
COMMENTS

Called ndhnumbers in the da Silva et al. link.
From Jon E. Schoenfield, Aug 19 2017: (Start)
If (following da Silva et al.) we refer to these numbers as "ndhnumbers" (meaning that they cannot be expressed as the difference of two "harmonic numbers" [which, in this context, are 3smooth numbers]), we could refer to the sequence of positive integers that are not in this sequence as "dhnumbers", and say that the set of positive integers <= 100 includes the 11 ndhnumbers listed at the link (i.e., a(1) = 41 through a(11) = 97) and 100  11 = 89 dhnumbers. Each of the 89 dhnumbers <= 100 can be written as the difference of two 3smooth numbers using no 3smooth number larger than 162 (which is required to obtain the difference 98 = 162  64). The table below shows results from checking every difference between two 3smooth numbers < 10^50 (which seems very nearly certain to capture all differences in [1,10^10]):
.
Number Number
of ndh of dh
numbers numbers
in in Largest 3smooth number required
k [1,10^k] [1,10^k] to obtain a dhnumber in [1,10^k]
= ======== ======== ==================================
1 0 10 12 = 3 + 9
2 11 89 162 = 64 + 98
3 522 478 13122 = 12288 + 834
4 8433 1567 531441 = 524288 + 7153
5 96065 3935 6377292 = 6291456 + 85836
6 991699 8301 68024448 = 67108864 + 915584
7 9984463 15537 688747536 = 679477248 + 9270288
8 99973546 26454 7346640384 = 7247757312 + 98883072
.
A101082 gives the numbers that cannot be written as a difference of 2smooth numbers (i.e., the powers of 2: A000079).
Numbers that cannot be written as a difference of 5smooth numbers (A051037) appear to be 281, 289, 353, 413, 421, 439, 443, 457, 469, 493, 541, 562, 563, 578, 581, 583, 641, 653, 661, 677, 683, 691, 701, 706, 707, 731, 733, 737, 751, 761, 769, 779, 787, 793, 803, 811, 817, 823, 826, 827, 829, 841, 842, 843, 853, 857, 867, 877, 878, 881, 883, 886, ...
Numbers that cannot be written as a difference of 7smooth numbers (A002473) appear to be 1849, 2309, 2411, 2483, 2507, 2531, 2629, 2711, 2753, 2843, 2851, 2921, 2941, 3139, 3161, 3167, 3181, 3217, 3229, 3251, 3287, 3289, 3293, 3323, 3379, 3481, 3487, 3541, 3601, 3623, 3653, 3697, 3698, 3709, 3737, 3739, 3803, 3827, 3859, 3877, 3901, 3923, 3947, ...
Numbers that cannot be written as a difference of 7smooth numbers (A051038) appear to be 9007, 10091, 10531, 10831, 11801, 12197, 12431, 12833, 12941, 13393, 13501, 13619, 13679, 13751, 13907, 13939, 14219, 14423, 14737, 14851, 14893, 15217, 15641, 15767, 15773, 15803, 15959, 16019, 16201, 16241, 16393, 16397, 16417, 16441, 16517, 16559, 16579, ...
(End)
