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A293652 a(n) is the smallest prime number whose a056240-type is n (see Comments).

%I A293652 #80 Jan 18 2019 04:40:05
%S A293652 5,211,4327,4547,25523,81611,966109,1654111,3851587,1895479,66407189,
%T A293652 134965049,129312889,425845151,35914507,504365461,2400397969,
%U A293652 8490141637,8429770031,20416021309,23555107819,23912414437
%N A293652 a(n) is the smallest prime number whose a056240-type is n (see Comments).
%C A293652 For a prime p >= 5 whose prime-index is m, the a056240-type of p is defined to be the unique integer k such that A288814(p) = prime(m-k)*A056240(prime(m)-prime(m-k)).
%C A293652 In other words, k is such that prime(n-k) is the greatest prime divisor of the smallest composite number whose sum of prime factors (taken with multiplicity) is prime(n).
%C A293652 The sequence lists the smallest prime of each successive a056240-type.
%C A293652 In the Examples section, the a056240-type k (=a(k)) of a prime p = prime(m) is indicated by p ~ k(g1,g2,...,gk) where gi = prime(m - i + 1) - prime(m - i). See also A295185.
%C A293652 For the values of the a056240-types of the primes 2, 3, 5, 7, ... see A299912. - _N. J. A. Sloane_, Mar 10 2018
%C A293652 a(20), a(21) > 14 * 10^9. Conjecture: a(k) > 14 * 10^9 for k > 22. - _David A. Corneth_, Mar 25 2018
%C A293652 a(20), a(21) computed on the basis of the above conjecture. Note that A321983 records the smallest composite number whose sum of prime divisors (with repetition) is a(n). - _David James Sycamore_, Nov 30 2018
%C A293652 a(23)..a(25) > 45.8 * 10^9. - _David A. Corneth_, Dec 02 2018
%H A293652 David A. Corneth, <a href="/A293652/a293652.gp.txt">PARI program</a>
%e A293652 a(1) = 5 since 6 = 3 * 2, the smallest composite number whose prime divisors add to 5, is a multiple of 3, the greatest prime < 5, so k=1; 5 ~ 1(2).
%e A293652 a(2) = 211 since 6501 = 3 * 11 * 197, the smallest composite whose prime divisors add to 211, and 197 < 199 < 211 is the second prime below 211, so k=2, and 211 ~ 2(12,2), and since no smaller prime has this property, a(2)=211.
%e A293652 a(3) = 4327 since 526809 = 3 * 41 * 4283, the smallest composite whose prime divisors add to 4327, and 4283 < 4289 < 4297 < 4327 is the third prime below 4327, so k=3, 4327 ~ 3(30,8,6) and since no smaller prime has this property, a(3)=4327. Likewise,
%e A293652 a(4) = 4547 ~ 4(24, 4, 2, 4),
%e A293652 a(5) = 25523 ~ 5(52, 2, 6, 6, 4),
%e A293652 a(6) = 81611 ~ 6(42, 6, 4, 6, 2, 4),
%e A293652 a(7) = 966109 ~ 7(68, 12, 16, 2, 22, 6, 14),
%e A293652 a(8) = 1654111 ~ 8(54, 14, 4, 6, 2, 4, 6, 2),
%e A293652 a(9) = 3851587 ~ 9(128, 16, 12, 2, 6, 10, 14, 10, 2),
%e A293652 a(10) = 1895479 ~ 10(120, 2, 6, 30, 4, 30, 14, 10, 2, 12),
%e A293652 a(11) = 66407189 ~ 11(120, 6, 6, 16, 14, 6, 4, 8, 10, 2, 4),
%e A293652 a(12) = 134965049 ~ 12(138, 10, 2, 22, 18, 20, 6, 12, 18, 16, 8, 10),
%e A293652 a(13) = 129312889 ~ 13(98, 60, 22, 18, 8, 4, 18, 12, 38, 24, 6, 4, 8),
%e A293652 a(14) = 425845151 ~ 14(148, 2, 42, 16, 50, 24, 12, 6, 4, 20, 6, 48, 10, 12),
%e A293652 a(15) = 35914859 ~ 15(126, 82, 8, 4, 18, 12, 8, 4, 14, 6, 16, 8, 6, 30, 10),
%e A293652 a(16) = 504365461 ~ 16(122, 42, 10, 14, 36, 4, 6, 6, 12, 48, 2, 6, 10, 20, 6, 6),
%e A293652 a(17) = 2400397969 ~ 17(122, 58, 8, 4, 18, 36, 2, 4, 6, 32, 10, 2, 16,12,18,32,12),
%e A293652 a(18) = 8490141637 ~ 18(126, 2, 82, 8, 52, 20, 34, 2, 10, 24, 8, 6,34,2,6,28,24,2),
%e A293652 a(19) = 8429770031 ~ 19(148, 26, 16, 18, 12, 2, 18, 18, 10,20,4,2,6,18,6,4,2,18,4),
%e A293652 a(20) = 20416021309 ~ 20(122, 4, 2, 64, 20, 40, 6, 12, 12, 20, 10, 6, 8, 10, 30, 2, 10, 38, 22, 140,
%e A293652 a(21) = 23555107819 ~ 21(192, 20, 156, 30, 18, 10, 2, 12, 58, 12, 12, 26, 28, 32, 4, 6, 12, 2, 6, 22, 2),
%e A293652 a(22) = 23912414437 ~ 22(344, 4, 12, 14, 40, 2, 4, 18, 2, 36, 10, 12, 2, 10, 26, 10, 24, 14, 40, 30, 14, 12).
%o A293652 (PARI) isok(k, n) = my(f=factor(k)); sum(j=1, #f~, f[j, 1]*f[j, 2]) == n;
%o A293652 snumbr(n) = my(k=2); while(!isok(k, n), k++); k; /* A056240 */
%o A293652 scompo(n) = forcomposite(k=4, , if (isok(k, n), return(k))); /* A288814 */
%o A293652 a(n) = {forprime(p=5,,ip = primepi(p); if (ip > n, x = scompo(p); fmax = vecmax(factor(x)[,1]); ifmax = primepi(fmax); if (ip - ifmax == n, y = fmax*snumbr(p - fmax;); if (y == x, return (p);););););} \\ _Michel Marcus_, Feb 17 2018
%o A293652 (PARI) \\ see Corneth link
%Y A293652 Cf. A056240, A288814, A289993, A295185, A299912, A300334, A300359, A321983.
%K A293652 nonn,more,hard
%O A293652 1,1
%A A293652 _David James Sycamore_, Feb 06 2018
%E A293652 a(7)-a(10) from _Michel Marcus_, Feb 23 2018
%E A293652 Name changed by _N. J. A. Sloane_, Mar 10 2018
%E A293652 a(11)-a(19) from _David A. Corneth_, Mar 24 2018, Mar 25 2018
%E A293652 a(20)-a(21) from _David James Sycamore_, Nov 30 2018
%E A293652 a(22) from _David A. Corneth_, Dec 02 2018