A026477 -id:A026477 - cp's OEIS frontend

A062090 a(1) = 1, a(n) = smallest odd number that does not divide the product of all previous terms.

1, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241

Offset: 1

Amarnath Murthy, Jun 16 2001

In A050150 but not here: [729, 15625, 59049, 117649, 531441]; here but not in A050150: [1, 6561, 390625]. - Klaus Brockhaus, Nov 01 2001

If "a(1) = 1," is removed from the definition, the subsequent terms remain the same, since 1 is the empty product. The resulting sequence then comprises the odd terms of A050376. - Peter Munn, Nov 03 2020

			After 13 the next term is 17 (not 15) as 15 = 3*5 divides the product of all the previous terms.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A026477, A062091, A050150 (a different sequence).
Odd terms of {1} U A050376.
Subsequence of A336882.

a062090 n = a062090_list !! (n-1)
a062090_list = f [1, 3 ..] [] where
   f (x:xs) ys = g x ys where
     g _ []     = x : f xs (x : ys)
     g 1 _      = f xs ys
     g z (v:vs) = g (z `div` gcd z v) vs
-- Reinhard Zumkeller, Aug 16 2013

a = {1}; Do[b = Apply[ Times, a]; k = 1; While[ IntegerQ[b/k], k += 2]; a = Append[a, k], { n, 2, 60} ]; a
nxt[{p_,on_}]:=Module[{c=on+2},While[Divisible[p,c],c+=2];{p*c,c}]; NestList[ nxt,{1,1},60][[All,2]] (* Harvey P. Dale, Jul 29 2021 *)

1 together with numbers of the form p^(2^k) where p is an odd prime and k is a nonnegative integer. [Corrected by Peter Munn, Nov 03 2020]

For n >= 2, a(n) = A336882(2^(n-2)). - Peter Munn, Nov 03 2020

Corrected and extended by Dean Hickerson, Jul 10 2001

A066724 a(1) = 1, a(2) = 2; for n > 1, a(n) is the least integer > a(n-1) such that the products a(i)*a(j) for 1 <= i < j <= n are all distinct.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, 23, 25, 29, 30, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 84, 89, 97, 101, 103, 107, 109, 113, 121, 127, 128, 131, 137, 139, 149, 151, 154, 157, 163, 167, 169, 173, 179, 180, 181, 191, 193, 197, 199, 211

Offset: 1

Robert E. Sawyer (rs.1(AT)mindspring.com), Jan 18 2002

The first 15 terms are the same as A026477; the first 13 terms are the same as A026416.

Contains all primes. - Ivan Neretin, Mar 02 2016

			a(7) is not 10 because we already have 10 = 2*5. Of course all primes appear. a(14) is not 24 because if it were there would be a repeat among the terms a(i)*a(j) for 1 <= i < j <= 14, namely 3*16 = 2*24.

Ivan Neretin, Table of n, a(n) for n = 1..1000

Cf. A000028, A026477, A026416, A050376, A084400.

Cf. A080431, A066720.

f[l_List] := Block[{k = 1, p = Times @@@ Subsets[l, {2}]},While[Intersection[p, l*k] != {}, k++ ];Append[l, k]];Nest[f, {1, 2}, 62] (* Ray Chandler, Feb 12 2007 *)

A026479 a(n) = least positive integer > a(n-1) and not a(i)a(j)a(k) for 1<=i

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 9, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 33, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 67, 69, 71, 73, 79, 81, 83, 87, 89, 93, 97, 101, 103, 107, 109, 111, 113, 121, 123, 127, 129, 131, 137, 139, 141, 149, 151, 157, 159, 163

Offset: 1

Clark Kimberling

nonn

Robert Price, Table of n, a(n) for n = 1..203

There are six related sequences: A026477: 1 <= i < j < k < n starting 1,2,3; A026478: 1 <= i <= j <= k < n starting 1,2,3; A026479: 1 <= i < j < k < n starting 1,2,4; A026480: 1 <= i <= j <= k < n starting 1,2,4; A026481: 1 <= i < j < k < n starting 1,3,4; A026482: 1 <= i <= j <= k < n starting 1,3,4.

a = {1, 2, 4}; no = {1 2 4};
Do[x = SelectFirst[Range[Last[a] + 1, 1000], ! MemberQ[no, #] &]; AppendTo[a, x]; no = Union[Times @@@ Subsets[a, {3}]], 200]; a (* Robert Price, May 26 2019 *)

a(31)-a(61) from Robert Price, May 26 2019

A026480 a(1) = 1, a(2) = 2, a(3) = 4, and for n > 3, a(n) is the least positive integer > a(n-1) not of form a(i)a(j)a(k) for 1 <= i <= j <= k < n.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 9, 11, 13, 15, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 51, 53, 57, 59, 61, 67, 69, 71, 73, 79, 83, 87, 89, 93, 97, 101, 103, 107, 109, 111, 113, 123, 127, 128, 129, 131, 137, 139, 141, 149, 151, 157, 159, 160, 163, 167, 173, 177, 179, 181

Offset: 1

Clark Kimberling

nonn

From David A. Corneth, Sep 24 2016 (Start):
Let t be a term of this sequence. Let v = (i, j, m) be a vector of three elements where i is A007814(t), j = A007949(t) and m = A001222(t) - i - j. Then possible vectors v for t up to 100000 are {[0, 0, 0], [0, 0, 1], [0, 0, 4], [0, 0, 7], [0, 1, 1], [0, 1, 4], [0, 2, 0], [0, 3, 0], [0, 4, 3], [0, 5, 3], [0, 6, 2], [0, 7, 2], [0, 8, 1], [0, 9, 1], [0, 10, 0], [1, 0, 0], [1, 0, 3], [1, 0, 6], [1, 1, 0], [1, 1, 3], [1, 1, 6], [1, 4, 2], [1, 5, 2], [1, 6, 1], [1, 7, 1], [1, 8, 0], [1, 9, 0], [2, 0, 0], [2, 3, 2], [2, 5, 1], [2, 7, 0], [3, 0, 2], [3, 0, 5], [3, 1, 2], [3, 1, 5], [3, 3, 1], [3, 4, 1], [3, 5, 0], [3, 6, 0], [4, 2, 1], [4, 2, 4], [4, 4, 0], [5, 0, 1], [5, 0, 4], [5, 1, 1], [5, 1, 4], [5, 2, 0], [5, 3, 0], [6, 0, 3], [6, 1, 0], [6, 1, 3], [7, 0, 0], [7, 3, 2], [8, 0, 2], [8, 1, 2], [8, 3, 1], [8, 5, 0], [9, 2, 1], [9, 4, 0], [10, 0, 1], [10, 1, 1], [10, 2, 0], [10, 3, 0], [11, 1, 0], [12, 0, 0]}.
(End)

David A. Corneth, Table of n, a(n) for n = 1..23405 (all terms up to 10^5).
David A. Corneth, PARI program

There are six related sequences: A026477: 1 <= i < j < k < n starting 1,2,3; A026478: 1 <= i <= j <= k < n starting 1,2,3; A026479: 1 <= i < j < k < n starting 1,2,4; A026480: 1 <= i <= j <= k < n starting 1,2,4; A026481: 1 <= i < j < k < n starting 1,3,4; A026482: 1 <= i <= j <= k < n starting 1,3,4.

a = {1, 2, 4}; no = {1 2 4};
Do[x = SelectFirst[Range[Last[a] + 1, 1000], ! MemberQ[no, #] &]; AppendTo[a, x]; no = Union[Times @@@ Tuples[a, {3}]], 60]; a (* Robert Price, May 26 2019 *)

Name corrected by Charles R Greathouse IV, Sep 23 2016

A026481 a(n) = least positive integer > a(n-1) and not a(i)a(j)a(k) for 1<=i

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 22, 23, 25, 26, 29, 31, 34, 37, 38, 41, 43, 46, 47, 49, 53, 58, 59, 61, 62, 67, 71, 73, 74, 79, 81, 82, 83, 86, 89, 94, 97, 101, 103, 106, 107, 109, 113, 118, 121, 122, 127, 131, 134, 137, 139, 142, 146, 149, 151, 157

Offset: 1

Clark Kimberling

There are six related sequences: A026477: 1 <= i < j < k < n starting 1,2,3; A026478: 1 <= i <= j <= k < n starting 1,2,3; A026479: 1 <= i < j < k < n starting 1,2,4; A026480: 1 <= i <= j <= k < n starting 1,2,4; A026481: 1 <= i < j < k < n starting 1,3,4; A026482: 1 <= i <= j <= k < n starting 1,3,4.

a = {1, 3, 4}; no = {1 3 4};
Do[x = SelectFirst[Range[Last[a] + 1, 1000], ! MemberQ[no, #] &]; AppendTo[a, x]; no = Union[Times @@@ Subsets[a, {3}]], 62]; a (* Robert Price, May 26 2019 *)

More terms from Todd Will (towill(AT)davidson.edu).

A026482 a(n) = least positive integer > a(n-1) and not equal to a(i)a(j)a(k) for 1<=i<=j<=k

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 41, 43, 46, 47, 53, 58, 59, 61, 62, 67, 71, 73, 74, 79, 81, 82, 83, 86, 89, 94, 97, 101, 103, 106, 107, 109, 113, 118, 122, 127, 131, 134, 135, 137, 139, 142, 146

Offset: 1

Clark Kimberling

There are six related sequences: A026477: 1 <= i < j < k < n starting 1,2,3; A026478: 1 <= i <= j <= k < n starting 1,2,3; A026479: 1 <= i < j < k < n starting 1,2,4; A026480: 1 <= i <= j <= k < n starting 1,2,4; A026481: 1 <= i < j < k < n starting 1,3,4; A026482: 1 <= i <= j <= k < n starting 1,3,4.

a = {1, 3, 4}; no = {1 3 4};
Do[x = SelectFirst[Range[Last[a] + 1, 1000], ! MemberQ[no, #] &]; AppendTo[a, x]; no = Union[Times @@@ Tuples[a, {3}]], 60]; a (* Robert Price, May 26 2019 *)

More terms from Naohiro Nomoto, Sep 15 2001

A079852 a(1) = 1, a(2) = 2, a(3) = 3 and a(n) is the smallest number such that all a(i)a(j)a(k) are different.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 210, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 289, 293, 307, 311, 313, 317, 331

Offset: 1

Amarnath Murthy, Feb 19 2003

nonn

Note that a(57) = 210 = 2*3*5*7, while 330 = 2*3*5*11 is not in the sequence. This demonstrates that this sequence is not determined by prime signature alone. - Charles R Greathouse IV, Oct 17 2015

Zak Seidov, Table of n, a(n) for n = 1..200

Cf. A000045, A079850, A066720, A026477.

f[l_List] := Block[{k = 1,p2 = Times @@@ Subsets[l, {2}], p3 = Times @@@ Subsets[l, {3}]},While[Intersection[p3, p2*k] != {}, k++ ]; Append[l, k]]; Nest[f, {1, 2, 3}, 62] (* Ray Chandler, Feb 12 2007 *)

Extended by Ray Chandler, Feb 12 2007

Typo in name fixed by Zak Seidov, Jul 06 2013

A062091 a(1) = 2, a(n)= smallest even number which does not divide the product of all previous terms.

Original entry on oeis.org

2, 4, 6, 10, 14, 18, 22, 26, 34, 38, 46, 50, 58, 62, 74, 82, 86, 94, 98, 106, 118, 122, 134, 142, 146, 158, 162, 166, 178, 194, 202, 206, 214, 218, 226, 242, 254, 262, 274, 278, 298, 302, 314, 326, 334, 338, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454

Offset: 1

Amarnath Murthy, Jun 16 2001

nonn

			After 10 the next term in the sequence is 14 (not 12) as 12 = 2*6 divides the product of all the previous terms.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A026477, A062090.

Block[{a = {2}, k = 4, P = 1}, Do[Set[P, P*a[[-1]]]; While[Mod[P, k] == 0, k += 2]; AppendTo[a, k], {i, 2, 56}]; a] (* Michael De Vlieger, Nov 04 2020 *)

2 and 4 together with numbers of the form 2*{p^(2^k)} where p is an odd prime and k is a nonnegative integer. [Corrected by Peter Munn, Nov 03 2020]

Conjecture: a(n) = 2*A062090(n-1) for n>=3. - R. J. Mathar, Jul 08 2025

More terms from Dean Hickerson, Jul 10 2001

A276382 a(1) = 1, and a(n) = a(n-1) + floor(3*n/2) + 1 for n >= 2.

Original entry on oeis.org

1, 5, 10, 17, 25, 35, 46, 59, 73, 89, 106, 125, 145, 167, 190, 215, 241, 269, 298, 329, 361, 395, 430, 467, 505, 545, 586, 629, 673, 719, 766, 815, 865, 917, 970, 1025, 1081, 1139, 1198, 1259, 1321, 1385, 1450, 1517, 1585, 1655, 1726, 1799, 1873, 1949, 2026

Offset: 1

Bob Selcoe, Sep 01 2016

Given 3 distinct numbers i, j and k whose prime signatures are exactly n 1's, then a(n) is the number of prime signatures for all permutations of i*j*k.
a(n) is the number of partitions of 3n such that there are no more than n-1 3's and no parts > 3.
a(3n+1) represents the number of prime signature sets whose members are excluded as terms in A026477, as a consequence of being products of three smaller terms whose prime signatures are exactly 3n+1 1's.
First differences are floor(3n/2) + 1 (A001651(n+1)); second differences are 1 when n is even and 2 when n is odd; third differences are 1 when n is even and -1 when n is odd.

			a(2)=5; the 5 prime signatures / partitions are: {3,1,1,1}, {2,2,2}, {2,2,1,1}, {2,1,1,1} and {1,1,1,1,1,1}.
G.f. = x + 5*x^2 + 10*x^3 + 17*x^4 + 25*x^5 + 35*x^6 + 46*x^7 + ... - _Michael Somos_, Sep 08 2023

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000040 (prime numbers), A001651, A026477.

Equals one less than A331952(n+1), two less than A077043(n+1), and three less than A084684(n+1). - Greg Dresden, Feb 22 2020

[((-1)^n+12*n+6*n^2-9)/8: n in [1..60]]; // Vincenzo Librandi, Sep 10 2016

a[1] = 1; a[n_] := a[n] = a[n - 1] + Floor[3 n/2] + 1 ; Array[a, 51] (* Michael De Vlieger, Sep 01 2016 *)
a[n_] := Floor[(3*(n+1)^2 - 7)/4]; (* Michael Somos, Sep 08 2023 *)

Vec(x*(1+3*x-x^3)/((1-x)^3*(1+x)) + O(x^60)) \\ Colin Barker, Sep 01 2016

{a(n) = (3*(n+1)^2 - 7)\4}; /* Michael Somos, Sep 09 2023 */

From Colin Barker, Sep 01 2016: (Start)
a(n) = ((-1)^n + 12*n + 6*n^2 - 9)/8 for n > 0.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n > 4.
G.f.: x*(1 + 3*x - x^3) / ((1-x)^3*(1+x)). (End)
a(n) = -1 + Sum_{k=1..n} floor((n+k+3)/2). - Wesley Ivan Hurt, Apr 01 2017
a(n) = a(-2-n) for all n in Z. - Michael Somos, Sep 08 2023
a(n) = floor((3*(n+1)^2 - 7)/4). - Michael Somos, Sep 09 2023

A026488 a(n) is the least positive integer > a(n-1) and not a(i)*a(j)-a(k) for 1 <= i <= j <= k <= n, where a(1) = 1.

Original entry on oeis.org

1, 3, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 82, 85, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, 130, 133, 136, 139, 142, 145, 148, 151, 154, 157, 160, 163, 166, 169, 172

Offset: 1

Clark Kimberling

nonn

Conjecture: This is the increasing sequence of numbers congruent to 1 mod 3, with 3 inserted.

Conjecture holds for n <= 1300. - Sean A. Irvine, Oct 12 2019

Cf. A026477, A016777.

Corrected and extended by Clark Kimberling, Oct 12 2019

cp's OEIS Frontend

A062090 a(1) = 1, a(n) = smallest odd number that does not divide the product of all previous terms.

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A066724 a(1) = 1, a(2) = 2; for n > 1, a(n) is the least integer > a(n-1) such that the products a(i)*a(j) for 1 <= i < j <= n are all distinct.

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A026479 a(n) = least positive integer > a(n-1) and not a(i)*a(j)*a(k) for 1<=i

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A026480 a(1) = 1, a(2) = 2, a(3) = 4, and for n > 3, a(n) is the least positive integer > a(n-1) not of form a(i)*a(j)*a(k) for 1 <= i <= j <= k < n.

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A026481 a(n) = least positive integer > a(n-1) and not a(i)*a(j)*a(k) for 1<=i

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A026482 a(n) = least positive integer > a(n-1) and not equal to a(i)*a(j)*a(k) for 1<=i<=j<=k

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A079852 a(1) = 1, a(2) = 2, a(3) = 3 and a(n) is the smallest number such that all a(i)*a(j)*a(k) are different.

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A062091 a(1) = 2, a(n)= smallest even number which does not divide the product of all previous terms.

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A026479 a(n) = least positive integer > a(n-1) and not a(i)a(j)a(k) for 1<=i

A026480 a(1) = 1, a(2) = 2, a(3) = 4, and for n > 3, a(n) is the least positive integer > a(n-1) not of form a(i)a(j)a(k) for 1 <= i <= j <= k < n.

A026481 a(n) = least positive integer > a(n-1) and not a(i)a(j)a(k) for 1<=i

A026482 a(n) = least positive integer > a(n-1) and not equal to a(i)a(j)a(k) for 1<=i<=j<=k

A079852 a(1) = 1, a(2) = 2, a(3) = 3 and a(n) is the smallest number such that all a(i)a(j)a(k) are different.