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).

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

Offset: 2

Author

David A. Corneth and Peter Munn, Oct 11 2020

Keywords

• nonn
• tabf

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

Examples

			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,
  ...
		

Links

• L. E. Dickson, Finiteness of the Odd Perfect and Primitive Abundant Numbers with n Distinct Prime Factors, Amer. J. Math., 35 (1913), 413-426.
• Index entries for sequences where any odd perfect numbers must occur

Crossrefs

A000040, A006530 are used to define this sequence.

Permutation of A006039.

A047802\{12}, A308710 are subsequences.

Cf. A000043, A000079, A000668, A035100, A059305, A061652, A338427.

Programs

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)))

Formula

A006530(T(n, k)) = A000040(n).

T(n, 1) = A308710(n-1) [provided there is no least deficient number that is not a power of 2, as described in A000079].

For m >= 1, T(A059305(m), 1) = A000668(m) * 2^(A000043(m)-1) = A000668(m) * A061652(m).