A354790 -id:A354790 - cp's OEIS frontend

A354791 Indices of nonprime terms in A354790.

1, 7, 14, 15, 17, 29, 31, 35, 59, 63, 64, 71, 79, 119, 120, 127, 129, 143, 159, 223, 239, 241, 255, 259, 260, 287, 288, 319, 320, 447, 479, 483, 484, 511, 512, 519, 521, 575, 577, 639, 641, 895, 959, 960, 967, 968, 969, 1023, 1025, 1039, 1043, 1044, 1151, 1152

Offset: 1

Michael De Vlieger and N. J. A. Sloane, Jul 17 2022

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..5715
Rémy Sigrist, C program

Cf. A354790, A354792.

C
```
See Links section.
```

from math import lcm, gcd
from itertools import count, islice
from collections import deque
from sympy import factorint
def A354791_gen(): # generator of terms
    aset, aqueue, c, b, f, i = {1}, deque([1]), 2, 1, True, 1
    yield 1
    while True:
        for m in count(c):
            if m not in aset and gcd(m,b) == 1 and all(map(lambda n:n<=1,fs:=factorint(m).values())):
                i += 1
                if len(fs) > 1:
                    yield i
                aset.add(m)
                aqueue.append(m)
                if f: aqueue.popleft()
                b = lcm(*aqueue)
                f = not f
                while c in aset:
                    c += 1
                break
A354791_list = list(islice(A354791_gen(),30)) # Chai Wah Wu, Jul 17 2022

More terms from Rémy Sigrist, Jul 17 2022

A354792 Nonprime terms in A354790 in order of appearance.

Original entry on oeis.org

1, 6, 35, 22, 39, 85, 14, 33, 65, 34, 133, 69, 319, 155, 481, 82, 119, 57, 253, 1247, 235, 403, 74, 287, 901, 177, 1159, 737, 1633, 2117, 215, 611, 2449, 166, 3293, 679, 697, 159, 1121, 671, 1541, 2059, 365, 4343, 1339, 5029, 3379, 158, 3071, 623, 1649, 4633

Offset: 1

Michael De Vlieger and N. J. A. Sloane, Jul 17 2022

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..5715
Rémy Sigrist, C program

Cf. A354790, A354791.

C
```
See Links section.
```

from math import lcm, gcd
from itertools import count, islice
from collections import deque
from sympy import factorint
def A354792_gen(): # generator of terms
    aset, aqueue, c, b, f = {1}, deque([1]), 2, 1, True
    yield 1
    while True:
        for m in count(c):
            if m not in aset and gcd(m,b) == 1 and all(map(lambda n:n<=1,fs:=factorint(m).values())):
                if len(fs) > 1:
                    yield m
                aset.add(m)
                aqueue.append(m)
                if f: aqueue.popleft()
                b = lcm(*aqueue)
                f = not f
                while c in aset:
                    c += 1
                break
A354792_list = list(islice(A354792_gen(),30)) # Chai Wah Wu, Jul 17 2022

More terms from Rémy Sigrist, Jul 17 2022

A355895 Even terms in A354790, divided by 2, in order of appearance.

Original entry on oeis.org

1, 3, 11, 7, 17, 41, 37, 83, 79, 109, 241, 239, 499, 491, 1117, 1109, 2539, 2531, 5623, 5591, 11471, 11467, 23369, 23357

Offset: 1

N. J. A. Sloane, Aug 27 2022

The even terms in A354790 appear at indices 2, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, ... and conjecturally at indices 2^k-1 for all k >= 3.

Cf. A354790, A355896.

a(16)-a(24) from Rémy Sigrist, Aug 29 2022

A355897 a(n) = index in A354790 of first nonprime term divisible by prime(n).

Original entry on oeis.org

7, 7, 14, 14, 15, 17, 29, 64, 71, 79, 119, 120, 127, 223, 239, 260, 287, 288, 319, 320, 447, 484, 511, 512, 519, 960, 967, 968, 969, 1044, 1151, 1152, 1155, 1156, 1279, 1280, 1283, 1387, 1791, 1792, 1919, 1920, 1921, 2048, 2051, 2052, 2079, 3872, 3875, 3876, 3879, 3880, 4095, 4096

Offset: 1

N. J. A. Sloane, Aug 28 2022

nonn

			The first few nonprime terms in B = A354790, and their prime factors are:
    n  B(n)=p*q   p     pi(p)     q     pi(q)
    7,   6,       2,      1,      3,      2
   14,   35,      5,      3,      7,      4
   15,   22,      2,      1,      11,     5
   17,   39,      3,      2,      13,     6
   29,   85,      5,      3,      17,     7
   31,   14,      2,      1,      7,      4
   35,   33,      3,      2,      11,     5
B(14) = A354790(14) = 35 = 5*7 is the first nonprime term divisible by 5 = prime(3), therefore a(3) = 14.

N. J. A. Sloane, Table of n, a(n) for n = 1..236

Cf. A354790, A354791, A354792.

A356803 a(n) = product of prohibited prime factors of A354790(n).

Original entry on oeis.org

1, 1, 2, 6, 15, 105, 385, 2310, 6006, 102102, 277134, 6374082, 16804398, 520936338, 3038795305, 66853496710, 190275336790, 7420738134810, 17897074325130, 769574195980590, 1903683537425670, 100895227483560510, 258818192240437830, 15787909726666707630, 36475515575402393490

Offset: 1

Michael De Vlieger, Sep 06 2022

nonn

Let s(n) = A354790(n), a squarefree number by definition. Prime p | s(n) implies p does not divide s(n+j), 1 <= j <= n. Therefore a(n) is the product of primes p that cannot divide s(n). a(n) = product of distinct primes that divide a(j) for floor((n+1)/2) <= j <= n-1. (After N. J. A. Sloane in A355057.)

Analogous to A355057.

			a(1) = 1;
a(2) = 1 since s(1) = 1, and (2-1)/2 is not an integer;
a(3) = a(2) * s(2) / s((3-1)/2) = 1 * 2 / 1 = 2;
a(4) = a(3) * s(3) = 2 * 3 = 6;
a(5) = a(4) * s(4) / s((5-1)/2) = 6 * 5 / 2 = 15;
a(6) = a(5) * s(5) = 15 * 7 = 105;
a(7) = a(6) * s(6) / s((7-1)/2) = 105 * 11 / 3 = 385; etc.

Michael De Vlieger, Table of n, a(n) for n = 1..585
Michael De Vlieger, Annotated plot of prime p | m at (n, pi(p)) for m = a(n) in blue and A354790(n) in red and n = 1..80.
Michael De Vlieger, Plot of prime p | m at (n, pi(p)) for m = a(n) in cyan and A354790(n) in red and n = 1..1024.

Cf. A354790, A355057.

Block[{s = Import["https://oeis.org/A354790/b354790.txt", "Data"][[1 ;; 26, -1]], ww, m = 1, t, w = 3, k = 3}, Reap[Do[m *= Times @@ FactorInteger[s[[If[# == 0, 1, #] &[i - 1]]]][[All, 1]]; If[IntegerQ[#] && # > 0, m /= Times @@ FactorInteger[s[[#]]][[All, 1]]] &[(i - 1)/2]; Sow[m], {i, Length[s] - 1}]][[-1, -1]] ]

a(n) = a(n-1) * s(n-1) / s((n-1)/2), where the last operation is only carried out iff (n-1)/2 is an integer.

A355894 Let A354790(n) = Product_{i >= 1} prime(i)^e(i); then a(n) is the concatenation, in reverse order, of e_1, e_2, ..., ending at the exponent of the largest prime factor of A354790(n); a(1)=0 by convention.

Original entry on oeis.org

0, 1, 10, 100, 1000, 10000, 11, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000, 1100, 10001, 100000000000, 100010, 1000000000000, 10000000000000, 100000000000000, 1000000000000000, 10000000000000000, 100000000000000000, 1000000000000000000, 10000000000000000000

Offset: 0

N. J. A. Sloane, Aug 25 2022

nonn

The terms of A354790 are squarefree, so here the exponents e_i are 0 or 1.

This bears the same relation to A354790 as A355893 does to A090252.

			The terms, right-justified, for comparison with A355892 and A355893, are:
   1 ...................................0
   2 ...................................1
   3 ..................................10
   4 .................................100
   5 ................................1000
   6 ...............................10000
   7 ..................................11
   8 ..............................100000
   9 .............................1000000
  10 ............................10000000
  11 ...........................100000000
  12 ..........................1000000000
  13 .........................10000000000
  14 ................................1100
  15 ...............................10001
  16 ........................100000000000
  17 ..............................100010
  18 .......................1000000000000
  19 ......................10000000000000
  20 .....................100000000000000
  21 ....................1000000000000000
  22 ...................10000000000000000
  23 ..................100000000000000000
  24 .................1000000000000000000
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..1051

Cf. A090252, A354790, A355893.

A355896 Multiples of 3 in A354790, divided by 3, in order of appearance.

Original entry on oeis.org

1, 2, 13, 11, 23, 19, 59, 53, 127, 113, 277, 271, 631, 619, 1409, 1399, 1823, 3967, 3947, 8783, 8779, 18257, 18253, 25873

Offset: 1

N. J. A. Sloane, Aug 27 2022

The multiples of 3 in A354790 appear at indices 3, 7, 17, 35, 71, 143, 287, 575, 1151, ... and conjecturally at indices 9*2^k-1 for all k >= 1.

Cf. A354790, A355895.

a(16)-a(24) from Rémy Sigrist, Aug 29 2022

A090252 The Two-Up sequence: a(n) is the least positive number not already used that is coprime to the previous floor(n/2) terms.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 9, 11, 13, 17, 8, 19, 23, 25, 21, 29, 31, 37, 41, 43, 47, 53, 16, 59, 61, 67, 71, 73, 55, 79, 27, 49, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 26, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 85, 121, 223, 227, 57, 229

Offset: 1

Amarnath Murthy, Nov 27 2003

nonn

a(n) is coprime to the next n terms. - David Wasserman, Oct 24 2005
All values up to a(1000000) are either prime powers or semiprimes; this suggests the sequence is unlikely to be a permutation of the integers.
It appears that a(n) is even iff n = 3*2^k-1 for some k (A083356). - N. J. A. Sloane, Nov 01 2014
The even terms in the present sequence are listed in A354255.
We have a(1) = 1 and a(2) = 2. At step k >= 2, the sequence is extended by adding two terms: a(2*k-1) = smallest unused number which is relatively prime to a(k), a(k+1), ..., a(2*k-2), and a(2*k) = smallest unused number which is relatively prime to a(k), a(k+1), ..., a(2*k-1). So at step k=2 we add a(3)=3, a(4)=5; at step k=3 we add a(5)=4, a(6)=7; and so on. - N. J. A. Sloane, May 21 2022
Comments from N. J. A. Sloane, May 23 2022: (Start)
Conjecture 1. A090252 is a subsequence of A354144 (prime powers and semiprimes).
Conjecture 2. The terms of A354144 that are missing from A090252 are 6, 10, 14, 15, 22, 33, 34, 35, 38, 39, 46, 51, 58, 62, 65, 69, 74, 77, 82, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 166, 177, 178, 183, 185, 187, 194, 201, 202, 203, 209, 213, 214, 215, 218, 219, 221, ...
But since there is no proof that any one of these numbers is really missing, this list cannot yet have an entry in the OEIS.
Let S_p = list of indices of terms in A090252 that are divisible by the prime p.
Conjecture 3. For a prime p, there are constants v_1, v_2, ..., v_K and c such that
   S_p = { v_1, v_2, ..., v_k, lambda*2^i - 1, i >= c}.
For example, from Michael S. Branicky's 10000-term b-file, it appears that:
   S_2 = { 3*2^k-1, k >= 0 }  cf. A083329
   S_3 = { 2^k-1, k >= 2 }  cf. A000225
   S_5 = { 4 then 15*2^k-1 k >= 0 }  cf. A196305
   S_7 = { 6, 15, then 33*2^k-1, k >= 0  }
   S_11 = { 8, 29, then 61*2^k-1, k >= 0 }
   S_13 = { 9, 47, 97*2^n-1, n >= 0 }
   S_17 = { 10, 59, 121*2^n-1, n >= 0 }
   S_19 = { 12, 63, 129*2^n-1, n >= 0 }
   S_23 = { 13, 65, 133*2^n-1, n >= 0 }
   S_29 = { 16, 121, 245*2^n-1, n >= 0 }
   S_31 = { 17, 131, 265*2^n-1, n >= 0 }
The initial primes p and the corresponding values of lambda are:
    p:   2   3   5   7   11   13   17   19   23   29   31
lambda:..3...1..15..33...61...97..121..129..133..245..265
(This sequence of lambdas does not seem to have any simpler explanation, is not in the OEIS, and cannot be since the terms shown are all conjectural.)
Conjecture 2 is a consequence of Conjecture 3. For example, 6 does not appear in A090252, since the sets S_2 and S_3 are disjoint.
Also 10 does not appear, since S_2 and S_5 are disjoint.
In fact 2*p for 3 <= p <= 11 does not appear, but 26 = 2*13 does appear since S_2 and S_13 have 47 in common.
Assuming the numbers that appear to be missing (see Conjecture 2) really are missing, the numbers that take a record number of steps to appear are 1, 2, 3, 4, 7, 8, 16, 26, 32, 64, 128, 206, 256, 478, 512, 933, ..., and the indices where they appear are 1, 2, 3, 5, 6, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 8191, .... These two sequences are not yet in the OEIS, and cannot be added since the terms are all conjectural.
(End)
From N. J. A. Sloane, Jun 06 2022 (Start)
Theorem: (a) a(n) <= prime(n-1) for all n >= 2 (cf. A354154).
  (b) A stronger upper bound is the following. Let c(n) = A354166(n) denote the number of nonprime terms among a(1) .. a(n). Note c(1)=1. Then a(n) <= prime(n-c(n)) for n <> 7 and 14.
It appears that a(n) = prime(n-c(n)) for almost all n. That is, this is the equation to the line in the graph that contains most of the terms.
For example, a(34886) = 408710 (see the b-file) = prime(34886 - A354166(34886)) = prime(34886 - 374) = prime(34512) = 408710.
Another example: Consider Russ Cox's table of the first N = 5764982 terms. We see that a(5764982) = 99999989 = prime(5761455) = prime(N - 3527) which agrees with c(N) = 3527 (from the first Russ Cox link).
(End)
If we consider the May 23 2022 comment, note the conjectured indices show near complete overlap with terms of A081026: 1, 2, 3, 5, 6, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 8191. - Bill McEachen, Aug 09 2024

Michael S. Branicky, Table of n, a(n) for n = 1..34886
Russ Cox, Table of nonprime entries in A090252: n, A090252(n), # of prime factors, n = 1..3527.
Russ Cox, Table of n, a(n) for n = 1..5764982, up to the first term that is greater than 10^8 [gzipped file]
Michael De Vlieger, Thomas Scheuerle, Rémy Sigrist, N. J. A. Sloane, and Walter Trump, The Binary Two-Up Sequence, arXiv:2209.04108 [math.CO], Sep 11 2022.
N. J. A. Sloane, Blog post about the Two-Up sequence, June 13 2022.
Hugo van der Sanden, Perl program to calculate this sequence and A249064 (requires Math::Pari)
Hugo van der Sanden, Faster Perl program on github, used to compute 10^9 terms. [Link changed by _N. J. A. Sloane_, Jun 19 2022]
Hugo van der Sanden, Table of nonprime entries in the first 10^9 terms of A090252 [See beginning of the file for description. The blog in the above link has comments from _Hugo van der Sanden_ describing the algorithm used to generate this table.]

Cf. A083329, A084937, A196305, A249064, A354146, A354148-A354151, A354154, A354159-A354167, A354255 (even terms), A355893.
See also A354169, A354764, A354765, A355057.
See A247665 for the case when the numbers are required to be at least 2. A353730 is another version.
For a squarefree analog, see A354790, A354791, A354792.

nn = 120; c[] = 0; a[1] = c[1] = 1; u = 2; Do[k = u; While[Nand[c[k] == 0, AllTrue[Array[a[i - #] &, Floor[i/2]], CoprimeQ[#, k] &]], k++]; Set[{a[i], c[k]}, {k, i}]; If[k == u, While[c[u] > 0, u++]], {i, 2, nn}]; Array[a, nn] (* _Michael De Vlieger, May 21 2022 *)

A090252_first(N, U=[0], L=List())=vector(N, i, for(k=U[1]+1, oo, setsearch(U, k) && next; foreach(L,m, gcd(k,m)>1 && next(2)); bitand(i,1) || listpop(L,1); listput(L,k); if( k>U[1]+1, U=setunion(U,[k]), U[1]++; while(#U>1 && U[2]==U[1]+1, U=U[^1]));break); L[#L]) \\ M. F. Hasler, Jun 14 2022

from math import gcd, prod
from itertools import count, islice
def agen(): # generator of terms
    alst = [1]; aset = {1}; yield 1
    mink = 2
    for n in count(2):
        k, prodall = mink, prod(alst[n-n//2-1:n-1])
        while k in aset or gcd(prodall, k) != 1: k += 1
        alst.append(k); aset.add(k); yield k
        while mink in aset: mink += 1
print(list(islice(agen(), 64))) # Michael S. Branicky, May 21 2022

More terms from David Wasserman, Oct 24 2005

A356804 a(n) is a binary encoded version of A356803(n).

Original entry on oeis.org

0, 0, 1, 3, 6, 14, 28, 31, 59, 123, 243, 499, 995, 2019, 2028, 2045, 4061, 4095, 8127, 16319, 32575, 65343, 130623, 261695, 523327, 1047615, 2095167, 4192319, 8386611, 8386679, 16775270, 16775279, 33550447, 67104879, 134213709, 134213727, 268427359, 536862815

Offset: 1

Michael De Vlieger, Sep 06 2022

Let S(n) = list of forbidden primes for A354790(n); A356803(n) is the product of these primes. Then a(n) = Sum of 2^(i-1) over all prime(i) in S(n).
Conversely, if a(n) has binary expansion a(n) = Sum b(i)*2^i, b(i) = 0 or 1, then S(n) consists of {prime(i+1) such that b(i) = 1}. (After comment by N. J. A. Sloane at A354765)
Analogous to A354765.

			For n = 7 the forbidden primes are 5, 7, 11 = prime(3), prime(4) and prime(5). Their product is A356803(7) = 385. Then a(7) = 2^2 + 2^3 + 2^4 = 28.

Michael De Vlieger, Table of n, a(n) for n = 1..3405

Cf. A354790, A356803.

Block[{s = Import["https://oeis.org/A354790/a354790.txt", "Data"][[1 ;; 25, -1]], m = 0}, Join[{0, 0}, Reap[Do[If[i > 1, m += Total[2^PrimePi@ FactorInteger[s[[i - 1]]][[All, 1]]]]; If[IntegerQ[#] && # > 0, m -= Total[2^PrimePi@ FactorInteger[s[[#]]][[All, 1]]]] &[(i - 1)/2]; Sow[m], {i, Length[s]}] ][[-1, -1, 3 ;; -1]]/2] ]

cp's OEIS Frontend

A354791 Indices of nonprime terms in A354790.

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A354792 Nonprime terms in A354790 in order of appearance.

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A355895 Even terms in A354790, divided by 2, in order of appearance.

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A355897 a(n) = index in A354790 of first nonprime term divisible by prime(n).

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A356803 a(n) = product of prohibited prime factors of A354790(n).

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Formula

A355894 Let A354790(n) = Product_{i >= 1} prime(i)^e(i); then a(n) is the concatenation, in reverse order, of e_1, e_2, ..., ending at the exponent of the largest prime factor of A354790(n); a(1)=0 by convention.

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A355896 Multiples of 3 in A354790, divided by 3, in order of appearance.

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A090252 The Two-Up sequence: a(n) is the least positive number not already used that is coprime to the previous floor(n/2) terms.

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A356804 a(n) is a binary encoded version of A356803(n).

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