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
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
Programs
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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
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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
Comments