A321152
n such that all n - p are practical numbers where p is a practical number in range n/2 <= p < n.
Original entry on oeis.org
2, 3, 4, 5, 6, 8, 10, 12, 14, 16, 18, 20, 24, 36, 48, 60, 72, 84, 96, 120
Offset: 1
a(13)=24, because the practical numbers p in the range 12 <= p < 24 are {12, 16, 18, 20}. Also the complementary set {12, 8, 6, 4} has all its members practical numbers. This is the 13th occurrence of such a number.
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PracticalQ[n_] := Module[{f, p, e, prod=1, ok=True}, If[n<1||(n>1&&OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p, e}=Transpose[f]; Do[If[p[[i]]>1+DivisorSigma[1, prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]];
plst[n_] := Select[Range[Ceiling[n/2], n-1], PracticalQ]; lst={}; Do[If[plst[n]!={}&&AllTrue[n-plst[n], PracticalQ], AppendTo[lst, n]], {n, 1, 10000}]; lst
A321334
n such that all n - s are squarefree numbers where s is a squarefree number in range n/2 <= s < n.
Original entry on oeis.org
2, 3, 4, 5, 6, 7, 8, 12, 13, 16, 36
Offset: 1
a(10)=16, because the squarefree numbers s in the range 8 <= s < 16 are {10, 11, 13, 14, 15}. Also the complementary set {6, 5, 3, 2, 1} has all its members practical numbers. This is the 10th occurrence of such a number.
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plst[n_] := Select[Range[Ceiling[n/2], n-1], SquareFreeQ]; lst={}; Do[If[plst[n]!={}&&AllTrue[n-plst[n], SquareFreeQ], AppendTo[lst, n]], {n, 1, 10000}]; lst
A321612
Numbers k such that all k - t are triangular numbers where t is a triangular number in range k/2 <= t < k.
Original entry on oeis.org
2, 4, 6, 7, 9, 13, 16, 21, 31
Offset: 1
a(9) = 31, because the triangular numbers in the range 16 <= p < 31 are {21}. Also the complementary set {10} has all its members triangular numbers. This is the 9th occurrence of such a number.
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TriangularQ[n_] := Module[{m=0}, While[n>m(m+1)/2, m++]; If[n==m(m+1)/2, True, False]]; plst[n_] := Select[Range[Ceiling[n/2], n-1], TriangularQ]; lst={}; Do[If[plst[n]!={}&&AllTrue[n-plst[n], TriangularQ], AppendTo[lst, n]], {n, 1, 200}]; lst
A322605
Numbers k such that all k - u are Ulam numbers (A002858) where u is an Ulam number in the range k/2 <= u < k.
Original entry on oeis.org
2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 17, 19, 24, 29, 34, 39, 44
Offset: 1
a(10)=12, because the Ulam numbers u in the range 6 <= u < 12 are {6, 8, 11}. Also the complementary set {6, 4, 1} has all its members Ulam numbers. This is the 10th occurrence of such a number.
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Ulam[n_] := Module[{ulams={1, 2}, p}, Do[AppendTo[ulams, p=Last[ulams]; While[p++; Length[DeleteCases[Intersection[ulams, p-ulams], p/2, 1, 1]]!=2]; p], {n-2}]; ulams]; ulst=Ulam[1000]; plst[n_] := Select[ulst, Ceiling[n/2]<=#
A374057
Integers k such that all k - p are primitive practical numbers where p is a primitive practical number in range k/2 <= p < k.
Original entry on oeis.org
2, 3, 4, 7, 8, 12, 21, 22, 26, 62, 72, 182
Offset: 1
182 is a term because the primitive practical numbers p in the range 91 <= p < 182 are {104, 140}. Also the complementary set {78, 42} has all its members primitive practical numbers.
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PracticalQ[n_] := Module[{f, p, e, prod=1, ok=True}, If[n<1||(n>1&&OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p, e}=Transpose[f]; Do[If[p[[i]]>1+DivisorSigma[1, prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]];
DivFreeQ[n_] := Module[{plst=First/@Select[FactorInteger[n], #[[2]]>1 &], m, ok=False}, Do[If[! PracticalQ[n/plst[[m]]], ok=True, ok=False; Break[]], {m, 1, Length@plst}]; ok];
PPracticalQ[n_] := PracticalQ[n]&&(SquareFreeQ[n]||DivFreeQ[n]);
plst[n_] := Select[Range[Ceiling[n/2], n-1], PPracticalQ]; lst={}; Do[If[plst[n]!={}&&AllTrue[n-plst[n], PPracticalQ], AppendTo[lst, n]], {n, 1, 10000}]; lst
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