A364054 -id:A364054 - cp's OEIS frontend

A366470 a(n) = A364054(n-1) mod prime(n-1).

1, 0, 1, 4, 4, 2, 2, 0, 15, 3, 1, 26, 26, 24, 24, 18, 18, 16, 16, 12, 12, 6, 81, 81, 73, 73, 71, 63, 57, 57, 29, 29, 23, 23, 13, 13, 1, 158, 154, 154, 148, 148, 138, 138, 134, 134, 122, 122, 118, 118, 114, 114, 112, 112, 106, 106, 100, 100, 94, 94, 92

Offset: 2

N. J. A. Sloane, Oct 21 2023

nonn

Chai Wah Wu, Table of n, a(n) for n = 2..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
Michael De Vlieger, Log log scatterplot of a(n), n = 2..2^20, showing a(n) = 0 instead as a(n) = 1/10, with a color function showing b(n) = 0 in black, b(n) = 1 in blue, b(n) = 2 in green, b(n) = 3 in chartreuse, and b(n) = 4 in red, where b() = A366475(). (The plot accentuates larger values of b(n) through larger size, so adjacent smaller values may be eclipsed.)
Chai Wah Wu, Graph of first 10^8 terms

Cf. A364054, A366475.

nn = 2^20; c[] := False; m[] := 0; j = 1; c[0] = c[1] = True;
  Monitor[Do[p = Prime[n - 1]; r = Mod[j, p];
    While[Set[k, p m[p] + r ]; c[k], m[p]++];
    Set[{a[n - 1], c[k], j}, {r, True, k}], {n, 2, nn + 1}], n];
Array[a, nn] (* Michael De Vlieger, Oct 27 2023 *)

from itertools import count, islice
from sympy import nextprime
def A366470_gen(): # generator of terms
    a, aset, p = 1, {0,1}, 2
    while True:
        yield a
        for b in count(a,p):
            if b not in aset:
                aset.add(b)
                a = b%(p:=nextprime(p))
                break
A366470_list = list(islice(A366470_gen(),30)) # Chai Wah Wu, Oct 22 2023

A364054(n) = A366475(n)*prime(n-1) + a(n) for n > 1. - Michael De Vlieger, Mar 06 2024

A366475 a(n) = (A364054(n) - A366470(n))/prime(n-1).

Original entry on oeis.org

1, 2, 2, 0, 1, 0, 1, 2, 2, 1, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 3, 0, 2, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 2, 2, 1, 2, 1, 2, 0, 1, 2, 0, 2, 0

Offset: 2

N. J. A. Sloane, Oct 26 2023

nonn

a(29) = 3. When, if ever, does 4 appear?

Answer: a(28025) = 4. - Michael De Vlieger, Oct 26 2023

			   n p(n-1)  x   y  a(n)  [x = A364054(n), y = A366470(n)]
   1   (1)   1   -   -    [a(n) = (x-y)/p(n-1)]
   2    2    3   1   1
   3    3    6   0   2
   4    5   11   1   2
   5    7    4   4   0
   6   11   15   4   1
   7   13    2   2   0
...

Michael De Vlieger, Table of n, a(n) for n = 2..65536
Michael De Vlieger, 2048 X 2048 raster showing a(n), n = 1..4194304 in rows of 2048 terms, left to right, then continued below for 2048 rows total. Color indicates terms as follows: black = 0, blue = 1, green = 2, gold = 3, red = 4.

Cf. A364054, A366470, A366477 (records).

nn = 2^20;
  c[] := False; m[] := 0; a[1] = j = 1; c[0] = c[1] = True;
  Monitor[Do[p = Prime[n - 1]; r = Mod[j, p];
    While[Set[k, p m[p] + r ]; c[k], m[p]++];
    Set[{a[n], b[n], c[k], j}, {k, m[p], True, k}], {n, 2, nn}], n];
Array[b, nn-1, 2] (* Michael De Vlieger, Oct 26 2023 *)

from itertools import count, islice
from sympy import nextprime
def A366475_gen(): # generator of terms
    a, aset, p = 1, {0,1}, 1
    while True:
        p = nextprime(p)
        b = a%p
        for i in count(0):
            if b not in aset:
                aset.add(b)
                a = b
                break
            b += p
        yield i
A366475_list = list(islice(A366475_gen(),30)) # Chai Wah Wu, Oct 27 2023

A368384 Records in A364054.

Original entry on oeis.org

1, 3, 6, 11, 15, 19, 38, 61, 63, 67, 71, 77, 83, 85, 164, 170, 174, 277, 384, 385, 389, 393, 399, 401, 407, 413, 417, 425, 826, 848, 1279, 1285, 1291, 1301, 1325, 1333, 1341, 1367, 1401, 1431, 1443, 1455, 1463, 1811, 1981, 1987, 2996, 3054, 3106, 3154, 3200, 3226, 3238, 3266, 3278, 4390, 4402, 4916, 7293, 7299, 7533, 7577

Offset: 1

N. J. A. Sloane, Mar 05 2024

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A364054, A368385.

nn = 2^20; c[] := False; m[] := 0; s = 0; a[1] = j = 1;
c[0] = c[1] = True;
{1}~Join~Reap[Monitor[Do[p = Prime[n - 1]; r = Mod[j, p];
     While[Set[k, p  m[p] + r]; c[k], m[p]++];
     Set[{c[k], j}, {True, k}];
If[k > s, s = k; Sow[s]], {n, 2, nn}], n]][[-1, 1]] (* Michael De Vlieger, Mar 05 2024 *)

A368385 Indices of records in A364054.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 22, 23, 25, 27, 28, 29, 63, 65, 67, 69, 71, 73, 75, 77, 79, 80, 83, 84, 86, 88, 91, 93, 95, 97, 100, 104, 108, 110, 112, 114, 151, 167, 169, 170, 177, 182, 189, 192, 195, 197, 200, 202, 305, 307, 352, 353, 355, 369, 375, 380, 385, 388, 391, 395, 397, 412, 416, 421, 429, 432

Offset: 1

N. J. A. Sloane, Mar 05 2024

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A364054, A368384.

nn = 2^20; c[] := False; m[] := 0; s = 0; a[1] = j = 1;
c[0] = c[1] = True;
{1}~Join~Reap[Monitor[Do[p = Prime[n - 1]; r = Mod[j, p];
     While[Set[k, p  m[p] + r]; c[k], m[p]++];
     Set[{c[k], j}, {True, k}];
If[k > s, s = k; Sow[n]], {n, 2, nn}], n] ][[-1, 1]] (* Michael De Vlieger, Mar 05 2024 *)

A366911 a(n) = (A364054(n+1) - A364054(n)) / prime(n) (where prime(n) denotes the n-th prime number).

Original entry on oeis.org

1, 1, 1, -1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, 1, 1, 1, -3, 2, -2, 1, -1, 1, -1, 2, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1

Offset: 1

Rémy Sigrist, Oct 27 2023

sign

a(n) is the number of steps of size prime(n) in going from A364054(n) to A364054(n+1).

			a(7) = (A364054(8) - A364054(7)) / prime(7) = (19 - 2) / 17 = 1.

N. J. A. Sloane, Table of n, a(n) for n = 1..9999
Rémy Sigrist, PARI program

Cf. A160357, A364054, A366912 (partial sums).

nn = 2^16; c[] := False; m[] := 0; j = 1; c[0] = c[1] = True;
  Monitor[Do[p = Prime[n - 1]; r = Mod[j, p];
    While[Set[k, p m[p] + r ]; c[k], m[p]++];
    Set[{a[n - 1], c[k], j}, {(k - j)/p, True, k}], {n, 2, nn + 1}], n];
Array[a, nn] (* Michael De Vlieger, Oct 27 2023 *)

PARI
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See Links section.
```

from itertools import count, islice
from sympy import nextprime
def A366911_gen(): # generator of terms
    a, aset, p = 1, {0,1}, 2
    while True:
        k, b = divmod(a,p)
        for i in count(-k):
            if b not in aset:
                aset.add(b)
                a, p = b, nextprime(p)
                yield i
                break
            b += p
A366911_list = list(islice(A366911_gen(),30)) # Chai Wah Wu, Oct 27 2023

A366477 a(n) = smallest k such that A366475(k) >= n, or -1 if no such k exists.

Original entry on oeis.org

2, 3, 29, 28025, 2467754261

Offset: 1

N. J. A. Sloane, Oct 26 2023

Cf. A364054, A366475.

nn = 2^16; c[] := False; m[] := 0; j = 1; c[0] = c[1] = True; q[_] := 0; s = -1;
  Monitor[Do[p = Prime[n - 1]; r = Mod[j, p];
    While[Set[k, p m[p] + r ]; c[k], m[p]++];
    (If[q[#] == 0, Set[q[#], n]]; If[# > s, s = #]) &[ m[p] ];
    Set[{c[k], j}, {True, k}], {n, 2, nn}], n];
Array[q, s] (* Michael De Vlieger, Oct 27 2023 *)

from itertools import count
from sympy import nextprime
def A366477(n):
    a, aset, p = 1, {0,1}, 1
    for i in count(2):
        p = nextprime(p)
        b = a%p
        for j in count(0):
            if b not in aset:
                aset.add(b)
                a = b
                break
            b += p
        if j>=n:
            return i # Chai Wah Wu, Oct 27 2023

a(4) = 28025 from Michael De Vlieger, Oct 26 2023, who also reports that 5 does not appear in the first 2^24 terms of A366475.

a(5) from Chai Wah Wu, Oct 28 2023

A366912 Partial sums of A366911: a(1) = 0, and for n > 0, a(n+1) = a(n) + A366911(n).

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 2, 3, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 6, 5, 6, 5, 6, 7, 8, 5, 7, 5, 6, 5, 6, 5, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 8, 6, 7, 8, 9, 8, 9, 8

Offset: 1

Rémy Sigrist, Oct 27 2023

nonn

By analogy with A064289, a(n) corresponds to the height of A364054(n) = number of addition steps - number of subtraction steps to produce it.

			a(5) = A366911(1) + A366911(2) + A366911(3) + A366911(4) = 1 + 1 + 1 - 1 = 2.

Rémy Sigrist, Colored scatterplot of the first 100000 terms of A364054 (where the color is function of a(n))
Rémy Sigrist, PARI program

Cf. A064289, A364054, A366911, A366913.

nn = 2^16; c[] := False; m[] := 0; j = 1; s = b[1] = 0;
  c[0] = c[1] = True;
  Monitor[Do[p = Prime[n - 1]; r = Mod[j, p];
    While[Set[k, p m[p] + r ]; c[k], m[p]++]; s += (k - j)/p;
    Set[{a[n - 1], b[n - 1], c[k], j}, {(k - j)/p, s, True, k}],
    {n, 2, nn + 1}], n];
Array[b, nn] (* Michael De Vlieger, Oct 27 2023 *)

PARI
```
See Links section.
```

from itertools import count, islice
from sympy import nextprime
def A366912_gen(): # generator of terms
    a, aset, p, c = 1, {0,1}, 2, 0
    while True:
        k, b = divmod(a,p)
        for i in count(-k):
            if b not in aset:
                aset.add(b)
                a, p = b, nextprime(p)
                yield c
                c += i
                break
A366912_list = list(islice(A366912_gen(),30)) # Chai Wah Wu, Oct 27 2023

a(n) = Sum_{k = 1..n-1} A366911(k).

A367290 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, a(n) and a(n+1) are congruent modulo the n-th prime number, and the least value not yet in the sequence appears as soon as possible.

Original entry on oeis.org

1, 5, 2, 17, 3, 69, 4, 310, 6, 558, 7, 193, 8, 869, 9, 2077, 10, 1780, 11, 3562, 12, 961, 13, 6155, 14, 2439, 15, 8255, 16, 6120, 18, 12464, 19, 9472, 20, 11195, 21, 4260, 22, 24070, 23, 16133, 24, 18360, 25, 19528, 26, 27456, 27, 25905, 28, 46395, 29, 6054

Offset: 1

Rémy Sigrist, Nov 12 2023

nonn

To build the sequence:
- we start with a(1) = 1, and repeatedly:
- let a(n) be the last known term and v the least value not yet in the sequence,
- if a(n) and v are congruent modulo the n-th prime number then a(n+1) = v,
- otherwise a(n+2) = v and a(n+1) is chosen as small as possible in such a way as to satisfy the required congruences (this is always possible as two consecutive prime numbers are coprime).
This sequence is a variant of A364054, and, by design, is guaranteed to be a permutation of the positive integers (with inverse A367291).

			The first terms are:
  n   a(n)   a(n) mod prime(n)  a(n+1) mod prime(n)
  --  -----  -----------------  -------------------
   1      1                  1                    1
   2      5                  2                    2
   3      2                  2                    2
   4     17                  3                    3
   5      3                  3                    3
   6     69                  4                    4
   7      4                  4                    4
   8    310                  6                    6
   9      6                  6                    6
  10    558                  7                    7
  11      7                  7                    7
  12    193                  8                    8
  13      8                  8                    8

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A364054, A367288, A367291 (inverse).

PARI
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See Links section.
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A368382 a(1) = 1; for n > 1, a(n) is the least positive integer not already in the sequence such that a(n) == a(n-1) (mod A004280(n)).

Original entry on oeis.org

1, 3, 6, 11, 4, 13, 2, 15, 30, 47, 9, 51, 5, 55, 28, 57, 26, 59, 24, 61, 22, 63, 20, 65, 18, 67, 16, 69, 14, 71, 12, 73, 10, 75, 8, 77, 148, 221, 146, 223, 144, 225, 142, 227, 53, 231, 49, 235, 45, 239, 41, 243, 37, 247, 33, 251, 29, 255, 25, 259, 21, 263, 17, 267, 140, 269, 7, 273, 138, 275, 136, 277, 134, 279, 132, 281

Offset: 1

N. J. A. Sloane, Mar 03 2024

nonn

Analogous to A364054, but whereas that sequence is based on the sequence of primes (2, 3, 5, 7, 11, ....), the present sequence is based on the sequence 2, 3, 5, 7, 9, 11, 13, 15, ... (2 together with the odd numbers >1, essentially A004280).

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

Cf. A004280.

Similar definitions: A005132, A006509, A364054.

from itertools import count, islice
def A368382_gen(): # generator of terms
    a, aset, p = 1, {0,1}, 2
    while True:
        yield a
        for b in count(a%p,p):
            if b not in aset:
                aset.add(b)
                a, p = b, 3 if p == 2 else p+2
                break
A368382_list = list(islice(A368382_gen(),40)) # Chai Wah Wu, Mar 05 2024

A366864 Numbers m such that A366470(m) > A366470(m-1).

Original entry on oeis.org

4, 5, 10, 13, 24, 39, 84, 168, 370, 836, 1998, 4622, 11284, 28151, 53565, 138230, 334125, 659741, 1716635, 3977282, 10430384, 27132843, 71588934, 189472352, 505341104, 1353331592

Offset: 1

Chai Wah Wu, Oct 25 2023

Inspired by N. J. A. Sloane's remark about the graph of A366470 consisting of decreasing segments. Terms mark the beginning of these segments in the graph of A366470. Appears to grow exponentially. Terms seem to be near the values of t_i described in Sloane's sketch at A364054.

Seqfan discussion, Re: Ali Sada's A364054.

Cf. A364054, A366470.

from itertools import count, islice
from sympy import nextprime
def A366864_gen(): # generator of terms
    a, aset, p = 1, {0,1}, 2
    for i in count(3):
        for b in count(a,p):
            if b not in aset:
                aset.add(b)
                c = b%(p:=nextprime(p))
                if c > a:
                    yield i
                a = c
                break
A366864_list = list(islice(A366864_gen(),20))

cp's OEIS Frontend

A366470 a(n) = A364054(n-1) mod prime(n-1).

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A366475 a(n) = (A364054(n) - A366470(n))/prime(n-1).

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A368384 Records in A364054.

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A368385 Indices of records in A364054.

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A366911 a(n) = (A364054(n+1) - A364054(n)) / prime(n) (where prime(n) denotes the n-th prime number).

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A366477 a(n) = smallest k such that A366475(k) >= n, or -1 if no such k exists.

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A366912 Partial sums of A366911: a(1) = 0, and for n > 0, a(n+1) = a(n) + A366911(n).

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A367290 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, a(n) and a(n+1) are congruent modulo the n-th prime number, and the least value not yet in the sequence appears as soon as possible.

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