A026416 -id:A026416 - cp's OEIS frontend

A026417 a(n) = least positive integer > a(n-1) and not a(i)*a(j) for 1<=i

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, 105, 106, 107, 108, 109, 113, 118, 120, 121, 122, 127

Offset: 1

Clark Kimberling

nonn

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

Cf. A026416 and references therein.

a = {1, 3}; used = {a[[1]]*a[[2]]}; Do[k = a[[-1]] + 1; While[MemberQ[used, k], k++]; used = Union[used, k*a]; AppendTo[a, k], {n, 3, 60}]; a (* Ivan Neretin, Mar 07 2016 *)

Name clarified by Robert C. Lyons, Feb 08 2025

A026422 a(n) = least positive integer > a(n-1) and not a(i)*a(j) for 1 <= i <= j < n.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 11, 12, 13, 17, 18, 19, 20, 23, 27, 28, 29, 30, 31, 32, 37, 41, 42, 43, 44, 45, 47, 48, 50, 52, 53, 59, 61, 63, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 83, 89, 92, 97, 98, 99, 101, 102, 103, 105, 107, 108, 109, 110

Offset: 1

Clark Kimberling

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Apart from initial term, same as A026424.

Cf. A026416 and references therein.

a[1]=1; a[n_] := a[n] = (t= Union[Flatten[Table[a[i] a[j], {i, 1, n-1}, {j, i, n-1}]]]; Do[If[FreeQ[t, k], an = k; Break[]], {k, a[n-1]+1, Last[t]+1}]; an); Array[a, 60] (* Jean-François Alcover, May 06 2011 *)
Select[Range[110], OddQ[Total[FactorInteger[#]][[2]]] &] (* T. D. Noe, May 07 2011 *)
g = 110; t = Array[1 &, g];
Table[If[t[[j]] == 1, t[[j*i]] = 0, t[[i*j]] = 1], {j, 2, g/2}, {i, 2, g/j}]; Flatten[Position[t, 1]] (* Horst H. Manninger, Mar 15 2023 *)

is(n)=bigomega(n)%2 || n==1 \\ Charles R Greathouse IV, Sep 16 2015

from math import prod, isqrt
from sympy import primerange, primepi, integer_nthroot
def A026422(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,m)) for m in range(2,x.bit_length()+1,2)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Apr 10 2025

Name corrected by Charles R Greathouse IV, Sep 16 2015

A026423 a(n) = least positive integer > a(n-1) and not a(i)*a(j) for 1<=i<=j<=n, n >= 2.

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 17, 19, 22, 23, 26, 27, 29, 31, 34, 37, 38, 41, 43, 45, 46, 47, 53, 54, 58, 59, 61, 62, 63, 67, 71, 72, 73, 74, 75, 79, 82, 83, 86, 89, 90, 94, 96, 97, 99, 101, 103, 105, 106, 107, 109, 113, 117, 118, 120

Offset: 1

Clark Kimberling

nonn

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

Cf. A026416 and references therein.

a = {1, 3}; used = {a[[1]]^2, a[[1]]*a[[2]], a[[2]]^2}; Do[k = a[[-1]] + 1; While[MemberQ[used, k], k++]; AppendTo[a, k]; used = Union[used, k*a], {n, 3, 60}]; a (* Ivan Neretin, Mar 07 2016 *)

A026427 a(n) = least positive integer > a(n-1) and not equal to a(i)*a(j) for 1<=i<=j<=n.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 17, 19, 22, 23, 26, 27, 29, 31, 34, 37, 38, 41, 43, 45, 46, 47, 53, 54, 58, 59, 61, 62, 63, 67, 71, 72, 73, 74, 75, 79, 82, 83, 86, 89, 90, 94, 96, 97, 99, 101, 103, 105, 106, 107, 109, 113, 117, 118, 120, 122, 125, 126, 127, 128, 131, 134

Offset: 1

Clark Kimberling

nonn

Starting from 3, include only those numbers that aren't the product of two numbers already included.
Numbers >3 not included: 9,12,15,18,21,24,30,33,39,42,51,57,66,69,78,....
Appears to be A026423 shifted left. - R. J. Mathar, Jun 24 2025

			9 is excluded because 9=3*3. 10 is included because 10 is not the product of any two of 3,4,5,6,7,8.

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

Cf. A066680.

Cf. A026416 and references therein.

f[s_List] := Block[{k = s[[ -1]] + 1, ss = Times @@@ Tuples[s, 2]}, While[MemberQ[ss, k], k++ ]; Append[s, k]]; Nest[f, {3}, 65] (* Robert G. Wilson v Sep 23 2006 *)

Edited by N. J. A. Sloane, Sep 14 2008 at the suggestion of R. J. Mathar

A026419 a(n) = least positive integer > a(n-1) and not a(i)*a(j) for 1<=i

Original entry on oeis.org

1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 37, 38, 39, 41, 43, 46, 47, 49, 51, 53, 57, 58, 59, 61, 62, 67, 69, 71, 73, 74, 79, 81, 82, 83, 86, 87, 89, 93, 94, 97, 101, 103, 106

Offset: 1

Clark Kimberling

nonn

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

Cf. A026416 and references therein.

a = {1, 4}; used = {a[[1]]*a[[2]]}; Do[k = a[[-1]] + 1; While[MemberQ[used, k], k++]; used = Union[used, k*a]; AppendTo[a, k], {n, 3, 60}]; a (* Ivan Neretin, Mar 07 2016 *)

Name clarified by Robert C. Lyons, Feb 08 2025

A026420 a(n) = least positive integer > a(n-1) and not a(i)*a(j) for 1<=i

Original entry on oeis.org

2, 4, 5, 6, 7, 9, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 33, 37, 39, 40, 41, 43, 47, 48, 49, 51, 53, 56, 57, 59, 61, 67, 69, 70, 71, 72, 73, 79, 81, 83, 87, 88, 89, 93, 97, 101, 103, 104, 107, 109, 110, 111, 113, 120, 121, 123, 127

Offset: 1

Clark Kimberling

nonn

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

Cf. A026416 and references therein.

a = {2, 4}; used = {a[[1]]*a[[2]]}; Do[k = a[[-1]] + 1; While[MemberQ[used, k], k++]; used = Union[used, k*a]; AppendTo[a, k], {n, 3, 60}]; a (* Ivan Neretin, Mar 07 2016 *)

Name clarified by Robert C. Lyons, Feb 08 2025

A026421 a(n) = least positive integer > a(n-1) and not a(i)*a(j) for 1<=i

Original entry on oeis.org

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, 105, 106, 107, 108, 109, 113, 118, 120, 121, 122, 127, 131

Offset: 1

Clark Kimberling

nonn

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

Cf. A026416 and references therein.

a = {3, 4}; used = {a[[1]]*a[[2]]}; Do[k = a[[-1]] + 1; While[MemberQ[used, k], k++]; used = Union[used, k*a]; AppendTo[a, k], {n, 3, 60}]; a (* Ivan Neretin, Mar 07 2016 *)

a(n) = A026417(n+1). - R. J. Mathar, May 28 2008

Name clarified by Robert C. Lyons, Feb 08 2025

A026425 a(n) = least positive integer > a(n-1) and not a(i)*a(j) for 1<=i<=j<=n, n >= 2.

Original entry on oeis.org

1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 26, 27, 29, 31, 33, 34, 37, 38, 39, 41, 43, 46, 47, 51, 53, 57, 58, 59, 61, 62, 67, 69, 71, 73, 74, 79, 82, 83, 86, 87, 89, 93, 94, 97, 101, 103, 106, 107, 109, 111, 113

Offset: 1

Clark Kimberling

nonn

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

Cf. A026416 and references therein.

a = {1, 4}; used = {a[[1]]^2, a[[1]]*a[[2]], a[[2]]^2}; Do[k = a[[-1]] + 1; While[MemberQ[used, k], k++]; AppendTo[a, k]; used = Union[used, k*a], {n, 3, 60}]; a (* Ivan Neretin, Mar 07 2016 *)

A026426 a(n) = least positive integer > a(n-1) and not a(i)*a(j) for 1<=i<=j<=n, n >= 2.

Original entry on oeis.org

2, 4, 5, 6, 7, 9, 11, 13, 15, 17, 19, 21, 23, 27, 29, 31, 32, 33, 37, 39, 40, 41, 43, 47, 48, 50, 51, 53, 56, 57, 59, 61, 67, 69, 70, 71, 72, 73, 79, 83, 87, 88, 89, 93, 97, 98, 101, 103, 104, 107, 109, 110, 111, 113, 120, 123, 125, 127

Offset: 1

Clark Kimberling

nonn

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

Cf. A026416 and references therein.

a = {2, 4}; used = {a[[1]]^2, a[[1]]*a[[2]], a[[2]]^2}; Do[k = a[[-1]] + 1; While[MemberQ[used, k], k++]; AppendTo[a, k]; used = Union[used, k*a], {n, 3, 66}]; a (* Ivan Neretin, Mar 07 2016 *)

A066512 Least nonnegative integer not the sum or product of any previous pair. a(1)=0.

Original entry on oeis.org

0, 0, 1, 2, 4, 7, 10, 13, 16, 19, 22, 25, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 62, 65, 68, 71, 74, 77, 80, 83, 86, 89, 94, 97, 103, 106, 109, 115, 118, 121, 126, 129, 135, 138, 141, 147, 150, 153, 158, 161, 164, 167, 170, 173, 176, 179, 182, 185, 193, 196, 199

Offset: 1

Brian Galebach, Jan 04 2002

			a(13)=30, which is not a(i)+a(j) or a(i)*a(j) for any distinct i,j < 13.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A026416, A033627.

from itertools import count, islice
def agen(): # generator of terms
    a, sums, products = [0], set(), set()
    yield from a
    for k in count(0):
        if k not in sums and k not in products:
            yield k
            sums.update(k+a[i] for i in range(len(a)))
            products.update(k*a[i] for i in range(len(a)))
            a.append(k)
        sums.discard(k)
        products.discard(k)
print(list(islice(agen(), 61))) # Michael S. Branicky, Jun 09 2025

cp's OEIS Frontend

A026417 a(n) = least positive integer > a(n-1) and not a(i)*a(j) for 1<=i

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A026422 a(n) = least positive integer > a(n-1) and not a(i)*a(j) for 1 <= i <= j < n.

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A026423 a(n) = least positive integer > a(n-1) and not a(i)*a(j) for 1<=i<=j<=n, n >= 2.

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A026427 a(n) = least positive integer > a(n-1) and not equal to a(i)*a(j) for 1<=i<=j<=n.

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A026419 a(n) = least positive integer > a(n-1) and not a(i)*a(j) for 1<=i

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A026420 a(n) = least positive integer > a(n-1) and not a(i)*a(j) for 1<=i

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A026421 a(n) = least positive integer > a(n-1) and not a(i)*a(j) for 1<=i

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A026425 a(n) = least positive integer > a(n-1) and not a(i)*a(j) for 1<=i<=j<=n, n >= 2.

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A026426 a(n) = least positive integer > a(n-1) and not a(i)*a(j) for 1<=i<=j<=n, n >= 2.

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