A100113 -id:A100113 - cp's OEIS frontend

A100114 a(n) = the unique squarefree m such that A100113(A100112(m))=A005117(n).

1, 2, 5, 7, 3, 13, 10, 19, 23, 11, 6, 30, 41, 14, 17, 47, 26, 57, 15, 61, 21, 29, 34, 66, 39, 22, 71, 33, 87, 46, 94, 31, 102, 35, 42, 55, 110, 115, 62, 37, 43, 123, 51, 38, 139, 151, 65, 77, 53, 158, 70, 170, 82, 86, 58, 178, 78, 59, 93, 83, 190, 195, 69, 206, 74, 101, 213

Offset: 1

Reinhard Zumkeller, Nov 07 2004

nonn

a(A100112(n)) and A100113(A100112(n)) define a pair of inverse permutations of the squarefree numbers: a(A100112(A100113(n))) = A100113(A100112(a(n))) = A005117(n);

A100116(n) = if n is squarefree then a(A100112(n)) else n.

Eric Weisstein's World of Mathematics, Squarefree

A100115 If n is squarefree then A100113(A100112(n)) else n.

Original entry on oeis.org

1, 2, 6, 4, 3, 15, 5, 8, 9, 10, 14, 12, 7, 21, 30, 16, 22, 18, 11, 20, 33, 39, 13, 24, 25, 26, 27, 28, 34, 17, 51, 32, 42, 35, 55, 36, 65, 70, 38, 40, 19, 57, 66, 44, 45, 46, 23, 48, 49, 50, 69, 52, 78, 54, 58, 56, 29, 87, 93, 60, 31, 62, 63, 64, 74, 37, 111, 68, 102, 82, 41, 72

Offset: 1

Reinhard Zumkeller, Nov 07 2004

nonn

Permutation of the natural numbers with inverse A100116;

a(A013929(n)) = A013929(n).

Index entries for sequences that are permutations of the natural numbers

Cf. A005117.

A152247 a(1) = 1, a(2) = 3; thereafter a(n) is the smallest odd positive integer not yet occurring in the sequence such that gcd(a(n), a(n-1)) > 1.

Original entry on oeis.org

1, 3, 9, 15, 5, 25, 35, 7, 21, 27, 33, 11, 55, 45, 39, 13, 65, 75, 51, 17, 85, 95, 19, 57, 63, 49, 77, 91, 105, 69, 23, 115, 125, 135, 81, 87, 29, 145, 155, 31, 93, 99, 111, 37, 185, 165, 117, 123, 41, 205, 175, 119, 133, 147, 129, 43, 215, 195, 141, 47, 235, 225, 153

Offset: 1

Leroy Quet, Nov 30 2008

nonn

From Matthew Vandermast, Nov 21 2009: (Start)
Odd analog of the EKG sequence. Cf. A064413.
In contrast to A064413, there are at least 2 different patterns by which primes > a(2) are introduced into the sequence. 5 is the first of many primes p that are immediately preceded in the sequence by 3p and immediately followed by 5p. For p = 7, 19, or 31, p is immediately preceded by 5p and immediately followed by 3p. (End)
In fact, based on the first 10000 terms, it appears that apart from the three exceptions 7, 19, and 31, primes p are always preceded by 3*p and followed by 5*p. The graph is very similar to the graph of the EKG sequence. - N. J. A. Sloane, Oct 29 2020

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A064413, A098550, A100113, A152248.

M:= 10000;
N:= 100000;
V:= Array(0..100000,0): # V = hit?
A[1]:= 1: # A = sequence
A[2]:= 3: V[3]:= 1:
for n from 3 to M do # get candidates S for next term
  sw:=-1;
  S:= {seq(seq(k*p, k=1..N/p), p=numtheory:-factorset(A[n-1]))};
  for s in sort(convert(S, list)) do
    if type(s,odd) and V[s] = 0 then
        A[n]:= s; V[s]:=1; sw := 1;  break; fi;
  od;
  if sw=-1 then lprint("n not found",n); break; fi;
od: # od n
[seq(A[i], i=1..1000]; # N. J. A. Sloane, Oct 29 2020

Extended by Ray Chandler, Dec 05 2008

A379560 a(1) = 1, a(2) = 4; for n > 2, a(n) is the smallest unused positive nonsquarefree number that shares a factor with a(n-1).

Original entry on oeis.org

1, 4, 8, 12, 9, 18, 16, 20, 24, 27, 36, 28, 32, 40, 25, 45, 48, 44, 50, 52, 54, 56, 49, 63, 60, 64, 68, 72, 75, 80, 76, 84, 81, 90, 88, 92, 96, 98, 100, 104, 108, 99, 117, 120, 112, 116, 124, 126, 128, 132, 121, 176, 136, 140, 125, 135, 144, 147, 150, 148, 152, 156, 153, 162, 160, 164, 168, 171, 180, 172, 184, 188, 192, 189, 175, 196, 198, 200, 204, 207, 216

Offset: 1

Scott R. Shannon, Dec 26 2024

nonn

			a(3) = 8 as a(2) = 4 and 8 is the smallest unused nonsquarefree number (8 = 2^3) that shares a factor with 4.

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Cf. A013929, A100113, A064413, A379248.

cp's OEIS Frontend

A100114 a(n) = the unique squarefree m such that A100113(A100112(m))=A005117(n).

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A100115 If n is squarefree then A100113(A100112(n)) else n.

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A152247 a(1) = 1, a(2) = 3; thereafter a(n) is the smallest odd positive integer not yet occurring in the sequence such that gcd(a(n), a(n-1)) > 1.

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A379560 a(1) = 1, a(2) = 4; for n > 2, a(n) is the smallest unused positive nonsquarefree number that shares a factor with a(n-1).

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