A260075 -id:A260075 - cp's OEIS frontend

A259801 Numbers such that it and its two neighbors are products of 8 distinct primes.

102099792179230, 117092756174954, 136745109677256, 162338633743714, 167791215874866, 178571623400554, 183789996331514, 188284244083286, 211843056257854, 217181576415166, 224685381821406, 230455538364206, 234115003437666, 247662164889294, 265223112108514, 265730468260830, 266665427846390, 267248859559214, 268021718391414, 274354628059534

Offset: 1

James G. Merickel, Jul 14 2015

nonn

A subsequence of A169834.
With bound set at 4*10^14, the linked-to PARI program completed its run in about 2 days (producing 48 terms). The program fixes prospective smallest 4 prime factors so their product is at or above the minimum possible of the largest of 3 products of 4 primes without overlap (A260075(4)=20553), doing bound-restricted testing for the larger 4 in turn for each of these smaller quadruples. This is just one of a variety of ways of fixing a prospective trio by specifying one member as being within a certain range and satisfying the criterion. The program mostly avoids duplicates but does not entirely. See the part of the corresponding program at A259350 immediately before the print command for a fix.
The efficiency the program seems to generate empirically would come from the specification of product of 4 smaller primes as greater than a certain value and whole product within a certain range. Running through all even products of 8 distinct primes between the cube root of the (3n)-th primorial and the bound given would be a simpler way but one not so statistically limited (with a proportionally larger number of candidates). Note: The author is not making a claim of maximal efficiency, just of marked improvements over some simpler approaches.
a(1)=A093550(8).

			102099792179229=3*13*19*53*83*131*181*1321, 102099792179230=2*5*17*43*127*229*283*1697, and 102099792179231=7*11*23*29*31*71*113*7993. No smaller collection meets the criterion, so a(1)=102099792179230.

James G. Merickel, Table of n, a(n) for n = 1..48
James G. Merickel, PARI program

Cf. A093550, A169834, A248201, A248202, A248203, A248204, A259349, A259350, A260075.

\\ See above link to PARI program generating terms under 4*10^14 (out of order and with some duplicates).

A260079 The smallest product of n distinct primes no larger than the (2n)-th to exceed the square root of the (2n)-th primorial.

Original entry on oeis.org

3, 15, 182, 3135, 81345, 2733549, 114535239, 5708795638, 342503171205, 23622238184727, 1793796113720611, 154171427826650234, 15259854588449248245, 1601991507050573600715, 177792170427340904920562, 22932432920285320919637458, 3164592660873444717893657954, 474678349195034622662784926530

Offset: 1

James G. Merickel, Jul 15 2015

nonn

This is similar to but distinct from the even-indexed terms of A060796, with a(n) differing from A060796(2n) at n=7, 10, 11, 12, 13 and 16 (with A060796(36) unavailable for comparison). A260075 is the analog by splitting the first 3n primes into 3 equal-sized sets (but not by giving the smallest product larger than the cube root of the corresponding primorial). The percentages by which a(n) exceeds the square root of the (2n)-th primorial are 22.5, 3.51, 5.03, 0.660, 1.13, 0.347, 0.136, 1.82*10^(-3), 8.54*10^(-3), 6.21*10^(-3), 9.28*10^(-4), 1.84*10^(-4), 1.71*10^(-4), 1.31*10^(-5), 1.94*10^(-6), 5.62*10^(-8), 2.93*10^(-7) and 4.50*10^(-8).

The below PARI program functions by checking for each set of n primes through the (2n-1)-st whether either its product or its product's cofactor in the (2n)-th primorial gives an improvement.

			If the primes 2 through 13 are split into equal-sized sets, the larger product of the two is not less than 182. This is the value for {2, 7, 13}, so a(3)=182.

James G. Merickel, Table of n, a(n) for n = 1..18

Cf. A060796, A260075.

{
p=vector(50,n,prime(n));i=1;
while(1,
  P=prod(j=1,2*i,p[j]);Q=sqrt(P);r=10^1000;
  m=prod(j=1,i,p[j]);a=vectorsmall(2*i-1);
  for(j=1,i,a[j]=1);b=vectorsmall(2*i-1);
  for(j=i,2*i-1,b[j]=1);while(1,
    if(m>Q,if(m

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A259801 Numbers such that it and its two neighbors are products of 8 distinct primes.

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