|
|
A338133
|
|
Primitive nondeficient numbers sorted by largest prime factor then by increasing size. Irregular triangle T(n, k), n >= 2, k >= 1, read by rows, row n listing those with largest prime factor = prime(n).
|
|
4
|
|
|
6, 20, 28, 70, 945, 1575, 2205, 88, 550, 3465, 5775, 7425, 8085, 12705, 104, 572, 650, 1430, 2002, 4095, 6435, 6825, 9555, 15015, 78975, 81081, 131625, 189189, 297297, 342225, 351351, 570375, 63126063, 99198099, 117234117, 272, 748, 1870, 2210, 5355, 8415, 8925, 11492
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
2,1
|
|
COMMENTS
|
For definitions and further references/links, see A006039, the main entry for primitive nondeficient numbers.
Rows are finite: row n is a subset of the divisors of any of the products formed by multiplying 2^(A035100(n)-1) by a member of the first n finite sets described in the Dickson reference.
Column 1 includes the even perfect numbers.
The largest number in rows 2..n (therefore the largest that is prime(n)-smooth) is A338427(n). - Peter Munn, Sep 07 2021
|
|
LINKS
|
|
|
FORMULA
|
T(n, 1) = A308710(n-1) [provided there is no least deficient number that is not a power of 2, as described in A000079].
|
|
EXAMPLE
|
Row 1 is empty as there exists no primitive nondeficient number of the form prime(1)^k = 2^k.
Row 2 is (6) as 6 is the only primitive nondeficient number of the form prime(1)^k * prime(2)^m = 2^k * 3^m that is a multiple of prime(2) = 3.
Irregular triangle T(n, k) begins:
n prime(n) row n
2 3 6;
3 5 20;
4 7 28, 70, 945, 1575, 2205;
5 11 88, 550, 3465, 5775, 7425, 8085, 12705;
...
See also the factorization of initial terms below:
6 = 2 * 3,
20 = 2^2 * 5,
28 = 2^2 * 7,
70 = 2 * 5 * 7,
945 = 3^3 * 5 * 7,
1575 = 3^2 * 5^2 * 7,
2205 = 3^2 * 5 * 7^2,
88 = 2^3 * 11,
550 = 2 * 5^2 * 11,
3465 = 3^2 * 5 * 7 * 11,
5775 = 3 * 5^2 * 7 * 11,
7425 = 3^3 * 5^2 * 11,
8085 = 3 * 5 * 7^2 * 11,
12705 = 3 * 5 * 7 * 11^2,
104 = 2^3 * 13,
572 = 2^2 * 11 * 13,
650 = 2 * 5^2 * 13,
1430 = 2 * 5 * 11 * 13,
2002 = 2 * 7 * 11 * 13,
4095 = 3^2 * 5 * 7 * 13,
...
|
|
PROG
|
(PARI) rownupto(n, u) = { my(res = List(), pr = primes(n), e = vector(n, i, logint(u, pr[i]))); vu = vector(n, i, [0, e[i]]); vu[n][1] = 1; forvec(x = vu, c = prod(i = 1, n, pr[i]^x[i]); if(c <= u && isprimitive(c), listput(res, c) ) ); Set(res) }
isprimitive(n) = { my(f = factor(n), c); if(sigma(f) < 2*n, return(0)); for(i = 1, #f~, c = n / f[i, 1]; if(sigma(c) >= c * 2, return(0) ) ); 1 }
for(i = 2, 7, print(rownupto(i, 10^9)))
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,tabf
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|