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

5, 211, 4327, 4547, 25523, 81611, 966109, 1654111, 3851587, 1895479, 66407189, 134965049, 129312889, 425845151, 35914507, 504365461, 2400397969, 8490141637, 8429770031, 20416021309, 23555107819, 23912414437

Offset: 1

Author

David James Sycamore, Feb 06 2018

Keywords

• nonn
• more
• hard

Comments

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)).

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).

The sequence lists the smallest prime of each successive a056240-type.

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.

For the values of the a056240-types of the primes 2, 3, 5, 7, ... see A299912. - N. J. A. Sloane, Mar 10 2018

a(20), a(21) > 14 * 10^9. Conjecture: a(k) > 14 * 10^9 for k > 22. - David A. Corneth, Mar 25 2018

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

a(23)..a(25) > 45.8 * 10^9. - David A. Corneth, Dec 02 2018

Examples

			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).
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.
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,
a(4) = 4547 ~ 4(24, 4, 2, 4),
a(5) = 25523 ~ 5(52, 2, 6, 6, 4),
a(6) = 81611 ~ 6(42, 6, 4, 6, 2, 4),
a(7) = 966109 ~ 7(68, 12, 16, 2, 22, 6, 14),
a(8) = 1654111 ~ 8(54, 14, 4, 6, 2, 4, 6, 2),
a(9) = 3851587 ~ 9(128, 16, 12, 2, 6, 10, 14, 10, 2),
a(10) = 1895479 ~ 10(120, 2, 6, 30, 4, 30, 14, 10, 2, 12),
a(11) = 66407189 ~ 11(120, 6, 6, 16, 14, 6, 4, 8, 10, 2, 4),
a(12) = 134965049 ~ 12(138, 10, 2, 22, 18, 20, 6, 12, 18, 16, 8, 10),
a(13) = 129312889 ~ 13(98, 60, 22, 18, 8, 4, 18, 12, 38, 24, 6, 4, 8),
a(14) = 425845151 ~ 14(148, 2, 42, 16, 50, 24, 12, 6, 4, 20, 6, 48, 10, 12),
a(15) = 35914859 ~ 15(126, 82, 8, 4, 18, 12, 8, 4, 14, 6, 16, 8, 6, 30, 10),
a(16) = 504365461 ~ 16(122, 42, 10, 14, 36, 4, 6, 6, 12, 48, 2, 6, 10, 20, 6, 6),
a(17) = 2400397969 ~ 17(122, 58, 8, 4, 18, 36, 2, 4, 6, 32, 10, 2, 16,12,18,32,12),
a(18) = 8490141637 ~ 18(126, 2, 82, 8, 52, 20, 34, 2, 10, 24, 8, 6,34,2,6,28,24,2),
a(19) = 8429770031 ~ 19(148, 26, 16, 18, 12, 2, 18, 18, 10,20,4,2,6,18,6,4,2,18,4),
a(20) = 20416021309 ~ 20(122, 4, 2, 64, 20, 40, 6, 12, 12, 20, 10, 6, 8, 10, 30, 2, 10, 38, 22, 140,
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),
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).
		

Links

• David A. Corneth, PARI program

Crossrefs

Cf. A056240, A288814, A289993, A295185, A299912, A300334, A300359, A321983.

Programs

PARI

isok(k, n) = my(f=factor(k)); sum(j=1, #f~, f[j, 1]*f[j, 2]) == n;
snumbr(n) = my(k=2); while(!isok(k, n), k++); k; /* A056240 */
scompo(n) = forcomposite(k=4, , if (isok(k, n), return(k))); /* A288814 */
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

PARI

\\ see Corneth link

Extensions

a(7)-a(10) from Michel Marcus, Feb 23 2018

Name changed by N. J. A. Sloane, Mar 10 2018

a(11)-a(19) from David A. Corneth, Mar 24 2018, Mar 25 2018

a(20)-a(21) from David James Sycamore, Nov 30 2018

a(22) from David A. Corneth, Dec 02 2018