A366470 -id:A366470 - cp's OEIS frontend

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

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

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))

A366868 a(n) = A366470(A366864(n)-1).

Original entry on oeis.org

0, 1, 0, 1, 6, 1, 6, 15, 24, 29, 22, 1, 0, 7, 6, 11, 48, 109, 0, 7, 66, 155, 54, 21, 16, 11

Offset: 1

Chai Wah Wu, Oct 25 2023

Terms of A366470 at the end of a segment. Some values, e.g. 0, 1, 6, 7, 11, occur multiple times. Will they occur infinitely often?

Cf. A364054, A366470, A366864, A366869.

from itertools import count, islice
from sympy import nextprime
def A366868_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 a
                a = c
                break
A366868_list = list(islice(A366868_gen(),20))

A366869 a(n) = A366470(A366864(n)).

Original entry on oeis.org

1, 4, 15, 26, 81, 158, 417, 990, 2491, 6402, 17363, 44450, 119773, 326786, 659957, 1845500, 4779649, 9921002, 27575339, 67458614, 187615521, 515594444, 1433794185, 3989181038, 11160791967, 31287537756

Offset: 1

Chai Wah Wu, Oct 25 2023

Terms of A366470 at the beginning of a segment (except for the first segment).

Cf. A364054, A366470, A366864, A366868.

from itertools import count, islice
from sympy import nextprime
def A366869_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 c
                a = c
                break
A366869_list = list(islice(A366869_gen(), 20))

A364054 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 prime(n-1)).

Original entry on oeis.org

1, 3, 6, 11, 4, 15, 2, 19, 38, 61, 32, 63, 26, 67, 24, 71, 18, 77, 16, 83, 12, 85, 164, 81, 170, 73, 174, 277, 384, 57, 283, 29, 160, 23, 162, 13, 315, 158, 321, 154, 327, 148, 329, 138, 331, 134, 333, 122, 345, 118, 347, 114, 353, 112, 363, 106, 369, 100, 371, 94, 375, 92, 385

Offset: 1

Ali Sada, Oct 19 2023

5 is the smallest positive integer missing from the first 1000 terms. Also in the interval a(100) to a(1000) there are no entries less than 100. (From W. Edwin Clark via SeqFan.)
Comments from N. J. A. Sloane, Oct 22 2023 (Start)
It appears that the graph of this sequence is dominated by pairs of diverging lines, as suggested by the sketch (see link). For example, around step n = 4619, a descending line is changing to a descending line around a(4619) = 65, a companion ascending line is coming to an end near a(4594) = 44518, and a strong ascending line is starting up around a(4620) = 88899.
It would be nice to have more terms, in order to get better estimates of the times t_i where these transitions happen, and heights alpha_i, beta_i, gamma_i where line breaks are.
The only well-defined points are the (t_i, alpha_i) where the descending lines end, as can be seen from the b-file, where the end point a(4619) = 65 is well-defined. The other transitions, where an ascending line changes to a descending line, are less obvious.  It would be nice to know more.
Can the t_i and alpha_i sequences be traced back to the start of the sequence? Of course the alpha_i sequence is not monotonic, and in particular we do not know at present if some alpha_i is equal to 5.
(End)
a(28149) = 7. - Chai Wah Wu, Oct 22 2023
Comment from N. J. A. Sloane, Mar 05 2024 (Start):
At present there is no OEIS entry for the inverse sequence, since it is not known if 5 appears here.
The initial values of the inverse sequence are
n.....1..2..3..4..5..6....7.....8..9..10..11... . . .
index.1..7..2..5..?..3..28149..81..?...?...4... . . . (End)

			For n = 2, prime(2-1) = prime(1) = 2; a(1) = 1, so a(1) mod 2 = 1, so a(2) is the least positive integer == 1 (mod 2) that has not yet appeared; 1 has appeared, so a(2) = 3.
For n = 3, prime(3-1) = 3; a(2) mod 3 = 0, so a(3) is the least unused integer == 0 mod 3, which is 6, so a(3) =  6.
For n = 4, prime(4-1) = 5; a(3) mod 5 = 1, and 6 has already been used, so a(4) = 11.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Sketch showing the main features of the graph
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
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 = empty product {1}, red = prime (A40), gold = composite prime power (A246547), bright green = primorial A2110(k), k > 1, light green = squarefree semiprime (A6881 \ {6}), dark green = squarefree composite (A120944 \ {A2110 U A6881}), blue = numbers neither squarefree nor composite (A126706 \ A286708 = A332785), purple = squareful numbers that are not prime powers (A286708).
Chai Wah Wu, Graph of first 10^8 terms

Cf. A006509, A125717.
For a(n-1) (mod prime(n-1)) see A366470.
Records: A368384, A368385.
See also A366475, A366477.

a[1] = 1; a[n_] := a[n] = Module[{p = Prime[n - 1], k = 2, s = Array[a, n - 1]}, While[! FreeQ[s, k] || ! Divisible[k - a[n - 1], p], k++]; k]; Array[a, 100] (* Amiram Eldar, Oct 20 2023 *)
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], c[k], j}, {k, True, k}], {n, 2, nn}], n];
Array[a, nn] (* Michael De Vlieger, Oct 26 2023, fast, based on congruence, avoids search *)

from itertools import count, islice
from sympy import nextprime
def A364054_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, nextprime(p)
                break
A364054_list = list(islice(A364054_gen(),30)) # Chai Wah Wu, Oct 22 2023

cp's OEIS Frontend

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

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

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A366868 a(n) = A366470(A366864(n)-1).

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A366869 a(n) = A366470(A366864(n)).

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A364054 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 prime(n-1)).

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