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A301590
Primes p such that there are no other solutions to A023900(x) = A023900(p) than a power of p.
4
2, 3, 5, 7, 11, 17, 19, 23, 29, 31, 43, 47, 53, 59, 67, 71, 79, 83, 101, 103, 107, 127, 131, 137, 139, 149, 151, 163, 167, 173, 179, 191, 197, 199, 211, 223, 227, 229, 239, 251, 257, 263, 269, 271, 283, 293, 307, 311, 317, 331, 347, 359, 367, 373, 379, 383, 389
OFFSET
1,1
COMMENTS
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
LINKS
M. F. Hasler, Table of n, a(n) for n = 1..10000, Sep 01 2021
EXAMPLE
2 is a term because there are no other solutions to A023900(x) = A023900(2) = -1 than other powers of 2.
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
PROG
(PARI) f(n) = sumdivmult(n, d, d*moebius(d)); /* A023900 */
isok(p, vp) = {for (k=p+1, p^2-1, if (f(k) == vp, return (0)); ); return (1); }
lista(nn) = {forprime(p=2, nn, vp = f(p); if (isok(p, vp), print1(p, ", ")); ); }
(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
CROSSREFS
Complement (within the primes) of A301591, which has A005383 \ {3, 5} as a subsequence. Appears to have A079151 \ {13} as subsequence.
Sequence in context: A042988 A167135 A129990 * A293194 A162566 A040085
KEYWORD
nonn
AUTHOR
Michel Marcus, Mar 24 2018
STATUS
approved