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

%I A351975 #28 Mar 07 2022 07:53:51
%S A351975 1,6,14,18,48,124,134,284,3135,4221,9594,16468,34825,557096,711676,
%T A351975 746464,1333334,2676977,6514063,11280468,16081252,35401658,53879547,
%U A351975 133333334,198485452,223856659,1333333334,2514095219,2956260256,3100811124,10912946218,19780160858
%N A351975 Numbers k such that A037276(k) == -1 (mod k).
%C A351975 Numbers k such that the concatenation of prime factors of k is 1 less than a multiple of k.
%C A351975 Contains 2*m for m in A093170.
%C A351975 Terms k where k-1 is prime include 6, 14, 18, 48 and 284. Are there others?
%H A351975 Martin Ehrenstein, <a href="/A351975/b351975.txt">Table of n, a(n) for n = 1..38</a>
%e A351975 a(4) = 48 is a term because 48=2*2*2*2*3 and 22223 == -1 (mod 48).
%p A351975 tcat:= proc(x,y) x*10^(1+ilog10(y))+y end proc:
%p A351975 filter:= proc(n) local F,t,i;
%p A351975 F:= map(t -> t[1]$t[2], sort(ifactors(n)[2],(a,b)->a[1]<b[1]));
%p A351975 t:= F[1];
%p A351975 for i from 2 to nops(F) do
%p A351975   t:= tcat(t,F[i])
%p A351975 od;
%p A351975   t mod n = n-1
%p A351975 end proc:
%p A351975 filter(1):= true:
%p A351975 select(filter, [$1..10^8]);
%o A351975 (Python)
%o A351975 from sympy import factorint
%o A351975 def A037276(n):
%o A351975     if n == 1: return 1
%o A351975     return int("".join(str(p)*e for p, e in sorted(factorint(n).items())))
%o A351975 def afind(limit, startk=1):
%o A351975     for k in range(startk, limit+1):
%o A351975         if (A037276(k) + 1)%k == 0:
%o A351975             print(k, end=", ")
%o A351975 afind(10**6) # _Michael S. Branicky_, Feb 27 2022
%o A351975 # adapted and corrected by _Martin Ehrenstein_, Mar 06 2022
%o A351975 (Python)
%o A351975 from itertools import count, islice
%o A351975 from sympy import factorint
%o A351975 def A351975_gen(startvalue=1): # generator of terms >= startvalue
%o A351975     for k in count(max(startvalue,1)):
%o A351975         c = 0
%o A351975         for d in sorted(factorint(k,multiple=True)):
%o A351975             c = (c*10**len(str(d)) + d) % k
%o A351975         if c == k-1:
%o A351975             yield k
%o A351975 A351975_list = list(islice(A351975_gen(),10)) # _Chai Wah Wu_, Feb 28 2022
%Y A351975 Cf. A037276, A093170.
%K A351975 nonn,base
%O A351975 1,2
%A A351975 _J. M. Bergot_ and _Robert Israel_, Feb 26 2022
%E A351975 a(24)-a(25) from _Michael S. Branicky_, Feb 27 2022
%E A351975 Prepended 1 and more terms from _Martin Ehrenstein_, Feb 28 2022