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

Author

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

Keywords

• nonn
• base

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?

Examples

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

Links

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

Crossrefs

Cf. A037276, A093170.

Programs

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]
				

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

Extensions

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

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