333334, 2676977, 6514063, 11280468, 16081252, 35401658, 53879547, 133333334, 198485452, 223856659, 1333333334, 2514095219, 2956260256, 3100811124, 10912946218, 19780160858

Offset

1,2

Comments

Numbers k such that the concatenation of prime factors of k is 1 less than a multiple of k.

Contains 2*m for m in A093170.

Terms k where k-1 is prime include 6, 14, 18, 48 and 284. Are there others?

Links

Martin Ehrenstein, Table of n, a(n) for n = 1..38

Example

a(4) = 48 is a term because 48=2*2*2*2*3 and 22223 == -1 (mod 48).

Maple

tcat:= proc(x, y) x*10^(1+ilog10(y))+y end proc:
filter:= proc(n) local F, t, i;
F:= map(t -> t[1]$t[2], sort(ifactors(n)[2], (a, b)->a[1]<b[1]));
t:= F[1];
for i from 2 to nops(F) do
  t:= tcat(t, F[i])
od;
  t mod n = n-1
end proc:
filter(1):= true:
select(filter, [$1..10^8]);

Prog

(Python)
from sympy import factorint
def A037276(n):
    if n == 1: return 1
    return int("".join(str(p)*e for p, e in sorted(factorint(n).items())))
def afind(limit, startk=1):
    for k in range(startk, limit+1):
        if (A037276(k) + 1)%k == 0:
            print(k, end=", ")
afind(10**6) # Michael S. Branicky, Feb 27 2022
# adapted and corrected by Martin Ehrenstein, Mar 06 2022
(Python)
from itertools import count, islice
from sympy import factorint
def A351975_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 == k-1:
            yield k
A351975_list = list(islice(A351975_gen(), 10)) # Chai Wah Wu, Feb 28 2022

Crossrefs

Cf. A037276, A093170.

Sequence in context: A053474 A072204 A037177 * A255217 A048747 A175678

Adjacent sequences: A351972 A351973 A351974 * A351976 A351977 A351978

Keyword

nonn,base

Author

J. M. Bergot and Robert Israel, Feb 26 2022

Extensions

a(24)-a(25) from Michael S. Branicky, Feb 27 2022

Prepended 1 and more terms from Martin Ehrenstein, Feb 28 2022

Status

approved