A299760 - OEIS

A299760

Primes p with index k >= 3 such that A288189(k) = A295185(k).

5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 59, 61, 71, 73, 79, 83, 89, 101, 103, 107, 109, 113, 131, 139, 151, 167, 173, 179, 181, 193, 197, 199, 227, 229, 233, 239, 241, 269, 271, 281, 283, 311, 313, 317, 349, 353, 359, 379, 383, 389, 401, 421, 433, 439, 443, 449, 461, 463, 467, 491, 503, 509, 523, 569, 571, 599, 601, 607

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OFFSET

1,1

COMMENTS

Let A,B,X respectively, represent A288189, A295185, A056240. For prime p with index k >= 3, A(p) = X(t)(rp-t) for some multiple r >= 1 of p, and some integer t such that rp-t is prime; then sopfr(A(p)) = rp. Similarly B(p) = X(g)(p-g) where g = p-q for some prime q < p, where q = p-g is the greatest prime divisor of A295185(p); then sopfr(B(p)) = p. A(p) < B(p) if r and t exist such that (rp-t) is prime, with X(t)(rp-t) < X(g)(p-g), otherwise r = 1, t = g and A(p) = B(p). So A(p) <= B(p) and this sequence lists primes p for which this equality holds. All primes for which g = 2 or 4 are in this sequence, since then both 2(p-2), 4(p-4) are < 3(2p-3), the minimum possible value for any r > 1, t of X(t)(rp-t). Equivocal results are found for g >= 6, though in the great majority of cases (up to k=400), g > 6 ==> A(p) < B(p).

LINKS

Table of n, a(n) for n=1..70.

EXAMPLE

p=29 is included because 2p-3 and 3p-2 are both composite so A(29) = 8(p-6) = 8(p-6) = 8*23 = 184 = B(29).

p=37 is not included since A(37) = 3(2p-3) = 213 whereas B(37) = X(6)(37-6) = 8*31 = 248, so A(37) < B(37). In both examples g=6.

PROG

(PARI) sopfr(k) = my(f=factor(k)); sum(j=1, #f~, f[j, 1]*f[j, 2]);

ap288189(p) = forcomposite(c=p, , if (!(sopfr(c) % p), return(c)));

ap295185(p) = forcomposite(c=p, , if (sopfr(c) == p, return(c)));

isok(p) = isprime(p) && (ap288189(p)==ap295185(p)); \\ Michel Marcus, Apr 14 2018

CROSSREFS

Cf. A001414, A056240, A288189, A295185, A302720.