A297150 -id:A297150 - cp's OEIS frontend

A298615 Let b(k) be A056240(k); this sequence lists numbers b(2n) such that there is at least one m > n for which b(2m) < b(2n) belongs to A297150.

161, 217, 329, 371, 427, 511, 581, 623, 1246, 791, 1417, 1243, 1469, 2071, 917, 973, 1507, 1529, 1057, 1099, 1169, 1211, 1267, 1969, 1991, 1393, 2167, 2189, 2587, 1477, 2954, 2321, 2743, 1631, 1687, 2629, 2651, 1757, 1799, 1841, 1897, 1981, 3091, 3113, 2051, 4102

Offset: 1

David James Sycamore, Jan 26 2018

nonn

For even number n, if n-5 and n-3 are both composite then A056240(n) belongs to this sequence. The union of terms in this sequence together with those in A288313 and A297150 combine to make A056240(2n), for n >= 3. A288313(n) = A056240(A298252(n)), A297150(n) = A056240(A297925(n)), and the terms of this sequence correspond to A056240(A298366). Distinct sequences A298252, A297925 and A298366 form a partition of the nonnegative even integers (A005843) >= 6. These partitions holds because any even integer n >= 6 is such that, either n-3 is prime (A298252), or n-5 is prime but n-3 is composite (A297925), or both n-5 and n-3 are composite (A298366).

			n=1, a(1) = A056240(A298366(1)) = A056240(30) = 161;
n=24, a(24) = A056240(A298366(24)) = A056240(190) = 1969.

Michel Marcus, Table of n, a(n) for n = 1..2000

Cf. A056240, A288313, A297150, A298252, A298366, A005843.

A056240(n, p=n-1, m=oo)=if(n<6 || isprime(n), n, n==6, 8, until(p<3 || (n-p=precprime(p-1))*p >= m=min(m, A056240(n-p)*p), ); m);
is_A298366(n)= !isprime(n-5) && !isprime(n-3) && !(n%2) && (n>5);
lista(nn) = {for (n=0, nn, if (is_A298366(n), print1(A056240(n), ", ")););} \\ Michel Marcus, Apr 03 2020

a(n) = A056240(A298366(n)).

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

Original entry on oeis.org

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).

cp's OEIS Frontend

A298615 Let b(k) be A056240(k); this sequence lists numbers b(2n) such that there is at least one m > n for which b(2m) < b(2n) belongs to A297150.

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