cp's OEIS Frontend

A290365 Numbers that cannot be written as a difference of 3-smooth numbers (A003586).

%I A290365 #53 Jul 06 2019 20:08:59
%S A290365 41,43,59,67,82,83,85,86,89,91,97,103,109,113,118,121,123,129,131,133,
%T A290365 134,137,145,149,151,155,157,163,164,166,167,169,170,172,173,177,178,
%U A290365 181,182,185,187,193,194,197,199,201,203,205,206,209,218,221,223,226
%N A290365 Numbers that cannot be written as a difference of 3-smooth numbers (A003586).
%C A290365 Called ndh-numbers in the da Silva et al. link.
%C A290365 From _Jon E. Schoenfield_, Aug 19 2017: (Start)
%C A290365 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 A290365 .
%C A290365     Number    Number
%C A290365     of ndh-   of dh-
%C A290365     numbers   numbers
%C A290365       in        in     Largest 3-smooth number required
%C A290365 k  [1,10^k]  [1,10^k]   to obtain a dh-number in [1,10^k]
%C A290365 =  ========  ========  ==================================
%C A290365 1         0        10          12 =          3 +        9
%C A290365 2        11        89         162 =         64 +       98
%C A290365 3       522       478       13122 =      12288 +      834
%C A290365 4      8433      1567      531441 =     524288 +     7153
%C A290365 5     96065      3935     6377292 =    6291456 +    85836
%C A290365 6    991699      8301    68024448 =   67108864 +   915584
%C A290365 7   9984463     15537   688747536 =  679477248 +  9270288
%C A290365 8  99973546     26454  7346640384 = 7247757312 + 98883072
%C A290365 .
%C A290365 A101082 gives the numbers that cannot be written as a difference of 2-smooth numbers (i.e., the powers of 2: A000079).
%C A290365 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 A290365 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 A290365 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 A290365 (End)
%H A290365 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 A290365 terms = 54;
%t A290365 A3586 = Select[Range[3000], FactorInteger[#][[-1, 1]] <= 3&];
%t A290365 dd = Union[#[[2]] - #[[1]]& /@ Subsets[A3586, {2}]];
%t A290365 Complement[Range[u[[-1]]], dd][[1 ;; terms]] (* _Jean-François Alcover_, Sep 28 2018 *)
%Y A290365 Cf. A000079, A002473, A003586, A051037, A051038, A101082.
%K A290365 nonn
%O A290365 1,1
%A A290365 _Michel Marcus_, Aug 03 2017
%E A290365 a(12)-a(54) from _Jon E. Schoenfield_, Aug 18 2017