A353187 -id:A353187 - cp's OEIS frontend

A371182 Lexicographically earliest strictly increasing sequence of positive integers in which every triple of consecutive digits contains only one pair that is not coprime.

1, 2, 4, 5, 6, 8, 10, 21, 22, 32, 33, 42, 52, 54, 55, 62, 72, 74, 76, 77, 82, 92, 93, 100, 102, 103, 104, 105, 106, 107, 108, 109, 130, 133, 136, 139, 160, 162, 163, 164, 166, 168, 169, 190, 193, 196, 199, 221, 223, 225, 227, 229, 232, 233, 234, 238, 239, 241, 243, 245

Offset: 1

Ivan N. Ianakiev, Mar 14 2024

Cf. A353187, A371060, A371183.

pairwiseQ[a_,b_,c_]:=Sort[{CoprimeQ[a,b],CoprimeQ[b,c],CoprimeQ[a,c]}]=={False,True,True}; a[1]=1;a[2]=2;a[3]=4;a[4]=5;a[5]=6;a[6]=8;a[7]=10;
a[n_]:=a[n]=Module[{k=a[n-1]+1,digits},
While[digits=Join[Take[IntegerDigits[a[n-1]],-2],IntegerDigits[k]];
Union[ pairwiseQ@@@Table[{digits[[i]],digits[[i+1]],digits[[i+2]]},
{i,1,Length[digits]-2}]]!={True},k++];k]; a/@Range[60]

A371183 Lexicographically earliest sequence of distinct terms in which every triple of consecutive digits contains only one pair that is not coprime.

Original entry on oeis.org

1, 2, 4, 3, 8, 9, 22, 5, 6, 21, 20, 10, 7, 17, 71, 70, 12, 23, 24, 14, 25, 26, 13, 31, 30, 15, 51, 50, 16, 27, 28, 18, 29, 32, 33, 19, 34, 38, 39, 42, 52, 41, 40, 100, 102, 103, 104, 105, 106, 107, 108, 109, 130, 120, 122, 54, 43, 35, 36, 53, 37, 62, 55, 44, 56, 45, 46, 58, 47, 48

Offset: 1

Ivan N. Ianakiev, Mar 14 2024

Cf. A353187, A371060, A371182.

pairwiseQ[a_,b_,c_]:=Sort[{CoprimeQ[a,b],CoprimeQ[b,c],CoprimeQ[a,c]}]=={False,True,True}; a[1]=1;a[2]=2;a[3]=4;
a[n_]:=a[n]=Module[{k=1,digits,tab=Table[a[i],{i,1,n-1}]},
While[digits=Join[Take[Flatten[IntegerDigits[Take[tab,-3]]],-3],IntegerDigits[k]];
MemberQ[tab,k]||Union[pairwiseQ@@@
Table[{digits[[i]],digits[[i+1]],digits[[i+2]]},
{i,1,Length[digits]-2}]]!={True},k++];k]; a/@Range[70]

A356007 Lexicographically earliest strictly increasing sequence of positive integers in which every triple of consecutive terms contains only one pair whose numbers of divisors are not coprime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 10, 11, 16, 17, 18, 64, 65, 66, 81, 82, 83, 100, 101, 102, 121, 122, 123, 144, 145, 146, 169, 170, 173, 196, 197, 199, 225, 226, 227, 256, 257, 258, 289, 290, 291, 324, 326, 327, 361, 362, 365, 400, 401, 402, 441, 442, 443, 484, 485, 487, 529, 530, 533, 576, 577, 579

Offset: 1

Ivan N. Ianakiev, Jul 23 2022

nonn

For every k>=0, every (3k+1)-th term is square, because a triple of terms with {odd,even,even} number of divisors must be followed by a similar triple.

Cf. A353187.

coQ[n_,m_]:=CoprimeQ[DivisorSigma[0,n],DivisorSigma[0,m]];
f[x_,y_,z_]:=Sort[{coQ[x,y],coQ[x,z],coQ[y,z]}];
next[n_,m_]:=Module[{k=m+1},While[f[n,m,k]!={False,True,True},k++];k];
a[1]=1;a[2]=2;a[n_]:=a[n]=next[a[n-2],a[n-1]];a/@Range[102]

cp's OEIS Frontend

A371182 Lexicographically earliest strictly increasing sequence of positive integers in which every triple of consecutive digits contains only one pair that is not coprime.

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A371183 Lexicographically earliest sequence of distinct terms in which every triple of consecutive digits contains only one pair that is not coprime.

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Author

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Mathematica

A356007 Lexicographically earliest strictly increasing sequence of positive integers in which every triple of consecutive terms contains only one pair whose numbers of divisors are not coprime.

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Mathematica