A121495 -id:A121495 - cp's OEIS frontend

A073251 Numbers k such that k, k+1 and k+2 are nonprime and squarefree.

33, 85, 93, 141, 185, 201, 213, 217, 253, 265, 285, 301, 321, 393, 445, 453, 469, 481, 517, 533, 553, 581, 589, 609, 633, 669, 697, 705, 713, 753, 777, 789, 793, 805, 813, 869, 893, 897, 901, 913, 921, 933, 957, 985, 993, 1001, 1005, 1041, 1045, 1065, 1113

Offset: 1

Reinhard Zumkeller, Jul 22 2002

nonn

k-1 and k+3 are not squarefree. Proof: k is odd, otherwise k or k+2 would be divisible by 4. Thus k+1 is even and not divisible by 4, hence k-1 and k+3 are divisible by 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000469, A005117, A007675, A073249, A073250, A121495.

f[upto_]:=Module[{pp=PrimePi[upto],n},lst=Partition[Complement[Range[upto], Prime[Range[pp]]],3,1];Transpose[Select[lst,And@@SquareFreeQ/@#&]][[1]]]; f[1200] (* Harvey P. Dale, Mar 21 2011 *)

isok1(k) = !isprime(k) && issquarefree(k); \\ A000469
isok(k) = isok1(k) && isok1(k+1) && isok1(k+2); \\ Michel Marcus, Mar 25 2021

Edited by Klaus Brockhaus, Aug 07 2006

A367695 Numbers k such that k and k+1 are both exponentially odd numbers (A268335).

Original entry on oeis.org

1, 2, 5, 6, 7, 10, 13, 14, 21, 22, 23, 26, 29, 30, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42, 46, 53, 54, 55, 56, 57, 58, 61, 65, 66, 69, 70, 73, 77, 78, 82, 85, 86, 87, 88, 93, 94, 95, 96, 101, 102, 103, 104, 105, 106, 109, 110, 113, 114, 118, 119, 122, 127, 128

Offset: 1

Amiram Eldar, Nov 27 2023

nonn

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 6, 48, 478, 4734, 47195, 471707, 4716892, 47168363, 471681183, 4716806520, ... . Apparently, the asymptotic density of this sequence exists and equals Product_{p prime} (1 - 2/(p*(p+1))) = 0.47168... (A307868).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A268335.
Cf. A307868.
Subsequences: A007674, A325058.
Similar sequences: A071318, A121495, A340152, A367696.

expOddQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], OddQ]; Select[Range[128], And @@ expOddQ /@ {#, # + 1} &]

isexpodd(n) = {my(f = factor(n)); for(i=1, #f~, if (!(f[i, 2] % 2), return (0))); 1;}
is(n) = isexpodd(n) && isexpodd(n+1)

A367696 Numbers k such that k and k+1 are both exponentially odious numbers (A270428).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 25, 28, 29, 30, 33, 34, 35, 36, 37, 38, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85

Offset: 1

Amiram Eldar, Nov 27 2023

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 8, 78, 762, 7615, 76113, 761127, 7611222, 76111895, 761119135, 7611190807, ... . Apparently, the asymptotic density of this sequence exists and equals 0.761119... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A270428.
Subsequences: A007674, A367697.
Similar sequences: A071318, A121495, A340152, A367695.

expOdQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], OddQ[DigitCount[#, 2, 1]] &]; Select[Range[100], And @@ expOdQ /@ {#, # + 1} &]

isexpod(n) = {my(f = factor(n)); for(i=1, #f~, if (!(hammingweight(f[i, 2]) % 2), return (0))); 1;}
is(n) = isexpod(n) && isexpod(n+1)

A240593 The smaller of a pair of consecutive composite squarefree numbers (A120944) without any prime number between them.

Original entry on oeis.org

14, 21, 33, 34, 38, 55, 57, 62, 65, 69, 74, 77, 85, 86, 91, 93, 94, 105, 110, 114, 115, 118, 119, 122, 129, 133, 141, 142, 143, 145, 154, 158, 159, 165, 174, 177, 182, 183, 185, 186, 187, 194, 201, 202, 203, 205, 206, 209, 213, 214, 215, 217, 218, 219, 221, 230, 235, 237, 246, 247, 253, 254, 258, 259, 265, 266, 273, 285, 286, 287, 290, 295, 298, 299

Offset: 1

Antonio Roldán, Apr 08 2014

nonn

Supersequence of A121495.

			62 is in the sequence because A120944(20)=62, A120944(21)=65, without primes between them.

Cf. A120944, A240592.

Select[Partition[Select[Range[400],CompositeQ[#]&&SquareFreeQ[#]&],2,1], PrimePi[ #[[1]]]==PrimePi[#[[2]]]&][[All,1]] (* Harvey P. Dale, Apr 12 2020 *)

freesqrcomp(n)=issquarefree(n)&&!isprime(n)
nextfqc(n)={local(k); k=n+1; while(!freesqrcomp(k), k+=1); return(k)}
{for(i=2, 1000, if(freesqrcomp(i) && (nextfqc(i)

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A073251 Numbers k such that k, k+1 and k+2 are nonprime and squarefree.

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A367695 Numbers k such that k and k+1 are both exponentially odd numbers (A268335).

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A367696 Numbers k such that k and k+1 are both exponentially odious numbers (A270428).

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A240593 The smaller of a pair of consecutive composite squarefree numbers (A120944) without any prime number between them.

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