A182002 -id:A182002 - cp's OEIS frontend

A258069 Nonnegative integers that can be computed using exactly eight 8's and the four basic arithmetic operations {+, -, *, /}.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 92

Offset: 1

Alois P. Heinz, May 18 2015

The smallest non-computable number here is 91. The largest computable number here is 8^8 = 16777216.

Alois P. Heinz, Table of n, a(n) for n = 1..1660

Cf. A171826, A171827, A171828, A171829, A258068, A258070, A258071.

Cf. A182002, A258097.

f:= proc(n) f(n):= `if`(n=1, {8}, {seq(seq(seq([x+y, x-y, x*y,
      `if`(y=0, [][], x/y)][], y=f(n-j)), x=f(j)), j=1..n-1)})
    end:
sort([select(z->z>=0 and is(z, integer), f(8))[]])[];

A258069(n=8, S=Vec([[n]], n))={for(n=2, n, S[n]=Set(concat(vector(n\2, k, Set(concat([Set(concat([[T+U, T-U, U-T, if(U, T/U), if(T, U/T), T*U] | T <- S[n-k]])) | U <- S[k]])))))); select(t->t>=0 && type(t)=="t_INT", S[n])} \\ A258069() yields this sequence, use optional arg to compute variants. - M. F. Hasler, Nov 24 2018

A258070 Nonnegative integers that can be computed using exactly nine 9's and the four basic arithmetic operations {+, -, *, /}.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67

Offset: 1

Alois P. Heinz, May 18 2015

The smallest non-computable number here is 195. The largest computable number here is 9^9 = 387420489.

Alois P. Heinz, Table of n, a(n) for n = 1..4769

Cf. A171826, A171827, A171828, A171829, A258068, A258069, A258071.

Cf. A182002, A258097.

f:= proc(n) f(n):= `if`(n=1, {9}, {seq(seq(seq([x+y, x-y, x*y,
      `if`(y=0, [][], x/y)][], y=f(n-j)), x=f(j)), j=1..n-1)})
    end:
sort([select(z->z>=0 and is(z, integer), f(9))[]])[];

A258070(n=9, S=Vec([[n]], n))={for(n=2, n, S[n]=Set(concat(vector(n\2, k, Set(concat([Set(concat([[T+U, T-U, U-T, if(U, T/U), if(T, U/T), T*U] | T <- S[n-k]])) | U <- S[k]])))))); select(t->t>=0 && type(t)=="t_INT", S[n])} \\ Requires at least 30 MB stack. (Use allocatemem()). A258070() yields this sequence, use optional arg to compute variants. - M. F. Hasler, Nov 26 2018

A258071 Nonnegative integers that can be computed using exactly ten 10's and the four basic arithmetic operations {+, -, *, /}.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67

Offset: 1

Alois P. Heinz, May 18 2015

The smallest non-computable number here is 443. The largest computable number here is 10^10 = 10000000000.

Alois P. Heinz, Table of n, a(n) for n = 1..15945

Cf. A171826, A171827, A171828, A171829, A258068, A258069, A258070.

Cf. A182002, A258097.

f:= proc(n) f(n):= `if`(n=1, {10}, {seq(seq(seq([x+y, x-y, x*y,
      `if`(y=0, [][], x/y)][], y=f(n-j)), x=f(j)), j=1..n-1)})
    end:
sort([select(z->z>=0 and is(z, integer), f(10))[]])[];

A171827 Nonnegative integers that can be made by using exactly four fours (4 4's) and the four basic operators {+, -, *, /}.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 15, 16, 17, 20, 24, 28, 32, 36, 48, 60, 64, 68, 80, 128, 256

Offset: 1

Sergio Pimentel, Dec 19 2009

Every integer can be made if other operators are allowed (e.g., 14 = 4!/4 + 4 + 4). The sequence is finite: a(26) = 4*4*4*4=256 is the last term.

			Example: a(11) = 12 since 12 = (4 - 4/4) * 4.

Wikipedia, Four Fours

Cf. A171826, A171828, A171829, A258068, A258069, A258070, A258071.

Cf. A182002, A258097.

A171827(n=4, S=Vec([[n]], n))={for(n=2, n, S[n]=Set(concat(vector(n\2, k, concat([concat([[T+U, T-U, U-T, if(U, T/U), if(T, U/T), T*U] | T <- S[n-k]]) | U <- S[k]]))))); [t|t <- S[n], t>=0 && type(t)=="t_INT"]} \\ A171827() yields this sequence, use optional arg to compute variants. - M. F. Hasler, Nov 24 2018

A171828 Nonnegative integers that can be made by using five fives (5 5's) and the four basic operators {+, -, *, /}.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 19, 20, 21, 23, 24, 25, 26, 27, 29, 30, 31, 35, 40, 45, 49, 50, 51, 55, 60, 70, 75, 80, 95, 100, 105, 115, 120, 124, 125, 126, 130, 135, 145, 150, 155, 175, 200, 225, 245, 250, 255, 275, 300, 375, 500, 600, 620, 630, 650, 750, 1250, 3125

Offset: 1

Sergio Pimentel, Dec 19 2009

Every integer can be made if other operators are allowed (i.e., 13 = (5!+5+5)/(5+5)). The sequence is finite: a(68) = 5*5*5*5*5 = 3125 is the last term.

			Example: a(14) = 14 = (5+5+5)-(5/5).

Wikipedia, Four Fours

Cf. A171826, A171827, A171829, A258068, A258069, A258070, A258071.

Cf. A182002, A258097.

A171828(n=5, S=Vec([[n]], n))={for(n=2, n, S[n]=Set(concat(vector(n\2, k, concat([concat([[T+U, T-U, U-T, if(U, T/U), if(T, U/T), T*U] | T <- S[n-k]]) | U <- S[k]]))))); select(t->t>=0 && type(t)=="t_INT", S[n])} \\ A171828() yields this sequence, use optional arg to compute variants. - M. F. Hasler, Nov 24 2018

A171829 Nonnegative integers that can be made by using six sixes (6 6's) and the four basic operators {+, -, *, /}.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 54, 60, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 84, 90, 96, 102, 107

Offset: 1

Sergio Pimentel, Dec 19 2009

More integers can be made if other operators are allowed (i.e., 22 = 6!/(6*6)+(6+6)/6). The sequence is finite: a(198) = 6*6*6*6*6*6 = 46656 is the last term.

See A258068 ff. for the integers that can be generated with the four basic operators and 7 7's, 8 8's, 9 9's, etc...

			49 is in the sequence: 49 = (6 + 6/6) * (6 + 6/6).

Alois P. Heinz, Table of n, a(n) for n = 1..198
Wikipedia, Four Fours

Cf. A171826, A171827, A171828, A258068, A258069, A258070, A258071.

Cf. A182002, A258097.

f:= proc(n) f(n):= `if`(n=1, {6}, {seq(seq(seq([x+y, x-y, x*y,
      `if`(y=0, [][], x/y)][], y=f(n-j)), x=f(j)), j=1..n-1)})
    end:
sort([select(z->z>=0 and is(z, integer), f(6))[]])[];
#  Alois P. Heinz, Aug 04 2013

f[1] = {6}; f[n_] := f[n] = Union @ Flatten @ Table[Table[Table[{x+y, x-y, x*y, If[y == 0, Null, x/y]}, {y, f[n-j]}], {x, f[j]}], {j, 1, n-1}];
Sort[Select[f[6], # >= 0 && IntegerQ[#]&]] (* Jean-François Alcover, Jun 01 2018, after Alois P. Heinz *)

A171829(n=6, S=Vec([[n]],n))={for(n=2, n, S[n]=Set(concat(vector(n\2, k, concat([concat([[T+U, T-U, U-T, if(U, T/U), if(T, U/T), T*U] | T <- S[k]]) | U <- S[n-k]]))))); select(t-> t>=0 && denominator(t)==1,S[n])} \\ A171829() yields this sequence. Optional args allow to compute variants. - M. F. Hasler, Nov 24 2018

Corrected and edited by Alois P. Heinz, Aug 03 2013

A258068 Nonnegative integers that can be computed using exactly seven 7's and the four basic arithmetic operations {+, -, *, /}.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 69, 70, 71

Offset: 1

Alois P. Heinz, May 18 2015

The smallest non-computable number here is 38. The largest computable number here is 7^7 = 823543.

Alois P. Heinz, Table of n, a(n) for n = 1..536

Cf. A171826, A171827, A171828, A171829, A258069, A258070, A258071.

Cf. A182002, A258097.

f:= proc(n) f(n):= `if`(n=1, {7}, {seq(seq(seq([x+y, x-y, x*y,
      `if`(y=0, [][], x/y)][], y=f(n-j)), x=f(j)), j=1..n-1)})
    end:
sort([select(z->z>=0 and is(z, integer), f(7))[]])[];

A258068(n=7, S=Vec([[n]], n))={for(n=2, n, S[n]=Set(concat(vector(n\2, k, Set(concat([Set(concat([[T+U, T-U, U-T, if(U, T/U), if(T, U/T), T*U] | T <- S[n-k]])) | U <- S[k]])))))); select(t->t>=0 && type(t)=="t_INT", S[n])} \\ A258068() yields this sequence, use optional arg to compute variants. - M. F. Hasler, Nov 24 2018

A171826 Nonnegative integers that can be made using exactly three threes (3 3's) and the four basic operators {+, -, *, /}.

Original entry on oeis.org

0, 2, 3, 4, 6, 9, 12, 18, 27

Offset: 1

Sergio Pimentel, Dec 19 2009

More integers can be made using 3 3's if more operators are allowed (i.e., 1 = 3^(3-3) or 5 = 3! - 3/3). Note that 1 = 3/3 but the sequence requires to use exactly three threes and it's not possible to "get rid" of the third three.

			a(1) = 0 = (3-3)*3, a(2) = 2 = 3 - 3/3, a(3) = 3 = 3*3/3,
a(4) = 4 = 3 + 3/3, a(5) = 6 = 3*3 - 3, a(6) = 9 = 3 + 3 + 3,
a(7) = 12 = 3*3 + 3, a(8) = 18 = (3+3)*3, a(9) = 27 = 3*3*3.

Wikipedia, Four Fours

Cf. A171827, A171828, A171829, A258068, A258069, A258070, A258071 (analog for four 4's, ..., ten 10's).

Cf. A182002, A258097.

vecextract([0..27],134484573) \\ Better programs to compute this are available in the related sequences, but they are much longer than this sequence itself. - M. F. Hasler, Nov 24 2018

A321996 Numbers that cannot be written using six 6's and operations +, -, * and/or /. Complement of A171829.

Original entry on oeis.org

22, 26, 46, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 68, 76, 80, 81, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 103, 104, 105, 106, 110, 111, 112, 113, 115, 116, 117, 118, 119, 121, 122, 123, 124, 125, 127, 128, 129, 130, 131, 133, 134

Offset: 1

M. F. Hasler, Nov 23 2018

nonn

A171829 contains all but 3 nonnegative integers below 50, which motivates to list the complement, even though A171829 is finite and its complement, this sequence, contains all integers above 9072 except for 15552 and 46656. (These are the last three terms of A171829.)

Cf. A171829 (complement), A171826, A171827, A171828, A258068, A258069, A258070, A258071 (analog for 3, ..., 10).

Cf. A182002, A258097.

A321997 Numbers that cannot be written using seven 7's and operations +, -, * and/or /. Complement of A258068.

Original entry on oeis.org

38, 66, 67, 68, 73, 74, 75, 76, 79, 80, 81, 87, 88, 108, 109, 115, 116, 117, 118, 121, 122, 123, 124, 125, 127, 128, 129, 130, 131, 132, 134, 135, 136, 137, 138, 142, 143, 151, 152, 156, 157, 158, 159, 160, 162, 163, 164, 165, 166, 167, 169, 170, 173, 174, 176, 177, 178, 179

Offset: 1

M. F. Hasler, Nov 23 2018

nonn

A258068 contains most nonnegative integers below 115, which motivates listing the complement, even though A258068 is finite and its complement, this sequence, contains all integers above 134456 except for 235298 and 823543. (These are the last three terms of A258068.)

Cf. A258068 (complement), A171826, A171827, A171828, A171829, A258069, A258070, A258071 (analog for 3, ..., 10).

Cf. A182002, A258097.

cp's OEIS Frontend

A258069 Nonnegative integers that can be computed using exactly eight 8's and the four basic arithmetic operations {+, -, *, /}.

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A258070 Nonnegative integers that can be computed using exactly nine 9's and the four basic arithmetic operations {+, -, *, /}.

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A258071 Nonnegative integers that can be computed using exactly ten 10's and the four basic arithmetic operations {+, -, *, /}.

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A171827 Nonnegative integers that can be made by using exactly four fours (4 4's) and the four basic operators {+, -, *, /}.

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A171828 Nonnegative integers that can be made by using five fives (5 5's) and the four basic operators {+, -, *, /}.

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A171829 Nonnegative integers that can be made by using six sixes (6 6's) and the four basic operators {+, -, *, /}.

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A258068 Nonnegative integers that can be computed using exactly seven 7's and the four basic arithmetic operations {+, -, *, /}.

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A171826 Nonnegative integers that can be made using exactly three threes (3 3's) and the four basic operators {+, -, *, /}.

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