

A289993


Primes p having the property that gpf(A288814(p)) < q, where q is greatest prime < p.


3



211, 541, 631, 673, 1693, 1801, 2879, 3181, 3271, 3299, 3343, 3571, 3943, 4177, 4327, 4441, 4547, 4561, 4751, 4783, 4813, 4861, 5147, 5261, 5381, 5431, 5501, 5779, 6029, 6197, 6421, 6469, 6521, 6599, 6673, 6883, 6947, 7103, 7283, 7321, 7369, 7477, 7573, 7603, 7789, 7901, 7963, 7993, 8419, 8443
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OFFSET

1,1


COMMENTS

For prime p in this sequence, b(p) = r*b(pr) where b(m) = A288814(m), and r = gpf(b(p)) is some prime < q. We can say that prime p_n (n>2)is of type k (>=1) if gpf(b(p_n)) = p_n  k. In the first ~4210 primes, most (~94%) are of type k=1, meaning that b(p) = q*b(pq). Data presented here lists the first 257 primes of type k > 1. These include 219 of type 2, 27 of type 3, 9 of type 4, and 2 (25523, 39293) of type 5. One type k=6 prime (81611) has also been found but is not included above because it is not consecutive to the above terms. It is not known if there are primes of type k > 6 (although it is tempting to conjecture that k may be unbounded). Prime gap pq, and pattern of gaps pr determines if p is in the sequence or not. Prime p is of type k > 2 only if pq is one of the even indices of A056240 on which A292081 is defined (12,18,24,28,30,36,...), and if there is a prime r < q < p such that b(pr) < b(pq). Based on above data the primes in this sequence are all "Overdue" (A060769).


LINKS

David A. Corneth, Table of n, a(n) for n = 1..31763 (terms up to 5*10^6, first 544 terms from Robert Israel)
David A. Corneth, PARI program


EXAMPLE

p=211 is a candidate for inclusion because pq = 211199 = 12, and b(12)=35 is a term in A292081. Since r=197 is the next prime below q, pr = 14 and b(14) = 33 < 35, 211 is in the sequence, of type 2. Conversely, p=809, which also has gap pq = 12, is not in the sequence because the only number n > 12 for which b(n) < b(12)=35 is n=14, and p14 = 795 is not prime. Therefore b(809) = 797*b(12) = 27895, and 809 is of type 1.


MAPLE

N:= 10^7: # to get terms before the first prime p>3 such that A288814(p) > N
Res:= NULL:
for x from 4 to N do
if isprime(x) then next fi;
F:= ifactors(x)[2];
p:= add(t[1]*t[2], t=F);
if not isprime(p) then next fi;
if not assigned(A288814[p]) then
A288814[p]:= x;
w:= max(seq(t[1], t=F));
if w < prevprime(p) then
Res:= Res, p
fi
fi
od:
pmax:= Res[1]:
Primes:= select(isprime, [seq(i, i=5..pmax, 2)]):
B:= remove(p > assigned(A288814[p]), Primes):
sort(select(`<`, [Res], min(B))); # Robert Israel, Oct 19 2017


PROG

(PARI) see PARI link.  David A. Corneth, Mar 23 2018


CROSSREFS

Cf. A000040, A056240, A288814, A288313, A060769, A292081, A001223.
Sequence in context: A073102 A076167 A217620 * A300334 A032659 A098674
Adjacent sequences: A289990 A289991 A289992 * A289994 A289995 A289996


KEYWORD

nonn,easy


AUTHOR

David James Sycamore, Sep 13 2017


EXTENSIONS

a(30)a(50) from Robert Israel, Oct 02 2017


STATUS

approved



