A302720 Primes with index k >= 3 such that A288189(prime(k)) < A295185(prime(k)).

37, 53, 67, 97, 127, 137, 149, 157, 163, 191, 211, 223, 251, 257, 263, 277, 293, 307, 331, 337, 347, 367, 373, 397, 409, 419, 431, 457, 479, 487, 499, 521, 541, 547, 557, 563, 577, 587, 593, 613, 631, 641, 653, 673, 691, 701, 709, 719, 727, 751, 757, 769, 787, 797, 809, 821, 839, 853, 877, 907, 919, 929, 937, 953, 967, 977

Offset: 1

Author

David James Sycamore, Apr 12 2018

Keywords

• nonn

Comments

Let A,B,X represent A288189, A295185, A056240 respectively. A(p) is defined for every prime, B(p) is defined for primes >= 5. For a prime p with index k >= 3, A(p) = X(t)(rp-t) for some multiple r of p, and some integer t such that rp-t is prime. Then Sopfr(A(p)) = Sopfr(X(t))+(rp-t) = t+rp-t = rp. B(p) = X(g)(p-g) where g = p-q for some prime q = p-g < p. q is the greatest prime divisor of A295185(p), so 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). A(p) is computed from the list of possible values in the list of inequalities: 3(2p-3) < 2(3p-2) < 5(2p-5) < 2(5p-2) < ... < X(g)(p-g), selecting the first (smallest) value of (rp-t) which is prime. If such a term exists and is < X(p)(p-g), then A(p) < B(p) and p is in this sequence. Otherwise A(p) = B(p) = X(p)(p-g) and p is in A299760.

Examples

			k=12, prime(12)=37, A288189(37) = 213 < 248 = A295185(37). 37 is the smallest prime with this property, so a(1)=37.
		

Crossrefs

Cf. A288189, A295185, A299760, A299760.

Programs

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)));
isokp(p) = (ap288189(p) < ap295185(p));
lista(nn) = forprime(p=5, nn, if (isokp(p), print1(p, ", "))); \\ Michel Marcus, May 13 2018

Extensions

a(53) corrected by Georg Fischer, Mar 20 2022