|
|
A337469
|
|
a(n) is the least k that is a multiple of A071395(n) (the n-th primitive abundant number) for which A003961(k) is abundant.
|
|
1
|
|
|
120, 420, 1320, 1560, 4080, 4560, 5520, 6960, 1650, 3432, 3900, 4488, 7524, 1890, 17760, 19680, 20640, 4290, 22560, 3150, 25440, 5610, 28320, 29280, 12012, 6270, 4410, 6630, 7410, 7590, 23256, 8970, 28152, 9570, 9690, 10230, 6930, 52440, 22620, 59160, 24180, 12210, 8190, 63240, 64320
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
A003961(k) replaces each prime factor of k with the next larger prime. Thus for all terms a(n), A003961(a(n)) is an odd abundant number (some of which are also primitive abundant numbers, starting with n = 1, 2, 9, 10, 12, ...).
|
|
LINKS
|
|
|
FORMULA
|
|
|
EXAMPLE
|
The table below shows a(n), for n less than 16, alongside A071395(n) and its prime factors, and the additional prime factors that are needed to produce a(n).
1 120 / (2 * 3) = 20 = 2^2 * 5,
2 420 / (2 * 3) = 70 = 2 * 5 * 7,
3 1320 / (3 * 5) = 88 = 2^3 * 11,
4 1560 / (3 * 5) = 104 = 2^3 * 13,
5 4080 / (3 * 5) = 272 = 2^4 * 17,
6 4560 / (3 * 5) = 304 = 2^4 * 19,
7 5520 / (3 * 5) = 368 = 2^4 * 23,
8 6960 / (3 * 5) = 464 = 2^4 * 29,
9 1650 / (3) = 550 = 2 * 5^2 * 11,
10 3432 / (2 * 3) = 572 = 2^2 * 11 * 13,
11 3900 / (2 * 3) = 650 = 2 * 5^2 * 13,
12 4488 / (2 * 3) = 748 = 2^2 * 11 * 17,
13 7524 / (3 * 3) = 836 = 2^2 * 11 * 19,
14 1890 / (2) = 945 = 3^3 * 5 * 7,
15 17760 / (3 * 5) = 1184 = 2^5 * 37, ...
|
|
MATHEMATICA
|
Map[Block[{k = 1}, While[DivisorSigma[1, #] <= 2 # &[Times @@ Map[#1^#2 & @@ # &, FactorInteger[k #] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}]], k++]; # k] &, Select[Range[5*10^3], DivisorSigma[1, #] > 2 # && Times @@ Boole@ Map[DivisorSigma[1, #] < 2 # &, Most@ Divisors@ #] == 1 &]] (* Michael De Vlieger, Oct 05 2020 *)
|
|
PROG
|
(PARI)
isA071395(n) = if(sigma(n) <= 2*n, 0, fordiv(n, d, if((d != n)&&(sigma(d) >= 2*d), return(0))); (1)); \\ After code in A071395
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA337386(n) = { my(x=A003961(n)); (sigma(x)>=2*x); };
for(n=1, 2^13, if(isA071395(n), i=0; for(k=1, oo, if(isA337386(k*n), i++; print1(k*n, ", "); break))));
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|