A299760 - OEIS

%I #16 Apr 26 2018 11:09:20

%S 5,7,11,13,17,19,23,29,31,41,43,47,59,61,71,73,79,83,89,101,103,107,

%T 109,113,131,139,151,167,173,179,181,193,197,199,227,229,233,239,241,

%U 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

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

%C 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).

%e 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).

%e 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.

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

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

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

%o isok(p) = isprime(p) && (ap288189(p)==ap295185(p)); \\ _Michel Marcus_, Apr 14 2018

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

%K nonn

%O 1,1

%A _David James Sycamore_, Feb 18 2018