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A290365 Numbers that cannot be written as a difference of 3-smooth numbers (A003586). 0
41, 43, 59, 67, 82, 83, 85, 86, 89, 91, 97, 103, 109, 113, 118, 121, 123, 129, 131, 133, 134, 137, 145, 149, 151, 155, 157, 163, 164, 166, 167, 169, 170, 172, 173, 177, 178, 181, 182, 185, 187, 193, 194, 197, 199, 201, 203, 205, 206, 209, 218, 221, 223, 226 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Called ndh-numbers 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 "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]):

.

    Number    Number

    of ndh-   of dh-

    numbers   numbers

      in        in     Largest 3-smooth number required

k  [1,10^k]  [1,10^k]   to obtain a dh-number 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 2-smooth numbers (i.e., the powers of 2: A000079).

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, ...

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, ...

Numbers that cannot be written as a difference of 7-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, ...

(End)

LINKS

Table of n, a(n) for n=1..54.

Natalia da Silva, Serban Raianu, Hector Salgado, Differences of Harmonic Numbers and the abc-Conjecture, arXiv:1708.00620 [math.NT], 2017.

MATHEMATICA

terms = 54;

A3586 = Select[Range[3000], FactorInteger[#][[-1, 1]] <= 3&];

dd = Union[#[[2]] - #[[1]]& /@ Subsets[A3586, {2}]];

Complement[Range[u[[-1]]], dd][[1 ;; terms]] (* Jean-Fran├žois Alcover, Sep 28 2018 *)

CROSSREFS

Cf. A000079, A002473, A003586, A051037, A051038, A101082.

Sequence in context: A139774 A007643 A259552 * A277071 A186401 A180548

Adjacent sequences:  A290362 A290363 A290364 * A290366 A290367 A290368

KEYWORD

nonn

AUTHOR

Michel Marcus, Aug 03 2017

EXTENSIONS

a(12)-a(54) from Jon E. Schoenfield, Aug 18 2017

STATUS

approved

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Last modified April 22 04:04 EDT 2019. Contains 322329 sequences. (Running on oeis4.)