cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A280770 Even terms in A280864, divided by 2, in order of appearance.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 11, 9, 10, 12, 14, 13, 17, 15, 19, 16, 23, 18, 21, 24, 29, 20, 26, 22, 31, 25, 39, 27, 37, 28, 34, 30, 41, 32, 43, 33, 47, 35, 38, 36, 46, 40, 51, 42, 52, 44, 53, 45, 59, 48, 61, 49, 68, 50, 57, 54, 67, 55, 58, 56, 76, 60, 66, 63, 71, 62, 74, 64, 73, 65, 79, 69, 83, 70, 89, 72, 97
Offset: 1

Views

Author

N. J. A. Sloane, Jan 17 2017

Keywords

Comments

It is known (see A280864) that this is a permutation of the natural numbers.

Links

Crossrefs

Cf. A280864, A280771.

A280771 Positions of even terms in A280864.

Original entry on oeis.org

2, 3, 5, 6, 8, 9, 12, 13, 15, 16, 18, 19, 21, 22, 25, 26, 29, 30, 32, 33, 37, 38, 40, 41, 45, 46, 50, 51, 53, 54, 56, 57, 60, 61, 64, 65, 67, 68, 71, 72, 75, 76, 78, 79, 81, 82, 84, 85, 88, 89, 92, 93, 96, 97, 99, 100, 102, 103, 105, 106, 109, 110, 112, 113, 116, 117, 122, 123, 126, 127, 129, 130, 133
Offset: 1

Views

Author

N. J. A. Sloane, Jan 17 2017

Keywords

Comments

By definition of A280864, this is a sequence of pairs of consecutive numbers, separated by gaps of size at least 1.

Links

Crossrefs

Cf. A280864, A280770.

A284725 a(n) = (1/3) * smallest multiple of 3 missing from [A280864(1), ..., A280864(n-1)].

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 11, 11, 11, 11, 11, 11, 11, 11, 11, 13, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 23, 23, 23, 27, 27, 27, 27, 27, 27, 27, 27
Offset: 1

Views

Author

N. J. A. Sloane, Apr 06 2017

Keywords

Comments

For k >= 1, n >= 1, let B_k(n) = smallest multiple of k missing from [A280864(1), ..., A280864(n-1)]. Sequence gives values of B_3(n)/3.
The analogous sequences B_k(n) for the EKG sequence A064413 were important for the analysis of that sequence, so they may also be useful for studying A280864.

Examples

			The initial terms of A280864 are 1,2,4,3,6,8,... The smallest missing multiple of 3 in [1,2,4,3,6] is 9, so a(6) = 9/3 = 3.

Links

Crossrefs

Cf. A280864, A064413, A284724, A280726.

Programs

  • Maple
    mex := proc(L)
local k;
for k from 1 do
if not k in L then
return k;
end if;
end do:
end proc:
read b280864;
k:=3; a:=[1,1]; ML:=[]; B:=1;
for n from 2 to 120 do
t:=b280864[n];
if (t mod k) = 0 then
ML:=[op(ML),t/k];
B:=mex(ML);
a:=[op(a),B];
else
a:=[op(a),B];
fi;
od:
a;
  • Mathematica
    terms = 85; rad[n_] := Times @@ FactorInteger[n][[All, 1]]; A280864 = Reap[present = 0; p = 1; pp = 1; Do[forbidden = GCD[p, pp]; mandatory = p/forbidden; a = mandatory; While[BitGet[present, a] > 0 || GCD[forbidden, a] > 1, a += mandatory]; Sow[a]; present += 2^a; pp = p; p = rad[a], terms]][[2, 1]];
Clear[a]; a[1] = 1; a[n_] := a[n] = For[b = 3 a[n - 1], True, b += 3, If[FreeQ[A280864[[1 ;; n - 1]], b], Return[b/3]]];
Array[a, terms] (* Jean-François Alcover, Nov 26 2017, after Rémy Sigrist's program for A280864 *)

A284726 a(n) = (1/4) * smallest multiple of 4 missing from [A280864(1), ..., A280864(n-1)].

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 20, 20, 20, 21, 21, 21, 22, 22
Offset: 1

Views

Author

N. J. A. Sloane, Apr 06 2017

Keywords

Comments

For k >= 1, n >= 1, let B_k(n) = smallest multiple of k missing from [A280864(1), ..., A280864(n-1)]. Sequence gives values of B_4(n)/4.
The analogous sequences B_k(n) for the EKG sequence A064413 were important for the analysis of that sequence, so they may also be useful for studying A280864.

Examples

			The initial terms of A280864 are 1,2,4,3,6,8,... The smallest missing multiple of 3 in [1,2,4,3,6] is 8, so a(6) = 8/4 = 2.

Links

Crossrefs

Cf. A280864, A064413, A284724, A280725.

Programs

  • Maple
    mex := proc(L)
local k;
for k from 1 do
if not k in L then
return k;
end if;
end do:
end proc:
read b280864;
k:=4; a:=[1,1]; ML:=[]; B:=1;
for n from 2 to 120 do
t:=b280864[n];
if (t mod k) = 0 then
ML:=[op(ML),t/k];
B:=mex(ML);
a:=[op(a),B];
else
a:=[op(a),B];
fi;
od:
a;
  • Mathematica
    terms = 84; rad[n_] := Times @@ FactorInteger[n][[All, 1]];
A280864 = Reap[present = 0; p = 1; pp = 1; Do[forbidden = GCD[p, pp]; mandatory = p/forbidden; a = mandatory; While[BitGet[present, a] > 0 || GCD[forbidden, a] > 1, a += mandatory]; Sow[a]; present += 2^a; pp = p; p = rad[a], terms]][[2, 1]];
Clear[a]; a[1] = 1;
a[n_] := a[n] = For[b = 4 a[n - 1], True, b += 4, If[FreeQ[A280864[[1 ;; n - 1]], b], Return[b/4]]];
Array[a, terms] (* Jean-François Alcover, Nov 26 2017, after Rémy Sigrist's program for A280864 *)

A372063 a(n) = A280864(n) - A280866(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 2, -23, 8, 11, 38, -3, -23, 11, -18, 9, 29, 17, -37, 14, -19, 22, -31, 4, 17, -11, -25, -17, 17, -51, 16, -39, 42, -29, 22, 7, 28, 2, 11, -6, -15, 1, -24, 5, 17, -18, 1, 9, -26, 25, 0, 0, 2, 25, -41, 59, -22, -11, -8, 41, 37, -3, 38, -12
Offset: 1

Views

Author

N. J. A. Sloane, May 09 2024

Keywords

Comments

A280864 and A280866 are closely related. There is a proof that A280866 is a permutation of the positive integers, but for A280864 this is only a conjecture. The present sequence compares them term-by-term; A372064 compares their inverses; and A372065 compares where the primes appear.

Links

Crossrefs

Cf. A280864, A280741, A280866, A304741, A372064, A372065.

A372065 a(n) = (index of prime(n) in A280864) - (index of prime(n) in A280866).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 1, 5, 5, 5, 0, -1, -2, 3, 13, 12, 13, 12, 9, 6, 6, 11, 11, 24, 28, 38, 74, 70, 55, 49, 58, 60, 60, 54, 46, 51, 50, 43, 54, 51, 44, 36, 36, 21, 34, 45, 39, 25, 25, 22, 24, 24, 25, 42, 44, 27, 38, 33, 26, 28, 22, 24, 32, 56, 72, 84, 63, 67, 107, 114, 111, 104, 108, 105, 93, 100, 100, 108, 105, 96, 94, 92
Offset: 1

Views

Author

N. J. A. Sloane, May 10 2024

Keywords

Comments

A372063 compares A280864 and A280866 term-by-term; A372064 compares their inverses; and the present sequence compares where the successive primes appear.
This is a subsequence of A372064.

Examples

			The 11th prime, 31, appears in A280864 at index 49, and in A280866 at index 43, so a(11) = 49 - 43 = 6.

Links

Crossrefs

Cf. A280864, A280741, A280866, A304741, A372063, A372064.

A280746 Index of 2^n in A280864.

Original entry on oeis.org

1, 2, 3, 6, 13, 30, 65, 127, 258, 530, 1009, 1988, 3942, 8004, 16195, 32389, 64923, 129870, 260326, 520787, 1043257, 2085202, 4171403, 8338443
Offset: 0

Views

Author

N. J. A. Sloane, Jan 13 2017

Keywords

Examples

			A280864(30) = 32, so a(5) = 30.

Crossrefs

Cf. A280864, A282024.

Extensions

a(17)-a(23) from Lars Blomberg, Jan 14 2017

A280755 Numbers n such that 3n appears earlier than 2n in A280864.

Original entry on oeis.org

65, 91, 95, 115, 119, 217, 231, 259, 299, 329, 371, 455, 511, 529, 539, 553, 623, 679, 707, 721, 749, 805, 973, 1043, 1057, 1211, 1337, 1351, 1379, 1631, 1799, 1883, 2839, 3017, 3077, 6601, 8449, 8897, 11237, 11509, 11921, 18683, 19193, 19411, 22043, 26719
Offset: 1

Views

Author

N. J. A. Sloane, Jan 16 2017

Keywords

Comments

Normally (at least as indicated by all the evidence so far), 2n appears earlier than 3n in A280864. The present sequence lists the exceptions.

Examples

			A280864(114) = 195 = 3*65 appears before A280864(130) = 130 = 2*65, so 65 is a term.
A280864(178) = 285 = 3*95 appears before A280864(190) = 190 = 2*95, so 95 is a term.
It is a bit surprising that both 130 and 190 are fixed points of A280864 (cf. A280754).

Links

Crossrefs

Cf. A280864, A280754.

A281369 Orbit of 25 under repeated applications of the map m -> A280864(m).

Original entry on oeis.org

25, 34, 27, 45, 52, 65, 64, 82, 84, 104, 67, 86, 121, 71, 94, 81, 102, 114, 195, 253, 295, 305, 249, 276, 151, 154, 164, 168, 170, 129, 146, 194, 184, 222, 236, 215, 189, 455, 526, 583, 600, 618, 915, 979, 996, 1032, 1463, 1541, 1544, 1752, 1887, 2348, 2378, 2195, 2485, 1279, 1298, 673, 591, 589
Offset: 1

Views

Author

N. J. A. Sloane, Feb 03 2017

Keywords

Comments

It is a strong conjecture that A280864 is a permutation of the positive integers. The orbit containing 25 appears to be the first infinite orbit. See A281370 for the orbit of 25 under the inverse permutation, and see A280864 for other orbits (there are several small cycles).

Links

Crossrefs

Cf. A280864, A281370.

A281370 Orbit of 25 under repeated applications of the inverse of the map m -> A280864(m).

Original entry on oeis.org

25, 35, 36, 33, 43, 66, 68, 60, 61, 95, 101, 149, 263, 454, 381, 411, 432, 423, 491, 910, 884, 856, 730, 707, 702, 682, 653, 1277, 2443, 2605, 2783, 2802, 2603, 2588, 2025, 2154, 1962, 1692, 1566, 1577, 1557, 1729, 970, 946, 912, 867, 786, 741, 490, 478, 422, 369, 404, 344, 337, 578, 568
Offset: 1

Views

Author

N. J. A. Sloane, Feb 03 2017

Keywords

Comments

It is a strong conjecture that A280864 is a permutation of the positive integers. The orbit containing 25 appears to be the first infinite orbit. See A281370 for the orbit of 25 under the inverse permutation, and see A280864 for other orbits (there are several small cycles).

Links

Crossrefs

Cf. A280864, A281369.
Previous Showing 11-20 of 54 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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