\n" if ((62*$ - cp's OEIS frontend

A386485 a(0) = 1; thereafter a(n) = 5n^2 - 5n + 2.

1, 2, 12, 32, 62, 102, 152, 212, 282, 362, 452, 552, 662, 782, 912, 1052, 1202, 1362, 1532, 1712, 1902, 2102, 2312, 2532, 2762, 3002, 3252, 3512, 3782, 4062, 4352, 4652, 4962, 5282, 5612, 5952, 6302, 6662, 7032, 7412, 7802, 8202, 8612, 9032, 9462, 9902, 10352, 10812, 11282, 11762, 12252, 12752, 13262, 13782, 14312

Offset: 0

N. J. A. Sloane, Aug 18 2025

Maximum number of regions that can be formed in the plane by drawing n regular pentagons (of any size). Differs from A062786 and A124080 by a small constant shift, but is included here because of its geometrical applications.

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A386480, A051890, A077588, A069894.

A386482 a(1)=1, a(2)=2; thereafter a(n) is either the greatest number k < a(n-1) not already used such that gcd(k, a(n-1)) > 1, or if no such k exists then a(n) is the smallest number k > a(n-1) not already used such that gcd(k, a(n-1)) > 1.

Original entry on oeis.org

1, 2, 4, 6, 3, 9, 12, 10, 8, 14, 7, 21, 18, 16, 20, 15, 5, 25, 30, 28, 26, 24, 22, 11, 33, 27, 36, 34, 32, 38, 19, 57, 54, 52, 50, 48, 46, 44, 42, 40, 35, 45, 39, 13, 65, 60, 58, 56, 49, 63, 51, 17, 68, 66, 64, 62, 31, 93, 90, 88, 86, 84, 82, 80, 78, 76, 74, 72, 70, 55, 75, 69, 23, 92, 94, 47, 141, 138, 136, 134, 132, 130, 128

Offset: 1

N. J. A. Sloane, Aug 15 2025, based on email messages from Geoffrey Caveney

Similar to the EKG sequence A064413, but whereas in that sequence a(n) is chosen to be as small as possible, here the primary goal is to choose a(n) to be less than a(n-1) and as close to it as possible. This sequence first differs from the EKG sequence at n = 8, where a(8) = k = 10 is closer to a(7) = 12 than A064413(8) = 8 is.
A significant difference from the EKG sequence is that the primes do not appear in their natural order. Also, it is not always true that a prime p is preceded by 2*p when it first appears. 4k+3 primes appear to be preceded by smaller multiples than 4k+1 primes.
It is conjectured that every positive number appears.
It is interesting to study what happens if the first two terms are taken to be 1,s, with s >= 2, or if the first s terms are taken to be 1,2,3,...,s, with s >= 2. Call two such sequences equivalent if they eventually merge. The 1,3 and 1,2,3 sequences merge with each other after half-a-dozen terms. But at present we do not know if they merge with the 1,2 sequence.
It appears that many sequences that start 1,s and 1,2,3,...,s with small s merge with one of the sequences 1,2 or 1,2,3 or 1,2,3,...,11.
[The preceding comments are from Geoffrey Caveney's emails.]
From Michael De Vlieger, Aug 15 2025: (Start)
There are long runs of terms with the same parity in this sequence. For example, beginning at a(481) = 948, there are 100 consecutive even terms. Starting with a(730076) = 1026330, there are 100869 consecutive even terms, followed by 36709 consecutive odd terms. Runs of even terms tend to be longer than those of odd.
There are long runs of first differences of -2 and -6 in this sequence, and that there appear to be three phases. The predominant (A) phase has a(n) = a(n-1)-2, the second (B) phase has a(n) = a(n-1)-6, and then there is a turbulent (C) phase [C] with varied differences.
Generally the even runs correspond to differences a(n)-a(n-1) = 2 and feature square-free terms separated by an odd number of terms in A126706. Phase [C] tends to be largely odd squarefree semiprimes and includes prime powers. (End)

Geoffrey Caveney, Emails to N. J. A. Sloane, Aug 13 2025 - Aug 15 2025.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16, showing primes in red, proper prime powers in gold, squarefree composites in green, and numbers that are neither squarefree nor prime powers in blue and purple, where purple represents powerful numbers that are not prime powers.

Cf. A064413 (EKG), A387072 (inverse), A387073 (record high points), A387074 (indices of record high points), A387075 (first differences), A387076 (primes in order of appearance), A387077 (indices of primes), A387078 (run lengths of consecutive odd and even terms), A387080 (variant that begins with 1,3).

See also A386483, A386484, A387087-A387090.

aList[n_] := Module[{an = 2, aset = <|2 -> True|>, m}, Reap[Sow[1]; Sow[an];
Do[m = SelectFirst[Range[an - 1, 2, -1], ! KeyExistsQ[aset, #] && GCD[#, an] > 1 & ];
If[MissingQ[m], m = NestWhile[# + 1 &, an + 1, !(! KeyExistsQ[aset, #] && GCD[#, an] > 1) & ]];
aset[m] = True; an = m; Sow[an], {n - 2}]][[2, 1]]]; aList[83]  (* Peter Luschny, Aug 15 2025 *)

PARI
```
\\ See Links section.
```

from math import gcd
from itertools import count, islice
def A386482_gen(): # generator of terms
    yield 1
    an, aset = 2, {2}
    while True:
        yield an
        m = next((k for k in range(an-1, 1, -1) if k not in aset and gcd(k, an) > 1), False)
        if not m: m = next(k for k in count(an+1) if k not in aset and gcd(k, an) > 1)
        an = m
        aset.add(an)
print(list(islice(A386482_gen(), 83))) # Michael S. Branicky, Aug 15 2025

A383445 Index of n in A377090, or -1 if n does not appear there.

Original entry on oeis.org

0, 3, 1, 5, 9, 15, 10, 16, 14, 17, 23, 21, 24, 22, 28, 31, 29, 36, 30, 35, 37, 43, 41, 47, 42, 48, 54, 49, 55, 61, 56, 62, 67, 63, 68, 66, 71, 75, 80, 76, 81, 79, 82, 89, 91, 88, 90, 92, 99, 93, 98, 100, 105, 111, 106, 112, 110, 113, 124, 122, 125, 123, 126, 128, 121, 127, 133, 138, 134, 139, 137, 147, 150, 148, 146, 149, 153

Offset: 0

N. J. A. Sloane, Apr 30 2025

nonn

It is conjectured (see A377090) that every positive integer appears exactly once either here or in A383446.

N. J. A. Sloane, Table of n, a(n) for n = 0..4996

Cf. A377090, A383446.

A382715 The term in A377091 that immediately precedes n, or 0 if n does not appear in A377091.

Original entry on oeis.org

0, 1, -1, 3, 4, -3, 6, 7, 13, 9, 10, 11, -12, 18, 14, 15, 16, -18, 20, 21, 25, 23, 19, 28, -24, 22, 26, 27, 30, 31, 32, -32, 29, 33, 34, 35, 36, 37, 38, 39, 40, -39, 42, 43, 44, 45, 46, 47, 48, 49, -49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, -59, 62, 63, 64, 65

Offset: 1

N. J. A. Sloane, Apr 09 2025

sign

It is a strong conjecture that every integer appears in A377091, so it is unlikely there will be a second 0 term.

			A377091(9) = -3 is followed by A377091(10) = 6, so a(6) = -3.

N. J. A. Sloane, Table of n, a(n) for n = 1..10060
N. J. A. Sloane, Table of n, a(n) for n = 1..125060

Cf. A377091, A379059, A379789, A382716, A382717, A382718.

A382155 a(n) = (n+1)! if n <= 2; thereafter a(n) = 4n if n even or 2n if n odd.

Original entry on oeis.org

1, 2, 6, 6, 16, 10, 24, 14, 32, 18, 40, 22, 48, 26, 56, 30, 64, 34, 72, 38, 80, 42, 88, 46, 96, 50, 104, 54, 112, 58, 120, 62, 128, 66, 136, 70, 144, 74, 152, 78, 160, 82, 168, 86, 176, 90, 184, 94, 192, 98, 200, 102, 208, 106, 216, 110, 224, 114, 232, 118, 240, 122, 248, 126, 256, 130, 264, 134, 272, 138, 280, 142, 288, 146, 296

Offset: 0

N. J. A. Sloane, Mar 23 2025

nonn

Let G denote the 2-dimensional grid obtained from the square grid Z X Z by deleting the vertices with both coordinates odd and the four edges at each of those vertices (see link). G has vertices with valency either 2 (one coordinate even and one odd, indicated by X) or 4 (both coordinates even, indicated by O). The present sequence is the coordination sequence of G with respect to a vertex of valency 2.

See A382154 for further information.

Paolo Xausa, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, The grid G (Each X-node is joined to two O-nodes, and each O-node to four X-nodes.)
N. J. A. Sloane, Illustrates the initial terms of the coordination sequence of G with respect to a vertex of degree 2. E.g. the 6 red vertices labeled 3 correspond to a(3) = 6, and the 16 blue vertices labeled 4 correspond to a(4) = 16.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Partial sums give A382156.

Cf. A382154, A319384.

Join[{1, 2, 6}, Riffle[4*# + 2, 8*(# + 1)]] & [Range[50]] (* Paolo Xausa, Mar 24 2025 *)

def A382155(n): return (1,2,6)[n] if n<3 else n<<(2>>(n&1)) # Chai Wah Wu, Mar 24 2025

G.f.: (-2*x^6+5*x^4+2*x^3+4*x^2+2*x+1)/(1-x^2)^2.

A382154 a(0) = 1; thereafter a(n) = 2n if n even or 4n if n odd.

Original entry on oeis.org

1, 4, 4, 12, 8, 20, 12, 28, 16, 36, 20, 44, 24, 52, 28, 60, 32, 68, 36, 76, 40, 84, 44, 92, 48, 100, 52, 108, 56, 116, 60, 124, 64, 132, 68, 140, 72, 148, 76, 156, 80, 164, 84, 172, 88, 180, 92, 188, 96, 196, 100, 204, 104, 212, 108, 220, 112, 228, 116, 236, 120, 244, 124, 252, 128, 260, 132, 268, 136, 276, 140, 284, 144, 292

Offset: 0

N. J. A. Sloane, Mar 23 2025

nonn

Let G denote the 2-dimensional grid obtained from the square grid Z X Z by deleting the vertices with both coordinates odd and the four edges at each of those vertices (see link). G has vertices with valency either 2 (one coordinate even and one odd, indicated by X) or 4 (both coordinates even, indicated by O). The present sequence is the coordination sequence of G with respect to a vertex of valency 4.

G arises in connection with the six-vertex lattice model of statistical mechanics (see Gorin-Nicoletti).

Gorin, Vadim, and Matthew Nicoletti. "Six-Vertex Model and Random Matrix Distributions," Bull. Amer. Math. Soc., 62:2 (2025), 175-234 (See Fig. 1.2).

Paolo Xausa, Table of n, a(n) for n = 0..10000
Vadim Gorin and Matthew Nicoletti, Six-Vertex Model and Random Matrix Distributions, arXiv:2309.12495 [math-ph], 2023-2024.
N. J. A. Sloane, The grid G (Each X-node is joined to two O-nodes, and each O-node to four X-nodes.)
N. J. A. Sloane, Illustrates the initial terms of the coordination sequence of G with respect to a vertex of degree 4. E.g. the 12 red vertices labeled 3 correspond to a(3) = 12.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Partial sums give A319384.

Cf. A382155, A382156.

Join[{1}, Riffle[8*# - 4, 4*#]] & [Range[50]] (* Paolo Xausa, Mar 24 2025 *)

def A382154(n): return n<<(1<<(n&1)) if n else 1 # Chai Wah Wu, Mar 24 2025

G.f.: (1+x^2)*(1+4*x+x^2)/(1-x^2)^2.

A381115 Composite terms in A381019 in order of appearance.

Original entry on oeis.org

4, 9, 8, 6, 25, 12, 10, 49, 15, 16, 14, 27, 20, 21, 22, 18, 35, 24, 169, 28, 33, 26, 85, 32, 57, 77, 30, 34, 39, 55, 38, 51, 40, 91, 36, 121, 42, 65, 44, 45, 529, 48, 119, 46, 95, 81, 143, 50, 63, 52, 54, 115, 56, 841, 187, 69, 62, 125, 87, 64, 133, 75, 58, 221

Offset: 1

N. J. A. Sloane, Feb 14 2025

nonn

Michael S. Branicky, Table of n, a(n) for n = 1..336

Cf. A381019, A381116, A381117.

nn = 500; c[_] = False; u = v = 2; a[1] = 1;
Monitor[Reap[
  Do[k = u;
    While[Or[c[k],
      ! CoprimeQ[k, Product[a[h], {h, n - Min[k, n - 1], n - 1}] ] ],
      If[k > n - 1, k = v, k++]];
    Set[{a[n], c[k]}, {k, True}];
    If[CompositeQ[k], Sow[k]];
    If[k == u, While[c[u], u++]];
If[k == v, While[Or[c[v], CompositeQ[v]], v++]], {n, 2, nn}] ][[-1, 1]], n] (* Michael De Vlieger, Feb 14 2025 *)

from math import gcd
from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    alst, aset, an, m = [1], {1}, 1, 2
    for n in count(2):
        if an > 3 and not isprime(an):
            yield an
        an = next(k for k in count(m) if k not in aset and all(gcd(alst[-j], k) == 1 for j in range(1, min(k, n-1)+1)))
        alst.append(an)
        aset.add(an)
        while m in aset: m += 1
print(list(islice(agen(), 64))) # Michael S. Branicky, Feb 14 2025

A379059 Index of n in A377091, or -1 if n does not appear there.

Original entry on oeis.org

0, 1, 2, 5, 6, 7, 10, 11, 12, 22, 23, 24, 25, 21, 33, 34, 35, 36, 32, 46, 45, 44, 48, 47, 52, 43, 49, 50, 51, 64, 63, 62, 61, 65, 66, 67, 68, 69, 70, 71, 72, 73, 82, 83, 84, 85, 86, 87, 88, 89, 90, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 188

Offset: 0

N. J. A. Sloane, Dec 26 2024

nonn

It is conjectured that every integer appears in A377091.
Conjecture 1: For n >= 10, |a(n) - 2*n| < 2*sqrt(n); for n >= 1000, |a(n) - 2*n| < 1.59*sqrt(n) (compare A379786). - N. J. A. Sloane, Jan 19 2025
Conjecture 2: lim sup |a(n) - 2*n|/sqrt(n) = sqrt(2) as n -> oo. - N. J. A. Sloane and Paolo Xausa, Feb 02 2025

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Paolo Xausa, Scatterplot of |a(n) - 2*n| for n <= 100000. The orange line is sqrt(2*n).

Cf. A377091, A379060, A379786.

A379057 a(0) = 0; thereafter a(n) is the least integer (in absolute value) not yet in the sequence such that the absolute difference between a(n-1) and a(n) is a square > 1; in case of a tie, preference is given to the positive value.

Original entry on oeis.org

0, 4, -5, -1, 3, -6, -2, 2, 6, -3, 1, 5, -4, -8, 8, 12, -13, -9, 7, 11, -14, -10, 15, 19, 10, 14, -11, -7, 9, 13, -12, -16, 20, 16, -20, -24, -15, -19, 17, 21, 25, 29, 33, 24, -25, -21, -17, -26, -22, -18, 18, 22, 26, -23, -27, -31, -35, -39, -30, 34, 30, -34

Offset: 0

N. J. A. Sloane, Dec 25 2024

sign

Inspired by A377091 and A277616.

Michael S. Branicky, Table of n, a(n) for n = 0..10000

Cf. A277616, A377091, A379058 (first differences), A380313 (partial sums).

A379057list[nmax_] := Module[{s, a, u = 1}, s[_] := False; s[0] = True; NestList[(While[s[u] && s[-u], u++]; a = u; While[s[a] || Abs[# - a] <= 1 || !IntegerQ[Sqrt[Abs[# - a]]], a = Boole[a < 0] - a]; s[a] = True; a) &, 0, nmax]];
A379057list[100] (* Paolo Xausa, Mar 21 2025 *)

from math import isqrt
from itertools import count, islice
def cond(n): return n > 1 and isqrt(n)**2 == n
def agen(): # generator of terms
    an, aset, m = 0, {0}, 1
    for n in count(0):
        yield an
        an = next(s for k in count(m) for s in [k, -k] if s not in aset and cond(abs(an-s)))
        aset.add(an)
        while m in aset and -m in aset: m += 1
print(list(islice(agen(), 62))) # Michael S. Branicky, Dec 25 2024

More terms from Michael S. Branicky, Dec 25 2024

A379053 Lexicographically earliest infinite sequence of distinct positive numbers with the property that n is a member of the sequence iff a(n) is not a prime.

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 9, 10, 12, 14, 13, 15, 16, 18, 20, 21, 19, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 23, 36, 37, 38, 39, 40, 42, 44, 45, 46, 48, 49, 43, 50, 51, 52, 54, 55, 53, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 61, 70, 72, 74, 75, 76, 77, 78, 71

Offset: 1

N. J. A. Sloane, Dec 17 2024

nonn

The sequence tells you exactly which terms of the sequence are either 1 or composite.

See the Comments in A379051 for further information.

Michael De Vlieger, Table of n, a(n) for n = 1..65536 [Terms 1 to 10000 from Scott R. Shannon]

Cf. A079000, A079253, A079254, A079313, A080032, A085925, A099797, A099798, A121053, A334067, A377901, A379051.

nn = 120; u = 3; v = {}; w = {}; c = 4;
{1}~Join~Reap[Do[
  If[MemberQ[w, n],
    k = c; w = DeleteCases[w, n],
    m = Min[{c, u, v}]; If[And[CompositeQ[m], n < m],
      AppendTo[v, n]];
      If[Length[v] > 0,
        If[v[[1]] == m,
        v = Rest[v]]]; k = m];
    AppendTo[w, k]; If[k == c, c++; While[PrimeQ[c], c++]]; Sow[k];
If[n + 1 >= u, u++; While[CompositeQ[u], u++]], {n, 2, nn}] ][[-1, 1]] (* Michael De Vlieger, Dec 17 2024 *)

When sorted, this appears to be the complement of [2, 5, 11, 17, 29, and prime(2*t+1), t >= 35]. - Scott R. Shannon, Dec 18 2024

More terms from Michael De Vlieger, Dec 17 2024

cp's OEIS Frontend

User: \n" if ((62*$

A386485 a(0) = 1; thereafter a(n) = 5*n^2 - 5*n + 2.

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A386482 a(1)=1, a(2)=2; thereafter a(n) is either the greatest number k < a(n-1) not already used such that gcd(k, a(n-1)) > 1, or if no such k exists then a(n) is the smallest number k > a(n-1) not already used such that gcd(k, a(n-1)) > 1.

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A383445 Index of n in A377090, or -1 if n does not appear there.

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A382715 The term in A377091 that immediately precedes n, or 0 if n does not appear in A377091.

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A382155 a(n) = (n+1)! if n <= 2; thereafter a(n) = 4*n if n even or 2*n if n odd.

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A382154 a(0) = 1; thereafter a(n) = 2*n if n even or 4*n if n odd.

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A381115 Composite terms in A381019 in order of appearance.

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A379059 Index of n in A377091, or -1 if n does not appear there.

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A379057 a(0) = 0; thereafter a(n) is the least integer (in absolute value) not yet in the sequence such that the absolute difference between a(n-1) and a(n) is a square > 1; in case of a tie, preference is given to the positive value.

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A386485 a(0) = 1; thereafter a(n) = 5n^2 - 5n + 2.

A382155 a(n) = (n+1)! if n <= 2; thereafter a(n) = 4n if n even or 2n if n odd.

A382154 a(0) = 1; thereafter a(n) = 2n if n even or 4n if n odd.