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%I #18 Dec 31 2023 16:49:46
%S 0,0,1,0,1,0,1,0,1,2,1,0,1,2,7,0,1,0,1,2,9,2,1,0,1,2,1,2,1,6,1,0,13,2,
%T 11,0,1,2,15,2,1,6,1,2,7,2,1,0,1,2,19,2,1,0,15,2,21,2,1,6,1,2,9,0,17,
%U 6,1,2,25,22,1,0,1,2,7,2,17,6,1,2,1,2,1,6,21,2,31,2,1,6,19,2,33,2,23,0,1,2,13
%N a(n) = (Product_{p is a prime factor of n} p) mod (Product_{p is a prime factor of n} p-1).
%C It can be shown that the unsolved problem of finding a composite solution n of the congruence n-1 = 1 mod phi(n) is equivalent to finding a squarefree composite number n such that a(n) = 1.
%H Antti Karttunen, <a href="/A089631/b089631.txt">Table of n, a(n) for n = 1..20000</a>
%H C. Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_248.htm">Puzzle 248</a>, The Prime Puzzles and Problems Connection.
%F a(n) = A007947(n) mod A173557(n). - _Antti Karttunen_, Feb 25 2018
%e Let n = 10; then Product_{p is a prime factor of 10} p = 2*5 = 10, Product_{p is a prime factor of 10} p-1 = (2-1) * (5-1) = 1*4 = 4, so a(10) = 10 mod 4 = 2.
%t p[n_] := Transpose[FactorInteger[n]][[1]]; t = Table[p[i], {i, 2, 100}]; r = {}; s = {}; For[i = 1, i <= 99, i++, r = Append[r, Product[t[[i]][[j]], {j, 1, Length[t[[i]]]}]]; s = Append[s, Product[t[[i]][[j]] - 1, {j, 1, Length[t[[i]]]}]]]; Mod[r, s]
%o (PARI)
%o A007947(n) = factorback(factorint(n)[, 1]); \\ This function from _Andrew Lelechenko_, May 09 2014
%o A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ This function from _Michel Marcus_, Oct 31 2017
%o A089631(n) = (A007947(n)%A173557(n)); \\ _Antti Karttunen_, Feb 25 2018
%Y Cf. A007947, A173557.
%K nonn
%O 1,10
%A _Joseph L. Pe_, Jan 04 2004
%E Term a(1) = 0 prepended by _Antti Karttunen_, Feb 25 2018
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