A337691 a(n) is the least positive integer divisible by exactly n primitive nondeficient numbers (A006039).

1, 6, 60, 140, 420, 3780, 17160, 28600, 40040, 138600, 120120, 180180, 300300, 360360, 600600, 1351350, 900900, 4144140, 1801800, 3063060, 5405400, 6126120, 8558550, 7657650, 19399380, 20720700, 17117100, 15315300, 29099070, 30630600, 45945900, 70450380, 91891800, 87297210

Offset: 0

Author

David A. Corneth, Antti Karttunen and Peter Munn, Sep 20 2020

Keywords

• nonn

Comments

a(10) starts a run of at least 31 terms divisible by 30030 = 13#, product of primes <= 13.

About 20% of known terms are not divisible by 4 (indices 0, 1, 15, 22, 23, 28, 33, 38, 40, ...). This contrasts with many sequences that require terms to have some higher measure of abundancy (cf. A002093, A004394, A004490), where almost all terms are divisible by 4. The possibility of nontrivial odd terms seems worth considering.

Examples

			The least nondeficient number, therefore the least primitive nondeficient number is 6. So a(1) = 6, as the smallest number divisible by exactly 1 primitive nondeficient number.
Table of n, a(n) and the relevant divisors starts:
  n    a(n)   divisors in A006039
  0       1   (none);
  1       6   6;
  2      60   6, 20;
  3     140   20, 28, 70;
  4     420   6, 20, 28, 70;
  5    3780   6, 20, 28, 70, 945;
  6   17160   6, 20, 88, 104, 572, 1430;
  7   28600   20, 88, 104, 550, 572, 650, 1430;
  8   40040   20, 28, 70, 88, 104, 572, 1430, 2002; ...
Note that a(6), a(7), a(8) are 3*5720, 5*5720, 7*5720.
		

Links

• David A. Corneth, Table of n, a(n) for n = 0..43
• David A. Corneth, Some more upper bounds on a(n) for n at most 110.

Crossrefs

A006039, A337690 are used to define this sequence.

See A000203 and A023196 for definitions of deficient and nondeficient.

Sequences with similar definitions: A091193, A335540, A338405.

Cf. A002093, A004394, A004490.

Programs

PARI

\\ Code for A337690 given under that entry.
A337691list(search_up_to_n) = { my(m=Map(),lista=List([]),t); for(n=1,search_up_to_n,if(!(n%(2^24)),print1("(",n,")")); t=A337690(n); if(!mapisdefined(m,t), mapput(m,t,n))); for(n=0,oo,if(mapisdefined(m,n,&t), listput(lista,t), return(Vec(lista)))); };
v337691 = A337691list(2^27);
A337691(n) = v337691[1+n];

Formula

a(n) = min({k integer : k >= 1 and A337690(k) = n}).