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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).

%I A338133 #33 Nov 29 2022 12:53:10
%S A338133 6,20,28,70,945,1575,2205,88,550,3465,5775,7425,8085,12705,104,572,
%T A338133 650,1430,2002,4095,6435,6825,9555,15015,78975,81081,131625,189189,
%U A338133 297297,342225,351351,570375,63126063,99198099,117234117,272,748,1870,2210,5355,8415,8925,11492
%N 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).
%C A338133 For definitions and further references/links, see A006039, the main entry for primitive nondeficient numbers.
%C A338133 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.
%C A338133 Column 1 includes the even perfect numbers.
%C A338133 The largest number in rows 2..n (therefore the largest that is prime(n)-smooth) is A338427(n). - _Peter Munn_, Sep 07 2021
%H A338133 L. E. Dickson, <a href="http://www.jstor.org/stable/2370405">Finiteness of the Odd Perfect and Primitive Abundant Numbers with n Distinct Prime Factors</a>, Amer. J. Math., 35 (1913), 413-426.
%H A338133 <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</a>
%F A338133 A006530(T(n, k)) = A000040(n).
%F A338133 T(n, 1) = A308710(n-1) [provided there is no least deficient number that is not a power of 2, as described in A000079].
%F A338133 For m >= 1, T(A059305(m), 1) = A000668(m) * 2^(A000043(m)-1) = A000668(m) * A061652(m).
%e A338133 Row 1 is empty as there exists no primitive nondeficient number of the form prime(1)^k = 2^k.
%e A338133 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.
%e A338133 Irregular triangle T(n, k) begins:
%e A338133   n   prime(n)  row n
%e A338133   2      3      6;
%e A338133   3      5      20;
%e A338133   4      7      28, 70, 945, 1575, 2205;
%e A338133   5     11      88, 550, 3465, 5775, 7425, 8085, 12705;
%e A338133   ...
%e A338133 See also the factorization of initial terms below:
%e A338133       6 = 2 * 3,
%e A338133      20 = 2^2 * 5,
%e A338133      28 = 2^2 * 7,
%e A338133      70 = 2 * 5 * 7,
%e A338133     945 = 3^3 * 5 * 7,
%e A338133    1575 = 3^2 * 5^2 * 7,
%e A338133    2205 = 3^2 * 5 * 7^2,
%e A338133      88 = 2^3 * 11,
%e A338133     550 = 2 * 5^2 * 11,
%e A338133    3465 = 3^2 * 5 * 7 * 11,
%e A338133    5775 = 3 * 5^2 * 7 * 11,
%e A338133    7425 = 3^3 * 5^2 * 11,
%e A338133    8085 = 3 * 5 * 7^2 * 11,
%e A338133   12705 = 3 * 5 * 7 * 11^2,
%e A338133     104 = 2^3 * 13,
%e A338133     572 = 2^2 * 11 * 13,
%e A338133     650 = 2 * 5^2 * 13,
%e A338133    1430 = 2 * 5 * 11 * 13,
%e A338133    2002 = 2 * 7 * 11 * 13,
%e A338133    4095 = 3^2 * 5 * 7 * 13,
%e A338133   ...
%o A338133 (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) }
%o A338133 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 }
%o A338133 for(i = 2, 7, print(rownupto(i, 10^9)))
%Y A338133 A000040, A006530 are used to define this sequence.
%Y A338133 Permutation of A006039.
%Y A338133 A047802\{12}, A308710 are subsequences.
%Y A338133 Cf. A000043, A000079, A000668, A035100, A059305, A061652, A338427.
%K A338133 nonn,tabf
%O A338133 2,1
%A A338133 _David A. Corneth_ and _Peter Munn_, Oct 11 2020