A289993 -id:A289993 - cp's OEIS frontend

A295185 a(n) is the smallest composite number whose prime divisors (with multiplicity) sum to prime(n); n >= 3.

6, 10, 28, 22, 52, 34, 76, 184, 58, 248, 148, 82, 172, 376, 424, 118, 488, 268, 142, 584, 316, 664, 1335, 388, 202, 412, 214, 436, 3729, 508, 1048, 274, 2919, 298, 1208, 1256, 652, 1336, 1384, 358, 3801, 382, 772, 394, 6501, 7385, 892, 454, 916, 1864, 478, 5061, 2008, 2056, 2104, 538, 2168, 1108, 562, 5943, 9669

Offset: 3

David James Sycamore, Nov 16 2017

nonn

Sequence is undefined for n=1,2 since no composites exist whose prime divisors sum to 2, 3. For n >= 3, a(n) = A288814(prime(n)) = prime(n-k)*B(prime(n) - prime(n-k)) where B=A056240, and k >= 1 is the "type" of prime(n), indicated as prime(n)~k(g1,g2,...,gk) where gi = prime(n-(i-1)) - prime(n-i); 1 <= i <= k. Thus: 5~1(2), 211~2(12,2), 4327~3(30,8,6) etc. The sequence relates to gaps between odd primes, and in particular to the sequence of k prime gaps below prime(n). The even-indexed terms of B are relevant, as are those of subsequences:
  C=A288313, 2,4 plus terms B(n) where n-3 is prime (A298252),
  D=A297150, terms B(n) where n-5 is prime and n-3 is composite (A297925) and
  E=A298615, terms B(n) where both n-3 and n-5 are composite (A298366).
  The above sequences of indices 2m form a partition of the even numbers and the corresponding terms B(2m) form a partition of the even-indexed terms of A056240. The union of D and E is the sequence A292081 = B-C.
Let g(n,t) = prime(n) - prime(n-t), t < n, and h(n,t) = g(n,t) - g(n,1), 1 < t < n. If g1=g(n,1) is a term in A298252 (g1-3 is prime), then B(g1) is a term in C, so k=1. If g1 belongs to A297925 or A298366 then B(g1) is a term in D or E and the value of k depends on subsequent gaps below prime(n), within a range dependent on g1.
Let range R1(g1) = u - g(n,1) where u is the index in B of the greatest term in C such that C(u) < B(g1). Let range R2(g1) = v-g(n,1) where v is the index in B of the greatest term in D such that D(v) <= B(g1). For all n, R2 < R1, and if g1 is a term in D then R2(g1)=0. Examples: R1(12)=2, R2(12)=0, R1(30)=26, R2(30)=6.
k >= 1 is the smallest integer such that B(g(n,k)) <= B(g(n,t)) for all t satisfying g1 <= g(n,t) <= g1 + R1(g1). For g1-3 prime, k=1. If g1-3 is composite, let z be least integer > 1 such that g(n,z)-3 is prime, and let w be least integer >= 1 such that g(n,w)-5 is prime. Then z "complies" if h(n,z) <= R1, and w "complies" if h(n,w) <= R2. If g1-5 is prime then R2=w=0 and only z is relevant.
B(g1) must belong to C,D or E. If in C (g1-3 is prime) then k=1. If in D (g1-5 is prime), k=z if z complies, otherwise k=1. If B(g1) is in E and z complies but not w then k=z, or if w complies but not z then k=w. If B(g1) is in E and z,w both comply then k=z if 3*(g(n,z)-3) < 5*(g(n,w)-5), otherwise k=w. If neither z nor w comply, then k=1.
Conjecture: For all n >= 3, a(n) >= A288189(n).

			5=prime(3), g(3,1)=5-3=2, a term in C; k=1, and a(3)=3*B(5-3)=3*2=6; 5~1(2).
17=prime(7), g(7,1)=17-13=4, a term in C; k=1, a(7)=13*B(17-13)=13*4=52; 17~1(4).
211=prime(47); g(47,1)=12, a term in D, R1=2, R2=0, k=z=2, a(47)=197*b(211-197)=197*33=6501; 211~2(12,2), and 211 is first prime of type k=2.
8923=prime(1109); g(1109,1)=30, a term in E. R1=26, R2=6, z=3 and w=2 both comply  but 3*(g(n,3)-3)=159 > 5*(g(n,2)-5)=155, so k=w=2. Therefore a(1109)=8887*b(8923-8887)=8887*b(36)=8887*155=1377485; 8923~2(30,6).
40343=prime(4232); g(4232,1)=54, a term in E. R1=58, R2=12,z=6 and w=3, both comply, 3*(g(n,z)-3)=309 and 5*(g(n,w)-5)=305 therefore k=w=3 and a(4232) = 40277*b(40343-40277)=40277*b(66)=40277*305=12284485; 40343~3(54,6,6).
81611=prime(7981); g(81611,1)=42, a term in D, R1=22, R2=0; z complies, k=z=6, a(7981)=81547*b(81611-81547)=81546*b(64)=81546*183=14923101; 81611~6(42,6,4,6,2,4) and is the first prime of type k=6.
If p is the greater of twin/cousin primes then p~1(2), p~1(4), respectively.

Giovanni Resta, Table of n, a(n) for n = 3..10000

Cf. A000040, A056240, A288814, A292081, A289993, A288313, A297150, A298615, A298252, A297925, A298366, A288189.

b[n_] := b[n] = Total[Times @@@ FactorInteger[n]];
a[n_] := For[k = 2, True, k++, If[CompositeQ[k], If[b[k] == Prime[n], Return[k]]]];
Table[a[n], {n, 3, 63}] (* Jean-François Alcover, Feb 23 2018 *)

a(n) = { my(p=prime(n)); forcomposite(x=6, , my(f=factor(x)); if(f[, 1]~*f[, 2]==p, return(x))); } \\ Iain Fox, Dec 08 2017

a(n) = A288814(prime(n)) = prime(n-k)*A056240(prime(n) - prime(n-k)) for some k >= 1 and prime(n-k) = gpf(A288814(prime(n)).

a(n) >= A288189(n).

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

Original entry on oeis.org

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

Offset: 1

David James Sycamore, Feb 06 2018

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

			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).

David A. Corneth, PARI program

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

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
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\\ see Corneth link
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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

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A295185 a(n) is the smallest composite number whose prime divisors (with multiplicity) sum to prime(n); n >= 3.

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

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