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

6, 28, 104, 496, 836, 1952, 2002, 3230, 4030, 5830, 7912, 8128, 8415, 8925, 11096, 17816, 32445, 35650, 45356, 45885, 46035, 47804, 51850, 55796, 61904, 63388, 66928, 76516, 77572, 77744, 83265, 83312, 91388, 93148, 101475, 107198, 111644, 114256, 130304, 131054, 133042

Offset

1,1

Comments

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.

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.

Links

Peter Munn, Table of n, a(n) for n = 1..5000

Peter Munn, PARI program

Index entries for sequences where any odd perfect numbers must occur

Index entries for sequences related to sigma(n)

Example

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.
12 is nondeficient, but is not primitive (as a multiple of nondeficient 6). So 12 is not listed.
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.
(The remaining examples explore a little about calculating relative abundancies.)
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).
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).
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.
      6  2 * 3                          2, 3
     28  2 * 2 * 7                      2, 6,  7
    104  2 * 2 *13 * 2                  2, 6, 13, 14
    496  2 * 2 * 2 * 2 * 31             2, 6, 14, 30,  31
    836  2 * 2 *11 *19                  2, 6, 11, 19
   1952  2 * 2 * 2 * 2 * 61 * 2         2, 6, 14, 30,  61,  62
   2002  2 * 7 *11 *13                  2, 7, 11, 13
   3230  2 * 5 *17 *19                  2, 5, 17, 19
   4030  2 * 5 *13 *31                  2, 5, 13, 31
   5830  2 * 5 *11 *53                  2, 5, 11, 53
   7912  2 * 2 * 2 *23 * 43             2, 6, 14, 23,  43
   8128  2 * 2 * 2 * 2 *  2 * 2 *127    2, 6, 14, 30,  62, 126, 127
   8415  3 * 5 *11 * 3 * 17             3, 5, 11, 12,  17
   8925  3 * 5 * 7 *17 *  5             3, 5,  7, 17,  30
  11096  2 * 2 * 2 *19 * 73             2, 6, 14, 19,  73
  17816  2 * 2 * 2 *17 *113             2, 6, 14, 17, 113
  32445  3 * 5 * 7 * 3 *103             3, 5,  7, 12, 103

Prog

(PARI) See Links section.

Crossrefs

Cf. A262228, A335557.

Subsequence of A006039, A023196.

A000396 is a subsequence.

Sequence in context: A276041 A134416 A266976 * A117999 A234617 A028379

Adjacent sequences: A352736 A352737 A352738 * A352740 A352741 A352742

Keyword

nonn

Author

Peter Munn, Mar 31 2022

Status

approved