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Primes p such that there are no other solutions to A023900(x) = A023900(p) than a power of p.
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%I #18 Oct 06 2021 15:18:54

%S 2,3,5,7,11,17,19,23,29,31,43,47,53,59,67,71,79,83,101,103,107,127,

%T 131,137,139,149,151,163,167,173,179,191,197,199,211,223,227,229,239,

%U 251,257,263,269,271,283,293,307,311,317,331,347,359,367,373,379,383,389

%N Primes p such that there are no other solutions to A023900(x) = A023900(p) than a power of p.

%C In the definition, A023900(p) = 1-p. One has sign(A023900(n)) = (-1)^A001221(n), so a different solution x can only exist if x has at least 3 distinct prime factors. The smallest number of the form p*q*r such that (p-1)*(q-1)*(r-1) = P-1 for primes p, q, r, P is 2*3*7 = 42, eliminating P = 13 = A301591(1) from this sequence. This is the case whenever (P+1)/2 = p > 3 is a prime (in A005382), whence P-1 = (2-1)*(3-1)*(p-1), which eliminates all P > 5 in A005383 from this sequence. - _M. F. Hasler_, Aug 14 2021

%H M. F. Hasler, <a href="/A301590/b301590.txt">Table of n, a(n) for n = 1..10000</a>, Sep 01 2021

%e 2 is a term because there are no other solutions to A023900(x) = A023900(2) = -1 than other powers of 2.

%e 13 is not a term because A023900(42) = -12 = A023900(13). Similarly, no P > 5 in A005383 is a term because A023900(P) = 1-P = (1-2)*(1-3)*(1-p) = A023900(2*3*p) with p = (P+1)/2. - _M. F. Hasler_, Aug 14 2021

%o (PARI) f(n) = sumdivmult(n, d, d*moebius(d)); /* A023900 */

%o isok(p, vp) = {for (k=p+1, p^2-1, if (f(k) == vp, return (0)); ); return (1); }

%o lista(nn) = {forprime(p=2, nn, vp = f(p); if (isok(p, vp), print1(p, ", ")); ); }

%o (PARI) select( {is_A301590(p)=!forcomposite(k=p+1, p^2-1, A023900(k)!=1-p|| return)&& isprime(p)}, primes([1,399])) \\ _M. F. Hasler_, Aug 14 2021

%Y Cf. A023900, A301374.

%Y Complement (within the primes) of A301591, which has A005383 \ {3, 5} as a subsequence. Appears to have A079151 \ {13} as subsequence.

%K nonn

%O 1,1

%A _Michel Marcus_, Mar 24 2018