A371513 a(n) is the smallest number m with n divisors d such that d^m mod m = d.

1, 2, 6, 42, 30, 105, 910, 561, 1365, 5005, 5565, 11305, 36465, 140505, 239785, 41041, 682465, 873145, 185185, 418285, 1683969, 2113665, 5503785, 1242241, 6697405, 8549905, 31932901, 11996985, 31260405, 30534805, 47031061, 825265, 27265161, 32306365, 55336645, 21662641

Offset: 0

Author

Juri-Stepan Gerasimov, Apr 10 2024

Keywords

• nonn

Examples

			a(0) = 1 with divisors {};
a(1) = 2 with divisor {1};
a(2) = 6 with divisors {1, 3};
a(3) = 42 with divisors {1, 7, 21};
a(4) = 30 with divisors {1, 6, 10, 15};
a(5) = 105 with divisors {1, 7, 15, 21, 35};
a(6) = 910 with divisors {1, 35, 65, 91, 130, 455};
a(7) = 561 with divisors {1, 3, 11, 17, 33, 51, 187};
a(8) = 1365 with divisors {1, 13, 21, 91, 105, 195, 273, 455};
a(9) = 5005 with divisors {1, 11, 55, 65, 77, 143, 385, 715, 1001};
a(10) = 5565 with divisors {1, 7, 15, 21, 35, 105, 265, 371, 1113, 1855};
a(11) = 11305 with divisors {1, 17, 19, 35, 85, 119, 323, 595, 665, 1615, 2261}.
		

Crossrefs

Cf. A182816, A272538, A279024, A371883, A371884.

Programs

Mathematica

f[n_] := DivisorSum[n, 1 &, PowerMod[#, n, n] == # &]; seq[max_] := Module[{t = Table[0, {max}], c = 0, n = 1, i}, While[c < max, i = f[n] + 1; If[i <= max && t[[i]] == 0, c++; t[[i]] = n]; n++]; t]; seq[18] (* Amiram Eldar, Apr 11 2024 *)

Python

from sympy import divisors
from itertools import count, islice
def f(n, divs): return sum(1 for d in divs if pow(d, n, n) == d%n)
def agen(verbose=False): # generator of terms
    adict, n = dict(), 0
    for k in count(1):
        divs = divisors(k)[1:]
        if len(divs) < n: continue
        v = f(k, divs)
        if v not in adict:
            adict[v] = k
            if verbose: print("FOUND", v, k)
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 15))) # Michael S. Branicky, Apr 10 2024, updated Apr 17 2024 after Jon E. Schoenfield

Extensions

a(12)-a(25) from Michael S. Branicky, Apr 10 2024

a(26)-a(35) from Jon E. Schoenfield, Apr 10 2024