A079000 -id:A079000 - cp's OEIS frontend

A080031 a(n) is taken to be the smallest positive integer not already present which is consistent with the condition "n is a member of the sequence if and only if a(n) is congruent to 2 mod 3".

1, 2, 5, 4, 8, 11, 7, 14, 17, 10, 20, 23, 13, 26, 29, 16, 32, 35, 19, 38, 41, 22, 44, 47, 25, 50, 53, 28, 56, 59, 31, 62, 65, 34, 68, 71, 37, 74, 77, 40, 80, 83, 43, 86, 89, 46, 92, 95, 49, 98, 101, 52, 104, 107, 55, 110, 113, 58, 116, 119, 61, 122, 125, 64, 128, 131, 67

Offset: 0

N. J. A. Sloane, Mar 14 2003

A permutation of all positive non-multiples of 3; also a permutation of A080030. - Matthew Vandermast, Mar 21 2003

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)

Cf. A079000, A079313, A080029, A080030.

a(3m)=3m+1, a(3m+1)=6m+2, a(3m+2)=6m+5. [corrected by Georg Fischer, Jun 08 2022]

More terms from Matthew Vandermast, Mar 21 2003

A080711 a(0) = 2; for n>0, a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is a multiple of 3".

Original entry on oeis.org

2, 4, 6, 7, 9, 10, 12, 15, 16, 18, 21, 22, 24, 25, 26, 27, 30, 31, 33, 34, 35, 36, 39, 40, 42, 45, 48, 51, 52, 53, 54, 57, 58, 60, 63, 66, 69, 70, 71, 72, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 90, 93, 96, 97, 98, 99, 102, 103, 105, 106, 107, 108, 109, 110, 111, 112

Offset: 0

N. J. A. Sloane, Mar 05 2003

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)
Index entries for sequences of the a(a(n)) = 2n family

Cf. A079000, A003605, A079253, A080710, A080712.

{a=2; m=[2]; for(n=1,68,print1(a,","); a=a+1; if(m[1]==n, while(a%3>0,a++); m=if(length(m)==1,[],vecextract(m,"2..")),if(a%3==0,a++)); m=concat(m,a))}

a(a(n)) = 3*(n+2).

More terms and PARI code from Klaus Brockhaus, Mar 06 2003

A099797 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is composite".

Original entry on oeis.org

2, 4, 5, 6, 8, 9, 11, 12, 14, 17, 18, 20, 23, 24, 29, 31, 32, 33, 37, 38, 41, 43, 44, 45, 47, 53, 59, 61, 62, 67, 68, 69, 70, 71, 73, 79, 80, 81, 83, 89, 90, 97, 98, 99, 100, 101, 102, 103, 107, 109, 113, 127, 128, 131, 137, 139, 149, 151, 152, 157, 158, 159, 163, 167

Offset: 1

Ray Chandler, Nov 02 2004

nonn

			a(1) cannot be 1 because 1 is not composite; it can be 2.

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)

Cf. A079000, A079254, A085925, A099798.

A099798 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is not composite".

Original entry on oeis.org

1, 2, 3, 6, 8, 11, 12, 13, 14, 15, 17, 19, 23, 29, 31, 32, 37, 38, 41, 42, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 59, 61, 62, 63, 64, 65, 67, 71, 72, 74, 79, 83, 84, 89, 97, 98, 101, 103, 107, 109, 113, 127, 131, 137, 138, 140, 141, 142, 149, 150, 151, 157, 163, 167

Offset: 1

Ray Chandler, Nov 02 2004

nonn

			a(4) cannot be 4 because 4 is composite; it cannot be 5, for then 4 is not in the sequence while a(4) is not composite; but 6 is possible.

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)

Cf. A079000, A079254, A085925, A099797.

A079255 a(n) is taken to be the smallest positive integer greater than a(n-1) such that the condition "n is in the sequence if and only if a(n) is odd and a(n+1) is even" can be satisfied.

Original entry on oeis.org

1, 4, 6, 9, 12, 15, 18, 20, 23, 26, 28, 31, 34, 36, 39, 42, 44, 47, 50, 53, 56, 58, 61, 64, 66, 69, 72, 75, 78, 80, 83, 86, 88, 91, 94, 97, 100, 102, 105, 108, 110, 113, 116, 119, 122, 124, 127, 130, 132, 135, 138, 140, 143, 146, 148, 151, 154, 157, 160, 162, 165, 168

Offset: 1

N. J. A. Sloane and Matthew Vandermast, Feb 04 2003

No two terms in the sequence are consecutive integers (see example for a(3)).

			a(2) cannot be odd; it also cannot be 2, since that would imply that a(2) was odd. 4 is the smallest value for a(2) that creates no contradiction. a(3) cannot be 5, which would imply that a(5) was odd because it is known from 4's being in the sequence that a(4) is odd and a(5) even. 6 is the smallest value for a(3) that creates no contradiction.

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)

Cf. A079000, A079259. First differences give A080428.

Cf. A026363, A080428.

With the convention A026363(0)=0 (offset is 1 for this sequence) we have a(n)=A026363(2n)+1; a(n)=(1+sqrt(3))*n+O(1). The sequence satisfies the meta-system for n>=2: a(a(n))=2*a(n)+2*n+2 ; a(a(n)-1)=2*a(n)+2*n-1 ; a(a(n)-2)=2*a(n)+2*n-4 which allows us to have all terms since first differences =2 or 3 only. a(n)=a(n-1)+3 if n is in A026363, a(n)=a(n-1)+2 otherwise (if n is in A026364). - Benoit Cloitre, Apr 23 2008

A079351 a(1)=3; for n > 1, a(n) is the smallest integer greater than a(n-1) consistent with the condition "n is in the sequence if and only if a(n) is congruent to 0 (mod 5)".

Original entry on oeis.org

3, 4, 5, 10, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115

Offset: 1

Benoit Cloitre, Feb 23 2003

nonn

Equivalently: unique monotonic sequence satisfying a(1)=3, a(a(n))=5n.

Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.
Index entries for sequences of the a(a(n)) = 2n family

Cf. A079000, A080589, A003605.

a(3*5^k + j) = 5^(k+1) + 3j + 2|j|, k >= 0, -2*5^k <= j < 2*5^k.

More terms from Matthew Vandermast, Mar 13 2003

A079352 a(1)=1, then a(n)=3a(n-1) if n is already in the sequence, a(n)=2a(n-1) otherwise.

Original entry on oeis.org

1, 2, 4, 12, 24, 48, 96, 192, 384, 768, 1536, 4608, 9216, 18432, 36864, 73728, 147456, 294912, 589824, 1179648, 2359296, 4718592, 9437184, 28311552, 56623104, 113246208, 226492416, 452984832, 905969664, 1811939328, 3623878656

Offset: 1

Benoit Cloitre, Feb 14 2003

nonn

Inspired by A079000. Cf. A064437.

a(n)=3*(3/2)^floor((log(n)-log(3))/log(2))*2^n

a(n+1)=3*a(n) for n=3 n of the form 3*2^k - 1, k>=2 . a(n+1)=2*a(n) otherwise. Hence a(n)=3*(3/2)^floor((log(n/3))/log(2))*2^n.

A079358 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is not a multiple of either 3 or 4.".

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 11, 12, 13, 14, 17, 19, 22, 24, 27, 29, 30, 31, 32, 33, 34, 36, 37, 39, 40, 41, 42, 43, 46, 47, 49, 50, 53, 54, 55, 58, 60, 61, 62, 65, 67, 70, 72, 75, 77, 79, 80, 82, 83, 84, 87, 89, 91, 94, 96, 99, 101, 102, 103, 106, 107, 108, 111, 113

Offset: 1

Matthew Vandermast, Feb 14 2003

A generalization of A079000 that, like A079000 itself, is based on a class of numbers comprising exactly one-half of the integers.

			a(3) cannot be 3 because that would imply that the third term is not a multiple of 3. 4 is the smallest possible value for a(3) that creates no contradiction; therefore a(3)=4 and the fourth term is the next member of the sequence that is not a multiple of 3 or 4.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Benoit Cloitre, N. J. A. Sloane and Matthew J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2. See also arXiv:math/0305308 [math.NT], 2003.

Cf. A079000.

Block[{c, k, nn, s}, nn = 2^16; c[] := False; k = 1; c[1] = True; s = {0, 3, 4, 6, 8, 9}; {1}~Join~Reap[Do[k++; Which[c[n], While[MemberQ[s, Mod[k, 12]], k++], k == n, If[MemberQ[s, Mod[k, 12]], k++], True, While[FreeQ[s, Mod[k, 12]], k++] ]; Sow[k]; c[k] = True, {n, 2, nn}] ][[-1, 1]] ] (* _Michael De Vlieger, Aug 10 2025 *)

a(42) onward corrected by Sean A. Irvine, Aug 10 2025

A080033 a(n) is taken to be the smallest positive integer not already present which is consistent with the condition "n is a member of the sequence if and only if a(n) is a multiple of 4".

Original entry on oeis.org

0, 2, 4, 5, 8, 12, 7, 16, 20, 10, 24, 13, 28, 32, 15, 36, 40, 18, 44, 21, 48, 52, 23, 56, 60, 26, 64, 29, 68, 72, 31, 76, 80, 34, 84, 37, 88, 92, 39, 96, 100, 42, 104, 45, 108, 112, 47, 116, 120, 50, 124, 53, 128, 132, 55, 136, 140, 58, 144, 61, 148, 152, 63, 156, 160, 66

Offset: 0

N. J. A. Sloane, Mar 14 2003

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)

Cf. A079000, A080030, A080032, A079313, A080034.

a(8m)=20m, a(8m+1)=8m+2, a(8m+2)=20m+4, a(8m+3)=8m+5, a(8m+4)=20m+8, a(8m+5)=20m+12, a(8m+6)=8m+7, a(8m+7)=20m+16.
From Chai Wah Wu, Sep 27 2016: (Start)
a(n) = 2*a(n-8) - a(n-16) for n > 15.
G.f.: x*(4*x^14 + x^13 + 8*x^12 + 12*x^11 + 3*x^10 + 16*x^9 + 6*x^8 + 20*x^7 + 16*x^6 + 7*x^5 + 12*x^4 + 8*x^3 + 5*x^2 + 4*x + 2)/(x^16 - 2*x^8 + 1). (End)

More terms from Matthew Vandermast, Mar 23 2003

A080591 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is congruent to 3 mod 4".

Original entry on oeis.org

1, 3, 4, 7, 11, 12, 13, 15, 16, 17, 18, 19, 23, 27, 28, 31, 35, 39, 43, 47, 48, 49, 50, 51, 52, 53, 54, 55, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 83, 87, 91, 95, 99, 103, 107, 111, 112, 113, 114, 115, 119, 123, 124

Offset: 0

N. J. A. Sloane, Feb 23 2003

The sequence of odd numbers shares many of the properties of this sequence.

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)
Index entries for sequences of the a(a(n)) = 2n family

a(n) = A080588(n+1) - 1. Cf. A079000.

a(a(n)) = 4n+3. a(2^k-1) = 2^(k+1)-1.

cp's OEIS Frontend

A080031 a(n) is taken to be the smallest positive integer not already present which is consistent with the condition "n is a member of the sequence if and only if a(n) is congruent to 2 mod 3".

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A080711 a(0) = 2; for n>0, a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is a multiple of 3".

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A099797 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is composite".

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A099798 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is not composite".

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A079255 a(n) is taken to be the smallest positive integer greater than a(n-1) such that the condition "n is in the sequence if and only if a(n) is odd and a(n+1) is even" can be satisfied.

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A079351 a(1)=3; for n > 1, a(n) is the smallest integer greater than a(n-1) consistent with the condition "n is in the sequence if and only if a(n) is congruent to 0 (mod 5)".

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A079352 a(1)=1, then a(n)=3*a(n-1) if n is already in the sequence, a(n)=2*a(n-1) otherwise.

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A079358 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is not a multiple of either 3 or 4.".

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A080033 a(n) is taken to be the smallest positive integer not already present which is consistent with the condition "n is a member of the sequence if and only if a(n) is a multiple of 4".

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A080591 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is congruent to 3 mod 4".

A079352 a(1)=1, then a(n)=3a(n-1) if n is already in the sequence, a(n)=2a(n-1) otherwise.