cp's OEIS Frontend

A352739 Primitive nondeficient numbers satisfying a stronger condition that compares abundancy with related numbers as detailed in the comments.

%I A352739 #19 May 27 2025 18:46:46
%S A352739 6,28,104,496,836,1952,2002,3230,4030,5830,7912,8128,8415,8925,11096,
%T A352739 17816,32445,35650,45356,45885,46035,47804,51850,55796,61904,63388,
%U A352739 66928,76516,77572,77744,83265,83312,91388,93148,101475,107198,111644,114256,130304,131054,133042
%N A352739 Primitive nondeficient numbers satisfying a stronger condition that compares abundancy with related numbers as detailed in the comments.
%C A352739 In this entry we list a primitive nondeficient number, m, if and only if there is no nondeficient number k = (m/p)*q whose abundancy is less than the abundancy of m, where p and q are prime numbers.
%C A352739 The primitive nondeficient number list (A006039) abbreviates the nondeficient number list (A023196) by omitting multiples of numbers that have already appeared. Here we abbreviate further, although typically by omission due to the lesser abundancy of a larger number. For instance, 315 * p is a primitive nondeficient number for prime p, 3 <= p <= 103; of these only 32445 = 315 * 103 is listed. See the example section for further detail.
%H A352739 Peter Munn, <a href="/A352739/b352739.txt">Table of n, a(n) for n = 1..5000</a>
%H A352739 Peter Munn, <a href="/A352739/a352739.txt">PARI program</a>
%H A352739 <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</a>
%H A352739 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%e A352739 6 is a perfect number, and a perfect number is a primitive nondeficient number with the least abundancy (1.0) for classification as nondeficient, from which we see there is no nondeficient k with lesser abundancy. So 6 meets the stronger condition to be listed here.
%e A352739 12 is nondeficient, but is not primitive (as a multiple of nondeficient 6). So 12 is not listed.
%e A352739 104 = 2^3*13 is nondeficient, but 52 & 8 (and so other proper divisors) are deficient. So 104 is primitive nondeficient. To evaluate the stronger condition, we replace a factor 2 by prime p_1 <> 2, giving k = 2^2 * 13 * p_1 (maybe k = 2^2 * 13^2), or the factor 13 by prime p_2 <> 13, giving k = 2^3 * p_2 (maybe k = 2^4). If p_1 > 11, p_2 = 2, or p_2 > 13, k is deficient. Otherwise (p_1 <= 11 or 2 < p_2 < 13, and) calculation shows abundancy of k is greater than the abundancy of 104. So there is no nondeficient k with lesser abundancy. So 104 is listed.
%e A352739 (The remaining examples explore a little about calculating relative abundancies.)
%e A352739 88 = 2^3*11 is nondeficient, but is not listed because 104 = 2^3*13 is nondeficient and 13 contributes a smaller factor to abundancy (14/13) than 11 does (12/11). Likewise, 272, 304, 368, 464 (each 16 times a prime) are omitted due to the lesser abundancy of 496 = 16*31 (which is listed).
%e A352739 70 is nondeficient, but is not listed due to the lesser abundancy of nondeficient 28. 70 = 2*7*5, whereas 28 = 2*7*2, and a second factor of 2 contributes a smaller factor to abundancy (7/6) than the first factor of 5 does (6/5).
%e A352739 The following table of terms shows prime factors listed in order of decreasing factor, (m+1)/m, that they contribute to the number's abundancy (e.g., a 2nd factor of 3 increases abundancy by 13/12, so appears after 11 in the factorization of 8415). The relevant values of m are shown on the right.
%e A352739       6  2 * 3                          2, 3
%e A352739      28  2 * 2 * 7                      2, 6,  7
%e A352739     104  2 * 2 *13 * 2                  2, 6, 13, 14
%e A352739     496  2 * 2 * 2 * 2 * 31             2, 6, 14, 30,  31
%e A352739     836  2 * 2 *11 *19                  2, 6, 11, 19
%e A352739    1952  2 * 2 * 2 * 2 * 61 * 2         2, 6, 14, 30,  61,  62
%e A352739    2002  2 * 7 *11 *13                  2, 7, 11, 13
%e A352739    3230  2 * 5 *17 *19                  2, 5, 17, 19
%e A352739    4030  2 * 5 *13 *31                  2, 5, 13, 31
%e A352739    5830  2 * 5 *11 *53                  2, 5, 11, 53
%e A352739    7912  2 * 2 * 2 *23 * 43             2, 6, 14, 23,  43
%e A352739    8128  2 * 2 * 2 * 2 *  2 * 2 *127    2, 6, 14, 30,  62, 126, 127
%e A352739    8415  3 * 5 *11 * 3 * 17             3, 5, 11, 12,  17
%e A352739    8925  3 * 5 * 7 *17 *  5             3, 5,  7, 17,  30
%e A352739   11096  2 * 2 * 2 *19 * 73             2, 6, 14, 19,  73
%e A352739   17816  2 * 2 * 2 *17 *113             2, 6, 14, 17, 113
%e A352739   32445  3 * 5 * 7 * 3 *103             3, 5,  7, 12, 103
%o A352739 (PARI) \\ See Links section.
%Y A352739 Cf. A262228, A335557.
%Y A352739 Subsequence of A006039, A023196.
%Y A352739 A000396 is a subsequence.
%K A352739 nonn
%O A352739 1,1
%A A352739 _Peter Munn_, Mar 31 2022