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a(n) is the smallest number m with n divisors d such that d^m mod m = d.
2

%I #44 May 12 2024 11:15:21

%S 1,2,6,42,30,105,910,561,1365,5005,5565,11305,36465,140505,239785,

%T 41041,682465,873145,185185,418285,1683969,2113665,5503785,1242241,

%U 6697405,8549905,31932901,11996985,31260405,30534805,47031061,825265,27265161,32306365,55336645,21662641

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

%e a(0) = 1 with divisors {};

%e a(1) = 2 with divisor {1};

%e a(2) = 6 with divisors {1, 3};

%e a(3) = 42 with divisors {1, 7, 21};

%e a(4) = 30 with divisors {1, 6, 10, 15};

%e a(5) = 105 with divisors {1, 7, 15, 21, 35};

%e a(6) = 910 with divisors {1, 35, 65, 91, 130, 455};

%e a(7) = 561 with divisors {1, 3, 11, 17, 33, 51, 187};

%e a(8) = 1365 with divisors {1, 13, 21, 91, 105, 195, 273, 455};

%e a(9) = 5005 with divisors {1, 11, 55, 65, 77, 143, 385, 715, 1001};

%e a(10) = 5565 with divisors {1, 7, 15, 21, 35, 105, 265, 371, 1113, 1855};

%e a(11) = 11305 with divisors {1, 17, 19, 35, 85, 119, 323, 595, 665, 1615, 2261}.

%t 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 *)

%o (Python)

%o from sympy import divisors

%o from itertools import count, islice

%o def f(n, divs): return sum(1 for d in divs if pow(d, n, n) == d%n)

%o def agen(verbose=False): # generator of terms

%o adict, n = dict(), 0

%o for k in count(1):

%o divs = divisors(k)[1:]

%o if len(divs) < n: continue

%o v = f(k, divs)

%o if v not in adict:

%o adict[v] = k

%o if verbose: print("FOUND", v, k)

%o while n in adict: yield adict[n]; n += 1

%o print(list(islice(agen(), 15))) # _Michael S. Branicky_, Apr 10 2024, updated Apr 17 2024 after _Jon E. Schoenfield_

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

%K nonn

%O 0,2

%A _Juri-Stepan Gerasimov_, Apr 10 2024

%E a(12)-a(25) from _Michael S. Branicky_, Apr 10 2024

%E a(26)-a(35) from _Jon E. Schoenfield_, Apr 10 2024