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%I #53 Jul 06 2019 20:08:59
%S 41,43,59,67,82,83,85,86,89,91,97,103,109,113,118,121,123,129,131,133,
%T 134,137,145,149,151,155,157,163,164,166,167,169,170,172,173,177,178,
%U 181,182,185,187,193,194,197,199,201,203,205,206,209,218,221,223,226
%N Numbers that cannot be written as a difference of 3-smooth numbers (A003586).
%C Called ndh-numbers in the da Silva et al. link.
%C From _Jon E. Schoenfield_, Aug 19 2017: (Start)
%C If (following da Silva et al.) we refer to these numbers as "ndh-numbers" (meaning that they cannot be expressed as the difference of two "harmonic numbers" [which, in this context, are 3-smooth numbers]), we could refer to the sequence of positive integers that are not in this sequence as "dh-numbers", and say that the set of positive integers <= 100 includes the 11 ndh-numbers listed at the link (i.e., a(1) = 41 through a(11) = 97) and 100 - 11 = 89 dh-numbers. Each of the 89 dh-numbers <= 100 can be written as the difference of two 3-smooth numbers using no 3-smooth 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 3-smooth numbers < 10^50 (which seems very nearly certain to capture all differences in [1,10^10]):
%C .
%C Number Number
%C of ndh- of dh-
%C numbers numbers
%C in in Largest 3-smooth number required
%C k [1,10^k] [1,10^k] to obtain a dh-number in [1,10^k]
%C = ======== ======== ==================================
%C 1 0 10 12 = 3 + 9
%C 2 11 89 162 = 64 + 98
%C 3 522 478 13122 = 12288 + 834
%C 4 8433 1567 531441 = 524288 + 7153
%C 5 96065 3935 6377292 = 6291456 + 85836
%C 6 991699 8301 68024448 = 67108864 + 915584
%C 7 9984463 15537 688747536 = 679477248 + 9270288
%C 8 99973546 26454 7346640384 = 7247757312 + 98883072
%C .
%C A101082 gives the numbers that cannot be written as a difference of 2-smooth numbers (i.e., the powers of 2: A000079).
%C Numbers that cannot be written as a difference of 5-smooth 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, ...
%C Numbers that cannot be written as a difference of 7-smooth 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, ...
%C Numbers that cannot be written as a difference of 11-smooth 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, ...
%C (End)
%H Natalia da Silva, Serban Raianu, Hector Salgado, <a href="https://arxiv.org/abs/1708.00620">Differences of Harmonic Numbers and the abc-Conjecture</a>, arXiv:1708.00620 [math.NT], 2017.
%t terms = 54;
%t A3586 = Select[Range[3000], FactorInteger[#][[-1, 1]] <= 3&];
%t dd = Union[#[[2]] - #[[1]]& /@ Subsets[A3586, {2}]];
%t Complement[Range[u[[-1]]], dd][[1 ;; terms]] (* _Jean-François Alcover_, Sep 28 2018 *)
%Y Cf. A000079, A002473, A003586, A051037, A051038, A101082.
%K nonn
%O 1,1
%A _Michel Marcus_, Aug 03 2017
%E a(12)-a(54) from _Jon E. Schoenfield_, Aug 18 2017
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