A026478 -id:A026478 - cp's OEIS frontend

A026477 a(1) = 1, a(2) = 2, a(3) = 3; and for n > 3, a(n) = smallest number > a(n-1) and not of the form a(i)a(j)a(k) for 1 <= i < j < k < n.

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, 120, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 168, 169, 173, 179, 181, 191, 193, 197, 199, 210, 211, 216, 223

Offset: 1

Clark Kimberling

nonn

From Bob Selcoe, Aug 25 2016: (Start)
Once a term with a given prime signature S (i.e., multiset of prime exponents) appears, then all numbers with the same prime signature follow. So either all or no terms with the same prime signature appear (first conjectured by Charles R Greathouse IV).
Proof:
i. Let S = {i,j,k..,y}, i>=j>=k.. be prime signatures, i.e., numbers of the form (p^i*q^j*r^k..*z^y) where {p,q,r,..,z} are distinct primes; denote S = {} as the signature for p^0 = 1.
ii. By definition, 1 and all primes p appear in the sequence; so terms where S = {k} (i.e., one-digit signatures denoting prime powers p^k) are k=0 and k = A026474(n) = {1,2,4,8,15,22,29..}, because k cannot be the sum of any combination of 3 smaller k. So the only prime power terms are p^0 = 1 and p^A026474(n), or S = {}, {1}, {2}, {4}, {8}, {15}, {22}, {29}...
iii. By induction, once S = {k} is determined, all other terms with the same prime signature appear (or do not) depending on the various combinations of signature exponents of previous terms with smaller signature sums. So for example, terms with the same two-digit signature S = {k,k} (i.e. (pq)^k) are constrained by the following: since p = {1} and q = {1} appear, then (pq) = {1,1} does not because in this case {1}*{1}*{} = {1,1}; since p^2 = {2} and q^2 = {2} appear, then (pq)^2 = {2,2} does not; since {4} appears but {3} does not, then {3,3} appears but {4,4} and {5,5} do not (the latter because {3,3}*{2}*{2} = {5,5} when p^2,q^2 = {2}). Note that {3,3} would not appear if {3,2} appeared because {3,2}*{1}*{} = {3,3} when q = {1}; but because p = {1} and p^2,q^2 = {2} appear, then {2}*{2}*{1} = {3,2} does not. {6,6} does not appear because {6,5} appears (by virtue of other constraints) and {6,5}*{1}*{} = {6,6} when q = {1}. Determining which signatures appear and which do not becomes increasingly complicated as the sequence increases.
Don Reble offered a proof on the Sequence Fans Mailing List which seems to be different (and certainly more formal) than mine. Perhaps mine is more of an "explanation" than a "proof"? (End)

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000028, A026416, A066724, A050376, A084400, A026474.

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, 3}; no = {1 2 3};
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 *)

list(lim)=my(v=List(),n,d,k); while(n++<=lim, d=divisors(n); for(i=1,#d-2, if(!setsearch(v,d[i]), next); for(j=i+1,#d-1, if(!setsearch(v,d[j]), next); k=n/(d[i]*d[j]); if(d[j]>=k, break); if(denominator(k)==1 && setsearch(v,k), next(3)))); listput(v,n)); Vec(v) \\ Charles R Greathouse IV, Sep 16 2015

More terms from Christian G. Bower, Nov 15 1999

A332821 One part of a 3-way classification of the positive integers. Numbers n for which A048675(n) == 1 (mod 3).

Original entry on oeis.org

2, 5, 9, 11, 12, 16, 17, 21, 23, 28, 30, 31, 39, 40, 41, 47, 49, 52, 54, 57, 59, 66, 67, 70, 72, 73, 75, 76, 83, 87, 88, 91, 96, 97, 100, 102, 103, 109, 111, 116, 126, 127, 128, 129, 130, 133, 135, 136, 137, 138, 148, 149, 154, 157, 159, 165, 167, 168, 169, 172, 175, 179, 180, 183, 184, 186, 190, 191, 197, 203, 211, 212

Offset: 1

Antti Karttunen and Peter Munn, Feb 25 2020

nonn

The positive integers are partitioned between A332820, this sequence and A332822.
For each prime p, the terms include exactly one of p and p^2. The primes alternate between this sequence and A332822. This sequence has the primes with odd indexes, those in A031368.
The terms are the even numbers in A332822 halved. The terms are also the numbers m such that 5m is in A332822, and so on for alternate primes: 11, 17, 23 etc. Likewise, the terms are the numbers m such that 3m is in A332820, and so on for alternate primes: 7, 13, 19 etc.
The numbers that are half of the even terms of this sequence are in A332820, which consists exactly of those numbers. The numbers that are one third of the terms that are multiples of 3 are in A332822, which consists exactly of those numbers. For larger primes, an alternating pattern applies as described in the previous paragraph.
If we take each odd term of this sequence and replace each prime in its factorization by the next smaller prime, the resulting number is in A332822, which consists entirely of those numbers.
The product of any 2 terms of this sequence is in A332822, the product of any 3 terms is in A332820, and the product of a term of A332820 and a term of this sequence is in this sequence. So if a number k is present, k^2 is in A332822, k^3 is in A332820, and k^4 is in this sequence.
If k is an even number, exactly one of {k/2, k, 2k} is in the sequence (cf. A191257 / A067368 / A213258); and generally if k is a multiple of a prime p, exactly one of {k/p, k, k*p} is in the sequence.

Positions of ones in A332823; equivalently, numbers in row 3k+1 of A277905 for some k >= 0.
Cf. A048675, A332820, A332822.
Comparable 2 or 3-way classifications: A000379/A000028, A001969/A000069, A003159/A036554, A005843/A005408, A028260/A026424, A191257/A067368/A213258, A325431/A325432, A329609/A329604/A332812.
Subsequences: intersection of A026478 and A066208, A031368 (prime terms), A033431\{0}, A052934\{1}, A069486, A099800, A167747\{1}, A244725\{0}, A244728\{0}, A338911 (semiprime terms).

Select[Range@ 212, Mod[Total@ #, 3] == 1 &@ Map[#[[-1]]*2^(PrimePi@ #[[1]] - 1) &, FactorInteger[#]] &] (* Michael De Vlieger, Mar 15 2020 *)

isA332821(n) =  { my(f = factor(n)); (1==((sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3)); };

{a(n) : n >= 1} = {2 * A332820(k) : k >= 1} U {A003961(A332822(k)) : k >= 1}.

{a(n) : n >= 1} = {A332822(k)^2 : k >= 1} U {A331590(2, A332820(k)) : k >= 1}.

A332822 One part of a 3-way classification of the positive integers. Numbers n for which A048675(n) == 2 (mod 3).

Original entry on oeis.org

3, 4, 7, 10, 13, 18, 19, 22, 24, 25, 29, 32, 34, 37, 42, 43, 45, 46, 53, 55, 56, 60, 61, 62, 71, 78, 79, 80, 81, 82, 85, 89, 94, 98, 99, 101, 104, 105, 107, 108, 113, 114, 115, 118, 121, 131, 132, 134, 139, 140, 144, 146, 150, 151, 152, 153, 155, 163, 166, 173, 174, 176, 181, 182, 187, 189, 192, 193, 194, 195, 199, 200, 204

Offset: 1

Antti Karttunen and Peter Munn, Feb 25 2020

nonn

The positive integers are partitioned between A332820, A332821 and this sequence.
For each prime p, the terms include exactly one of p and p^2. The primes alternate between this sequence and A332821. This sequence has the primes with even indexes, those in A031215.
The terms are the even numbers in A332820 halved. The terms are also the numbers m such that 5m is in A332820, and so on for alternate primes: 11, 17, 23 etc. Likewise, the terms are the numbers m such that 3m is in A332821, and so on for alternate primes: 7, 13, 19, 29 etc.
If we take each odd term of this sequence and replace each prime in its factorization by the next smaller prime, we get the same set of numbers as we get from halving the even terms of this sequence, and A332821 consists exactly of those numbers. The numbers that are one third of the terms that are multiples of 3 are in A332820, which consists exactly of those numbers. The numbers that are one fifth of the terms that are multiples of 5 constitute A332821, and for larger primes, an alternating pattern applies as described in the previous paragraph.
The product of any 2 terms of this sequence is in A332821, the product of any 3 terms is in A332820, and the product of a term of A332820 and a term of this sequence is in this sequence. So if a number k is present, k^2 is in A332821, k^3 is in A332820, and k^4 is in this sequence.
If k is an even number, exactly one of {k/2, k, 2k} is in the sequence (cf. A191257 / A067368 / A213258); and generally if k is a multiple of a prime p, exactly one of {k/p, k, k*p} is in the sequence.

Positions of terms valued -1 in A332823; equivalently, numbers in row 3k-1 of A277905 for some k >= 1.
Cf. A048675, A332820, A332821.
Comparable 2 or 3-way classifications: A000379/A000028, A001969/A000069, A003159/A036554, A005843/A005408, A028260/A026424, A191257/A067368/A213258, A325431/A325432, A329609/A329604/A332812.
Subsequences: intersection of A026478 and A066207, A031215 (prime terms), A033430\{0}, A117642\{0}, A169604, A244727\{0}, A244729\{0}, A338910 (semiprime terms).

Select[Range@ 204, Mod[Total@ #, 3] == 2 &@ Map[#[[-1]]*2^(PrimePi@ #[[1]] - 1) &, FactorInteger[#]] &] (* Michael De Vlieger, Mar 15 2020 *)

isA332822(n) =  { my(f = factor(n)); (2==((sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3)); };

{a(n) : n >= 1} = {2 * A332821(k) : k >= 1} U {A003961(A332821(k)) : k >= 1}.

{a(n) : n >= 1} = {A332821(k)^2 : k >= 1} U {A331590(2, A332821(k)) : k >= 1}.

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

A323520 Numbers of the form p^(k^2) where p is prime and k >= 0.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 16, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263

Offset: 1

Gus Wiseman, Jan 17 2019

nonn

Cf. A000290, A000961, A001248, A001597, A026478, A050376, A277562, A323519, A323526, A323528.

Select[Range[100],#==1||PrimePowerQ[#]&&IntegerQ[Sqrt[FactorInteger[#][[1,2]]]]&]

A323525 Number of ways to arrange the parts of a multiset whose multiplicities are the prime indices of n into a square matrix.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 1, 0, 6, 4, 0, 12, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 36, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 84, 0, 0, 72, 0, 0, 0, 0, 0, 0, 0, 0, 126, 252, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Jan 17 2019

nonn

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			The a(9) = 6 matrices:
  [1 1] [1 2] [1 2] [2 1] [2 1] [2 2]
  [2 2] [1 2] [2 1] [1 2] [2 1] [1 1]
The a(38) = 9 matrices:
  [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 2] [1 2 1] [2 1 1]
  [1 1 1] [1 1 1] [1 1 1] [1 1 2] [1 2 1] [2 1 1] [1 1 1] [1 1 1] [1 1 1]
  [1 1 2] [1 2 1] [2 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1]

The positions of 0's are numbers whose sum of prime indices is not a perfect square (A323527).
The positions of 1's are primes indexed by squares (A323526).
Cf. A008480, A026478, A056239, A103198, A112798, A120732, A318284, A318762, A323519, A323528, A323531.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,Reverse[primeMS[n]]];
Table[If[IntegerQ[Sqrt[Total[primeMS[n]]]],Length[Permutations[nrmptn[n]]],0],{n,100}]

If A056239(n) is a perfect square, a(n) = A318762(n). Otherwise, a(n) = 0.

A323519 a(n) is the number of ways to fill a square matrix with the multiset of prime factors of n, if the number of prime factors (counted with multiplicity) is a perfect square, and a(n) = 0 otherwise.

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 4, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 6, 1, 0, 0, 4, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 4, 0, 4, 0, 0, 1, 12, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 12, 0, 0

Offset: 1

Gus Wiseman, Jan 17 2019

nonn

			The a(60) = 12 matrices:
  [2 2] [2 2] [2 3] [2 3] [2 5] [2 5] [3 2] [3 2] [3 5] [5 2] [5 2] [5 3]
  [3 5] [5 3] [2 5] [5 2] [2 3] [3 2] [2 5] [5 2] [2 2] [2 3] [3 2] [2 2]

Positions of 0's are A323521.
Positions of 1's are A323520.
Cf. A000290, A008480, A026478, A056239, A089299, A103198, A112798, A120732, A323433, A323525, A323531.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[IntegerQ[Sqrt[PrimeOmega[n]]],Length[Permutations[primeMS[n]]],0],{n,100}]

If A001222(n) is a perfect square, then a(n) = A008480(n). Otherwise, a(n) = 0.

cp's OEIS Frontend

A026477 a(1) = 1, a(2) = 2, a(3) = 3; and for n > 3, a(n) = smallest number > a(n-1) and not of the form a(i)*a(j)*a(k) for 1 <= i < j < k < n.

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A332821 One part of a 3-way classification of the positive integers. Numbers n for which A048675(n) == 1 (mod 3).

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A332822 One part of a 3-way classification of the positive integers. Numbers n for which A048675(n) == 2 (mod 3).

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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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A323520 Numbers of the form p^(k^2) where p is prime and k >= 0.

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A323525 Number of ways to arrange the parts of a multiset whose multiplicities are the prime indices of n into a square matrix.

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Examples

A026477 a(1) = 1, a(2) = 2, a(3) = 3; and for n > 3, a(n) = smallest number > a(n-1) and not of the form a(i)a(j)a(k) for 1 <= i < j < k < n.

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