A280864 -id:A280864 - cp's OEIS frontend

A358267 a(1) = 1, a(2) = 2. Thereafter:(i). If no prime divisor of a(n-1) divides a(n-2), a(n) is the least novel multiple of the squarefree kernel of a(n-1). (ii). If some (but not all) prime divisors of a(n-1) do not divide a(n-2), a(n) is the least of the least novel multiples of all such primes. (iii). If every prime divisor of a(n-1) also divides a(n-2), a(n) = u, the least unused number.

1, 2, 4, 3, 6, 8, 5, 10, 12, 9, 7, 14, 16, 11, 22, 18, 15, 20, 24, 21, 28, 26, 13, 17, 34, 30, 25, 19, 38, 32, 23, 46, 36, 27, 29, 58, 40, 35, 42, 33, 44, 48, 39, 52, 50, 45, 51, 68, 54, 57, 76, 56, 49, 31, 62, 60, 55, 66, 63, 70, 64, 37, 74, 72, 69, 92, 78, 65

Offset: 1

David James Sycamore, Nov 06 2022

nonn

Let a(n-2) = i, a(n-1) = j. The sequence is generated from divisor relationships j->i, ranging from coprime: gcd(i,j) =1, to partial: 1 < gcd(i,j) < j, to total: gcd(i,j) = j, using conditions described in the definition.
The first 26 terms are the same as those of A280864 and A280866.
A prime cannot occur consequent to condition (i). a(n) = prime p either because p|a(n-1) but not a(n-2); see (ii), or because every prime divisor of a(n-1) also divides a(n-2), as when for example a(n-1) is a prime power q^k and q|a(n-2), which forces a(n) = u prime, see (iii).
If a(n) = u = p from condition (iii), a(n+1) = 2*p. If p|a(n-1)-> a(n) = p we see m*p->p->u (and u may of course be prime, as in ...,13,17,...). 13 is the first prime to appear consequent to condition (ii), see Example. Consecutive primes appear often: (13,17); (53,59); (61,67); ... Sequence is conjectured to be a permutation of the positive integers with primes appearing slowest, and in natural order.
Local minima consist of 1 and the primes p, while 4p dominates the maxima as n increases. - Michael De Vlieger, Nov 06 2022

			a(1) = 1, a(2) = 2 and since 2|a(2) but not a(1), and no other primes are involved, a(3) = 4, the least novel multiple of 2, the squarefree kernel of 2 (by (i)).
Every prime divisor of a(3) = 4 also divides a(2) = 2, thus a(4) = 3, the least unused number (by (iii)).
a(23) = 13 because 13|a(22) = 26, but does not divide a(21) = 28 (by (ii)). Then since every prime divisor of a(23) also divides a(22), a(24) = 17, the least unused term (by (iii)). This is the first occasion of consecutive primes.
a(25) = 34, a(26) = 30 and there are two primes (3,5) which divide 30 but not 34. At this point the least novel multiples of 3 and 5 are 27 and 25 respectively, so a(27) = 25 (by (ii)). This is the first departure from A280864/A280866, which both have a(27) = 45.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Scatterplot of a(n), n = 1..2^20
Michael De Vlieger, Annotated scatterplot of a(n) n = 1..200, showing primes p in red, 2p in light blue, 3p in green, and 4p in dark blue.
Michael De Vlieger, Log-log scatterplot of a(n) n = 1..2^14, showing maxima in red, local minima in blue, highlighting primes p in green and other prime powers in gold.

Cf. A280864, A280866, A352187, A357942, A357963, A357992, A357994.

Block[{a, c, f, g, k, m, q, u, nn}, nn = 120; c[] = False; q[] = 1; Array[Set[{a[#], c[#]}, {#, True}] &, 3]; q[2] = 2; u = 3; Do[m = FactorInteger[a[n - 1]][[All, 1]]; f = Select[m, CoprimeQ[#, a[n - 2]] &]; If[AllTrue[m, Mod[a[n - 2], #] == 0 &], k = u, Set[{k, q[#1]}, {#2, #2/#1}] & @@ First@ MinimalBy[Map[{#, Set[g, q[#]]; While[c[g #], g++]; # g} &, f], Last] ]; Set[{a[n], c[k]}, {k, True}]; If[k == u, While[c[u], u++]], {n, 3, nn}]; Array[a, nn] ] (* Michael De Vlieger, Nov 06 2022 *)

More terms from Michael De Vlieger, Nov 06 2022

A372064 a(n) = A280741(n) - A304741(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, 6, 0, -3, 0, 0, 0, 1, 0, -3, 0, 5, 0, 5, 1, 0, 0, 5, 0, -5, 9, 9, -3, 0, 5, -32, 5, 5, 0, -1, 5, -2, 6, -14, 5, -23, 5, 3, 9, -18, 5, 13, 5, 12, 1, 5, 5, -26, -23, 13, -1, -2, 5, 12, 5, -1, 5, 4, -4, 9, 6, 5, -16, 13, 5, 17, 7, 6, -2, 5, 9

Offset: 1

N. J. A. Sloane, May 09 2024

sign

A372063 compares A280864 and A280866 term-by-term; the present sequence compares their inverses; and A372065 compares where the primes appear.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

Cf. A280864, A280741, A280866, A304741, A372063, A372065.

A375029 Lexicographically least increasing sequence such that for any prime number p, any run of consecutive multiples of p has length exactly 2.

Original entry on oeis.org

1, 2, 4, 5, 10, 12, 15, 20, 22, 33, 36, 38, 57, 60, 70, 77, 88, 90, 105, 112, 114, 171, 172, 258, 261, 290, 300, 303, 404, 406, 609, 612, 646, 665, 700, 702, 741, 760, 770, 847, 848, 954, 957, 1276, 1278, 1491, 1498, 1712, 1713, 3426, 3428, 4285, 4290, 5148

Offset: 1

Rémy Sigrist, Jul 28 2024

nonn

This sequence is a variant of A280864.

			The first terms, alongside their prime factors, are:
  n   a(n)  Prime factors
  --  ----  --------------------
   1     1
   2     2   2
   3     4   2
   4     5       5
   5    10   2   5
   6    12   2 3
   7    15     3 5
   8    20   2   5
   9    22   2       11
  10    33     3     11
  11    36   2 3
  12    38   2                19
  13    57     3              19
  14    60   2 3 5
  15    70   2   5 7
  16    77         7 11
  17    88   2       11

Cf. A280864.

{ p = 0; r = 1; m = 1; for (n = 1, 54, forstep (v = ceil((p+1)/m)*m, oo, m, if (gcd(v, r)==m, print1 (v", "); r = vecprod(factor(p = v)[,1]~); m = r / m; break;););); }

A375030 Irregular triangle T(n, k), n > 0, k = 1..A373797(n), read by rows; the n-th row corresponds to the lexicographically earliest sequence S of A373797(n) distinct integers in the range 1..n such that for any prime number p, any run of consecutive multiples of p in S has length exactly 2.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 1, 2, 4, 1, 2, 6, 3, 1, 2, 6, 3, 1, 2, 4, 3, 6, 8, 1, 2, 4, 3, 6, 8, 1, 2, 4, 3, 9, 5, 10, 8, 1, 2, 4, 3, 9, 5, 10, 8, 1, 2, 4, 3, 6, 8, 5, 10, 12, 9, 1, 2, 4, 3, 6, 8, 5, 10, 12, 9, 1, 2, 4, 3, 6, 10, 5, 7, 14, 12, 9, 1, 2, 4, 3, 6, 8, 5, 15, 12, 14, 7

Offset: 1

Rémy Sigrist at the suggestion of Peter Luschny, Jul 28 2024

			Triangle T(n, k) begins:
    1;
    1;
    1;
    1, 2, 4;
    1, 2, 4;
    1, 2, 6, 3;
    1, 2, 6, 3;
    1, 2, 4, 3, 6, 8;
    1, 2, 4, 3, 6, 8;
    1, 2, 4, 3, 9, 5, 10, 8;
    1, 2, 4, 3, 9, 5, 10, 8;
    1, 2, 4, 3, 6, 8, 5, 10, 12, 9;
    1, 2, 4, 3, 6, 8, 5, 10, 12, 9;
    1, 2, 4, 3, 6, 10, 5, 7, 14, 12, 9;
    1, 2, 4, 3, 6, 8, 5, 15, 12, 14, 7;
    1, 2, 4, 3, 6, 8, 5, 10, 12, 9, 7, 14, 16;
    1, 2, 4, 3, 6, 8, 5, 10, 12, 9, 7, 14, 16;
    1, 2, 4, 3, 6, 8, 5, 15, 9, 16, 14, 7, 12, 18;
    1, 2, 4, 3, 6, 8, 5, 15, 9, 16, 14, 7, 12, 18;
    ...

Rémy Sigrist, Table of n, a(n) for n = 1..386 (rows for n = 1..31 flattened)
Rémy Sigrist, PARI program.
Peter Luschny, Maple program.

Cf. A280864, A373797.

Maple
```
# See Links section.
```
PARI
```
\\ See Links section.
```

A375024 a(n) is the length of the largest sequence S of distinct integers in the range 1..n such that for any prime number p, any run of consecutive multiples of p in S has length exactly 2, and two consecutive terms in S have some common prime factor.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 7, 7, 7, 7, 7, 7, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 13, 13, 16, 16, 16, 16, 19, 19, 19, 19, 19, 19

Offset: 1

Rémy Sigrist, Jul 28 2024

Sequences like A280864 can be split into segments of consecutive terms with properties similar to the sequences S that we are considering here.

			Some solutions for small n:
  n   a(n)  Solution S
  --  ----  --------------------------------------------------------------
   1     1  1
   4     2  2,4
   6     3  2,6,3
  10     4  3,6,10,5
  15     7  3,6,10,15,12,14,7
  21    10  3,6,10,15,12,14,21,18,20,5
  33    13  3,6,10,15,12,14,21,18,22,33,24,20,5
  35    16  3,6,10,15,12,14,21,18,20,35,28,22,33,24,26,13
  39    19  3,6,10,15,12,14,21,18,20,35,28,22,33,24,26,39,36,34,17
  45    22  5,10,6,15,20,12,21,14,18,33,22,24,39,26,36,45,40,28,35,30,42,7

Rémy Sigrist, C++ program

Cf. A280864, A283832, A373797.

a(n) <= A373797(n).

a(p) = a(p-1) for any prime number p.

A282061 Lexicographically earliest sequence of distinct terms such that, for any prime p, any run of consecutive multiples of p has length exactly 2, and the terms in such a run have the same p-adic valuation.

Original entry on oeis.org

1, 2, 6, 3, 4, 12, 15, 5, 7, 14, 10, 35, 21, 24, 8, 9, 18, 22, 11, 13, 26, 30, 105, 28, 20, 45, 36, 44, 33, 39, 52, 60, 165, 77, 42, 66, 55, 40, 56, 63, 72, 88, 99, 90, 70, 91, 65, 80, 16, 17, 34, 38, 19, 23, 46, 50, 25, 27, 54, 58, 29, 31, 62, 74, 37, 32, 96

Offset: 1

Rémy Sigrist, Feb 05 2017

This sequence is similar to A280864, with an additional constraint on the p-adic valuation of consecutive multiples of any prime p. However, the graphs of those two sequences are quite different.

			See Links section.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Illustration of the first terms
Rémy Sigrist, PARI program for A282061

Cf. A280864.

A285190 Records in A283832.

Original entry on oeis.org

1, 2, 3, 4, 10, 17, 20, 44, 52, 54, 78, 100, 102, 113, 135, 139, 140, 162, 180, 195, 259, 270, 288, 334, 336

Offset: 1

N. J. A. Sloane, Apr 25 2017

Cf. A280864, A283832, A285191.

A285191 Where records occur in A283832.

Original entry on oeis.org

1, 2, 3, 4, 6, 16, 29, 32, 73, 185, 208, 248, 394, 521, 1349, 1431, 1564, 1901, 2149, 2462, 2613, 8288, 9987, 10858, 24526, 27276, 40736, 53726, 104154, 134561, 160874, 173744

Offset: 1

N. J. A. Sloane, Apr 25 2017

Cf. A280864, A283832, A285190.

a(26)-a(32) from Lars Blomberg, Jun 06 2017

A338350 Lexicographically earliest infinite sequence of distinct odd terms such that, for any prime p, any run of consecutive multiples of p has length exactly 2.

Original entry on oeis.org

1, 3, 9, 5, 15, 21, 7, 11, 33, 27, 13, 39, 45, 25, 17, 51, 57, 19, 23, 69, 63, 35, 55, 77, 49, 29, 87, 75, 65, 91, 105, 135, 31, 93, 81, 37, 111, 99, 121, 41, 123, 117, 143, 165, 195, 169, 43, 129, 141, 47, 53, 159, 147, 119, 85, 95, 133, 161, 115, 125, 59

Offset: 1

N. J. A. Sloane, Oct 29 2020

nonn

A version of A280864 but only using odd numbers.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A338350

Cf. A064413, A152247, A280864, A338444,

PARI
```
See Links section.
```

More terms from Rémy Sigrist, Oct 30 2020

A339316 a(1) = 2; for n > 1, a(n) = smallest composite number not occurring earlier which does not share a factor with a(n-1).

Original entry on oeis.org

2, 9, 4, 15, 8, 21, 10, 27, 14, 25, 6, 35, 12, 49, 16, 33, 20, 39, 22, 45, 26, 51, 28, 55, 18, 65, 24, 77, 30, 91, 32, 57, 34, 63, 38, 69, 40, 81, 44, 75, 46, 85, 36, 95, 42, 115, 48, 119, 50, 87, 52, 93, 56, 99, 58, 105, 62, 111, 64, 117, 68, 121, 54, 125, 66, 133, 60, 143, 70, 123, 74, 129

Offset: 1

Scott R. Shannon, Nov 30 2020

nonn

The sequence excludes primes as otherwise the terms would simply be all the ordered integers >= 2. The terms appear to cluster around two lines; the lower line is a(n) ~ n while the upper lines starts with a gradient of approximately 2 and then slowly flattens. It is possible this gradient approaches 1 as n->infinity.

			a(2) = 9, as a(1) = 2 thus a(2) cannot contain 2 as a factor and cannot be a prime. The lowest unused composite matching these criteria is 9.
a(3) = 4, as a(2) = 9 and thus a(3) cannot contain 3 as a factor and cannot be a prime. The lowest unused composite matching these criteria is 4.
a(4) = 15, as a(3) = 4 and thus a(4) cannot contain 2 as a factor and cannot be a prime. The lowest unused composite matching these criteria is 15.

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

Cf. A337687, A000961, A064413, A336957, A098550, A020639, A280864.

isok(k, fprec, v) = {if (!isprime(k) && #select(x->(x==k), v) == 0, #setintersect(Set(factor(k)[,1]), fprec) == 0;);}
lista(nn) = {my(va= vector(nn)); va[1] = 2; for (n=2, nn, my(k=2, fprec = Set(factor(va[n-1])[,1])); while (! isok(k, fprec, va), k++); va[n] = k;); va;} \\ Michel Marcus, Nov 30 2020

from sympy import isprime, primefactors as pf
def aupton(terms):
  alst, aset = [2], {2}
  for n in range(2, terms+1):
    m, prevpf = 4, set(pf(alst[-1]))
    while m in aset or isprime(m) or set(pf(m)) & prevpf != set(): m += 1
    alst.append(m); aset.add(m)
  return alst
print(aupton(72)) # Michael S. Branicky, Feb 09 2021

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A372064 a(n) = A280741(n) - A304741(n).

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A375029 Lexicographically least increasing sequence such that for any prime number p, any run of consecutive multiples of p has length exactly 2.

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A375030 Irregular triangle T(n, k), n > 0, k = 1..A373797(n), read by rows; the n-th row corresponds to the lexicographically earliest sequence S of A373797(n) distinct integers in the range 1..n such that for any prime number p, any run of consecutive multiples of p in S has length exactly 2.

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A375024 a(n) is the length of the largest sequence S of distinct integers in the range 1..n such that for any prime number p, any run of consecutive multiples of p in S has length exactly 2, and two consecutive terms in S have some common prime factor.

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A282061 Lexicographically earliest sequence of distinct terms such that, for any prime p, any run of consecutive multiples of p has length exactly 2, and the terms in such a run have the same p-adic valuation.

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A285190 Records in A283832.

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A285191 Where records occur in A283832.

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A338350 Lexicographically earliest infinite sequence of distinct odd terms such that, for any prime p, any run of consecutive multiples of p has length exactly 2.

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A339316 a(1) = 2; for n > 1, a(n) = smallest composite number not occurring earlier which does not share a factor with a(n-1).

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