A280864 -id:A280864 - cp's OEIS frontend

A280741 Position of n in A280864.

1, 2, 4, 3, 7, 5, 11, 6, 10, 8, 14, 9, 23, 12, 17, 13, 24, 16, 28, 18, 20, 15, 31, 19, 35, 22, 34, 21, 39, 26, 49, 30, 43, 25, 36, 33, 55, 29, 44, 41, 63, 37, 66, 46, 27, 32, 70, 38, 48, 51, 59, 45, 87, 54, 42, 57, 74, 40, 91, 61, 95, 50, 58, 65, 52, 68, 104

Offset: 1

N. J. A. Sloane, Jan 12 2017

nonn

Inverse function to A280864. - N. J. A. Sloane, Jun 06 2024

Lars Blomberg and N. J. A. Sloane, Table of n, a(n) for n = 1..50440 [First 10000 terms from Lars Blomberg]

Cf. A280864, A280742.

Records: A372058, A372059.

More terms from Lars Blomberg, Jan 13 2017

A280742 Position where n-th prime appears in A280864.

Original entry on oeis.org

2, 4, 7, 11, 14, 23, 24, 28, 31, 39, 49, 55, 63, 66, 70, 87, 91, 95, 104, 121, 128, 132, 136, 140, 145, 149, 162, 166, 186, 198, 212, 256, 259, 262, 263, 276, 287, 291, 301, 312, 320, 331, 335, 351, 355, 359, 368, 376, 380, 400, 415, 421, 428, 444, 448, 454

Offset: 1

N. J. A. Sloane, Jan 12 2017

nonn

This sequence is very similar to A372058, the difference being caused by the presence of 1 and 25 in A372059. - N. J. A. Sloane, Jun 06 2024

David Suteu and N. J. A. Sloane, Table of n, a(n) for n = 1..5175 [First 679 terms from David Suteu]

Cf. A280864, A280741, A280745, A372058, A372059..

a(9) inserted and terms a(11) and beyond added by Daniel Suteu, Jan 13 2017

A280738 After S(n)=A280864(n) has been computed, let p(n) = product of distinct primes shared by S(n-1) and S(n); let q(n) = product of distinct primes in S(n) but not in S(n-1); and let r(n) = smallest number not yet in S. Sequence gives p(n).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 5, 2, 3, 1, 7, 2, 1, 11, 2, 3, 5, 2, 3, 7, 2, 13, 1, 17, 2, 15, 1, 19, 2, 1, 23, 2, 3, 1, 5, 7, 6, 1, 29, 2, 5, 11, 3, 13, 2, 11, 7, 1, 31, 2, 5, 13, 6, 1, 37, 2, 7, 3, 17, 2, 15, 1, 41, 2, 1, 43, 2, 33, 1, 47, 2, 35, 3, 19, 2, 3, 23, 2, 5

Offset: 1

N. J. A. Sloane, Jan 12 2017

nonn

We use the convention that an empty product is 1.
By decree, gcd(S(n+1),p(n)) = 1, gcd(S(n+1),q(n)) = q(n) = p(n+1), S(n+1) >= r(n).
By definition, all terms are squarefree. Let {i,j,k} be distinct fixed positive numbers. Conjecture: All squarefree numbers appear infinitely often, and all terms a(n) = j are immediately preceded and followed infinitely often by all terms a(n-1) = i and a(n+1) = k. If so, then A280864 is a permutation of the natural numbers. - Bob Selcoe, Apr 04 2017

Rémy Sigrist, Table of n, a(n) for n = 1..100000

Cf. A280864, A280740, A280741, A280742, A280743, A280744.

Cf. A005117 (squarefree numbers).

More terms from Rémy Sigrist, Jan 14 2017

A280774 Numbers k such that gcd(A280864(k), A280864(k+1)) = 1.

Original entry on oeis.org

1, 3, 6, 10, 13, 23, 27, 30, 34, 38, 48, 54, 62, 65, 69, 86, 90, 94, 103, 120, 127, 131, 135, 139, 144, 148, 161, 165, 185, 197, 211, 255, 258, 262, 275, 286, 290, 300, 311, 319, 330, 334, 350, 354, 358, 367, 375, 379, 399, 414, 420, 427, 443, 447, 453, 457, 465, 486, 500, 504, 530, 534, 549, 555, 559

Offset: 1

N. J. A. Sloane, Jan 19 2017

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..25317

Cf. A280864.

First differences give A283832.

A280740 After S(n)=A280864(n) has been computed, let p(n) = product of distinct primes shared by S(n-1) and S(n); let q(n) = product of distinct primes in S(n) but not in S(n-1); and let r(n) = smallest number not yet in S. Sequence gives r(n).

Original entry on oeis.org

2, 3, 3, 5, 5, 5, 7, 7, 7, 7, 11, 11, 11, 13, 13, 13, 13, 13, 13, 13, 13, 13, 17, 19, 19, 19, 19, 23, 23, 23, 25, 25, 25, 25, 29, 29, 29, 29, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 37, 37, 37, 37, 37, 37, 41, 41, 41, 41, 41, 41, 41, 41, 43, 43, 43, 47, 47, 47, 47, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53

Offset: 1

N. J. A. Sloane, Jan 12 2017

nonn

We use the convention that an empty product is 1.
By decree, gcd(S(n+1),p(n)) = 1, gcd(S(n+1),q(n)) = q(n) = p(n+1), S(n+1) >= r(n). (Note p(n) is as defined above; it is not the n-th prime.)
Conjecture: except for the four terms equal to 25, a(n) is always a prime, and all the primes appear and in their natural order.
The conjecture is true for n up to 10^7. - Lars Blomberg Jan 14 2017

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

Cf. A280864, A280738, A280741, A280742, A280743, A280744.

A280743 Records in A280864.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 14, 16, 22, 24, 28, 34, 45, 46, 48, 58, 77, 78, 82, 86, 99, 105, 121, 135, 136, 165, 195, 325, 455, 459, 465, 605, 637, 651, 715, 777, 897, 987, 1001, 1495, 1573, 2275, 2387, 2415, 2421, 2451, 2493, 2919, 3129, 3171, 3633, 3689, 4011, 4053, 4137, 4179, 4879, 4893

Offset: 1

N. J. A. Sloane, Jan 12 2017

nonn

Lars Blomberg, Table of n, a(n) for n = 1..1289 [First 226 terms from N. J. A. Sloane]

Cf. A280864, A280738, A280740, A280741, A280742, A280744.

A280744 Positions of records in A280864.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 12, 13, 15, 19, 21, 25, 27, 32, 38, 40, 47, 53, 64, 67, 69, 73, 86, 90, 99, 107, 114, 131, 189, 290, 304, 311, 354, 366, 387, 435, 520, 553, 563, 598, 817, 885, 1333, 1361, 1615, 1634, 1651, 1655, 1765, 1776, 2031, 2068, 2248, 2261, 2314, 2700, 2704, 2712, 2993, 3128, 3670

Offset: 1

N. J. A. Sloane, Jan 12 2017

nonn

Lars Blomberg, Table of n, a(n) for n = 1..1289 (First 226 terms from N. J. A. Sloane)

Cf. A280864, A280738, A280740, A280741, A280742, A280743.

A280754 Numbers n such that A280864(n) = n.

Original entry on oeis.org

1, 2, 54, 100, 130, 190, 392, 486, 608, 623, 799, 1448, 1614, 6169, 7807, 8149, 24403, 28945, 37665, 40395, 43559, 46117, 46259, 119663, 121149, 153317, 214459, 262759, 306637, 318605, 318815, 365029, 387471, 394597, 403431, 439125, 450051, 483511, 506807

Offset: 1

N. J. A. Sloane, Jan 16 2017

nonn

Lars Blomberg, Table of n, a(n) for n = 1..64, based on 10^7 terms of A280864

Cf. A280864.

More terms from Lars Blomberg, Jan 16 2017

A284724 a(n) = (1/2) * smallest even number missing from [A280864(1), ..., A280864(n-1)].

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 5, 5, 6, 7, 7, 7, 8, 9, 9, 9, 10, 10, 12, 13, 13, 13, 15, 15, 15, 15, 16, 16, 16, 16, 18, 18, 18, 20, 20, 20, 20, 20, 20, 20, 20, 22, 22, 22, 22, 22, 25, 25, 25, 25, 25, 27, 27, 27, 28, 28, 28, 30, 30, 30, 30, 32, 32, 32, 32, 33, 33, 33, 35, 35, 35, 35, 36, 36, 36, 36, 40, 40, 40, 42

Offset: 1

N. J. A. Sloane, Apr 06 2017

nonn

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_2(n)/2.

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.

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

Rémy Sigrist, Table of n, a(n) for n = 1..10000
J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The EKG sequence, arXiv:math/0204011 [math.NT], 2002.
J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The EKG Sequence, Exper. Math. 11 (2002), 437-446.

Cf. A280864, A064413, A284725, A284726.

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:=2; 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;

terms = 80; 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 = 2a[n-1], True, b += 2, If[FreeQ[A280864[[1 ;; n-1]], b], Return[b/2]]];
Array[a, terms] (* Jean-François Alcover, Nov 23 2017, after Rémy Sigrist program for A280864 *)

A280745 Primes p such that A280864(k)=p for some k and A280864(k-1)=m*p for some m>1.

Original entry on oeis.org

13, 139, 379, 397, 647, 661, 967, 983, 997, 1021, 1063, 1109, 1129, 1187, 1201, 1223, 1231, 1249, 1289, 1297, 1307, 1453, 1481, 1487, 1499, 1543, 1553, 1597, 1607, 1613, 1621, 1637, 1667, 1697, 1723, 1759, 1789, 1831, 1867, 1873, 1879, 1907, 1933, 2011, 2029, 2069, 2083, 2089, 2141, 2309

Offset: 1

N. J. A. Sloane, Jan 13 2017

nonn

Conjecture: m is always 2.
The conjecture is true for n up to 10^7.
These primes are exceptional, because it appears that usually a prime p in A280864 is followed by 2p, whereas for these primes p is preceded by 2p.

			13 is a term because A280864(23)=13 and A280864(22)=26.

Lars Blomberg, Table of n, a(n) for n = 1..10000 (first 1683 terms from N. J. A. Sloane)

Cf. A280864, A280745.

cp's OEIS Frontend

A280741 Position of n in A280864.

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A280742 Position where n-th prime appears in A280864.

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A280738 After S(n)=A280864(n) has been computed, let p(n) = product of distinct primes shared by S(n-1) and S(n); let q(n) = product of distinct primes in S(n) but not in S(n-1); and let r(n) = smallest number not yet in S. Sequence gives p(n).

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A280774 Numbers k such that gcd(A280864(k), A280864(k+1)) = 1.

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A280740 After S(n)=A280864(n) has been computed, let p(n) = product of distinct primes shared by S(n-1) and S(n); let q(n) = product of distinct primes in S(n) but not in S(n-1); and let r(n) = smallest number not yet in S. Sequence gives r(n).

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A280743 Records in A280864.

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A280744 Positions of records in A280864.

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A280754 Numbers n such that A280864(n) = n.

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A284724 a(n) = (1/2) * smallest even number missing from [A280864(1), ..., A280864(n-1)].

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Maple

Mathematica

A280745 Primes p such that A280864(k)=p for some k and A280864(k-1)=m*p for some m>1.

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