A191257 -id:A191257 - cp's OEIS frontend

A332820 Integers in the multiplicative subgroup of positive rationals generated by the products of two consecutive primes and the cubes of primes. Numbers k for which A048675(k) is a multiple of three.

1, 6, 8, 14, 15, 20, 26, 27, 33, 35, 36, 38, 44, 48, 50, 51, 58, 63, 64, 65, 68, 69, 74, 77, 84, 86, 90, 92, 93, 95, 106, 110, 112, 117, 119, 120, 122, 123, 124, 125, 141, 142, 143, 145, 147, 156, 158, 160, 161, 162, 164, 170, 171, 177, 178, 185, 188, 196, 198, 201, 202, 208, 209, 210, 214, 215, 216, 217, 219, 221, 225

Offset: 1

Antti Karttunen and Peter Munn, Feb 25 2020

nonn

The positive integers are partitioned between this sequence, A332821 and A332822, which list the integers in respective cosets of the subgroup.
As the sequence lists the integers in a multiplicative subgroup of the positive rationals, the sequence is closed under multiplication and, provided the result is an integer, under division.
It follows that for any n in this sequence, all powers n^k are present (k >= 0), as are all cubes.
If we take each odd term of this sequence and replace each prime in its factorization by the next smaller prime, the resulting numbers are a permutation of the full sequence; and if we take the square root of each square term we get the full sequence.
There are no primes in the sequence, therefore if k is present and p is a prime, k*p and k/p are absent (noting that k/p might not be an integer). This property extends from primes to all terms of A050376 (often called Fermi-Dirac primes), therefore to squares of primes, 4th powers of primes etc.
The terms are the even numbers in A332821 halved. The terms are also the numbers m such that 5m is in A332821, and so on for alternate primes: 11, 17, 23 etc. Likewise, the terms are the numbers m such that 3m is in A332822, and so on for alternate primes: 7, 13, 19 etc.
The numbers that are half of the even terms of this sequence are in A332822, which consists exactly of those numbers. The numbers that are one third of the terms that are multiples of 3 are in A332821, which consists exactly of those numbers. These properties extend in a pattern of alternating primes as described in the previous paragraph.
If k is an even number, exactly one of {k/2, k, 2k} is in the sequence (cf. A191257 / A067368 / A213258); and generally if k is a multiple of a prime p, exactly one of {k/p, k, k*p} is in the sequence.
If m and n are in this sequence then so is m*n (the definition of "multiplicative semigroup"), while if n is in this sequence, and x is in the complement A359830, then n*x is in A359830. This essentially follows from the fact that A048675 is totally additive sequence. Compare to A329609. - Antti Karttunen, Jan 17 2023

Positions of zeros in A332823; equivalently, numbers in row 3k of A277905 for some k >= 0.
Cf. A048675, A195017, A332821, A332822, A353350 (characteristic function), A353348 (its Dirichlet inverse), A359830 (complement).
Comparable 2 or 3-way classifications: A000379/A000028, A001969/A000069, A003159/A036554, A005843/A005408, A028260/A026424, A191257/A067368/A213258, A325431/A325432, A329609/A329604/A332812.
Subsequences: A000578\{0}, A006094, A090090, A099788, A245630 (A191002 in ascending order), A244726\{0}, A325698, A338471, A338556, A338907.
Subsequence of {1} U A268388.

Select[Range@ 225, Or[Mod[Total@ #, 3] == 0 &@ Map[#[[-1]]*2^(PrimePi@ #[[1]] - 1) &, FactorInteger[#]], # == 1] &] (* Michael De Vlieger, Mar 15 2020 *)

isA332820(n) =  { my(f = factor(n)); !((sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3); };

{a(n) : n >= 1} = {1} U {2 * A332822(k) : k >= 1} U {A003961(a(k)) : k >= 1}.
{a(n) : n >= 1} = {1} U {a(k)^2 : k >= 1} U {A331590(2, A332822(k)) : k >= 1}.
From Peter Munn, Mar 17 2021: (Start)
{a(n) : n >= 1} = {k : k >= 1, 3|A048675(k)}.
{a(n) : n >= 1} = {k : k >= 1, 3|A195017(k)}.
{a(n) : n >= 1} = {A332821(k)/2 : k >= 1, 2|A332821(k)}.
{a(n) : n >= 1} = {A332822(k)/3 : k >= 1, 3|A332822(k)}.
(End)

New name from Peter Munn, Mar 08 2021

A297845 Encoded multiplication table for polynomials in one indeterminate with nonnegative integer coefficients. Symmetric square array T(n, k) read by antidiagonals, n > 0 and k > 0. See comment for details.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 4, 1, 1, 5, 9, 9, 5, 1, 1, 6, 7, 16, 7, 6, 1, 1, 7, 15, 25, 25, 15, 7, 1, 1, 8, 11, 36, 11, 36, 11, 8, 1, 1, 9, 27, 49, 35, 35, 49, 27, 9, 1, 1, 10, 25, 64, 13, 90, 13, 64, 25, 10, 1, 1, 11, 21, 81, 125, 77, 77, 125, 81

Offset: 1

Rémy Sigrist, Jan 10 2018

For any number n > 0, let f(n) be the polynomial in a single indeterminate x where the coefficient of x^e is the prime(1+e)-adic valuation of n (where prime(k) denotes the k-th prime); f establishes a bijection between the positive numbers and the polynomials in a single indeterminate x with nonnegative integer coefficients; let g be the inverse of f; T(n, k) = g(f(n) * f(k)).
This table has many similarities with A248601.
For any n > 0 and m > 0, f(n * m) = f(n) + f(m).
Also, f(1) = 0 and f(2) = 1.
The function f can be naturally extended to the set of positive rational numbers: if r = u/v (not necessarily in reduced form), then f(r) = f(u) - f(v); as such, f is a homomorphism from the multiplicative group of positive rational numbers to the additive group of polynomials of a single indeterminate x with integer coefficients.
See A297473 for the main diagonal of T.
As a binary operation, T(.,.) is related to A306697(.,.) and A329329(.,.). When their operands are terms of A050376 (sometimes called Fermi-Dirac primes) the three operations give the same result. However the rest of the multiplication table for T(.,.) can be derived from these results because T(.,.) distributes over integer multiplication (A003991), whereas for A306697 and A329329, the equivalent derivation uses distribution over A059896(.,.) and A059897(.,.) respectively. - Peter Munn, Mar 25 2020
From Peter Munn, Jun 16 2021: (Start)
The operation defined by this sequence can be extended to be the multiplicative operator of a ring over the positive rationals that is isomorphic to the polynomial ring Z[x]. The extended function f (described in the author's original comments) is the isomorphism we use, and it has the same relationship with the extended operation that exists between their unextended equivalents.
Denoting this extension of T(.,.) as t_Q(.,.), we get t_Q(n, 1/k) = t_Q(1/n, k) = 1/T(n, k) and t_Q(1/n, 1/k) = T(n, k) for positive integers n and k. The result for other rationals is derived from the distributive property: t_Q(q, r*s) = t_Q(q, r) * t_Q(q, s); t_Q(q*r, s) = t_Q(q, s) * t_Q(r, s). This may look unusual because standard multiplication of rational numbers takes on the role of the ring's additive group.
There are many OEIS sequences that can be shown to be a list of the integers in an ideal of this ring. See the cross-references.
There are some completely additive sequences that similarly define by extension completely additive functions on the positive rationals that can be shown to be homomorphisms from this ring onto the integer ring Z, and these functions relate to some of the ideals. For example, the extended function of A048675, denoted A048675_Q, maps i/j to A048675(i) - A048675(j) for positive integers i and j. For any positive integer k, the set {r rational > 0 : k divides A048675_Q(r)} forms an ideal of the ring; for k=2 and k=3 the integers in this ideal are listed in A003159 and A332820 respectively.
(End)

			Array T(n, k) begins:
  n\k|  1   2   3    4    5    6    7     8    9    10
  ---+------------------------------------------------
    1|  1   1   1    1    1    1    1     1    1     1  -> A000012
    2|  1   2   3    4    5    6    7     8    9    10  -> A000027
    3|  1   3   5    9    7   15   11    27   25    21  -> A003961
    4|  1   4   9   16   25   36   49    64   81   100  -> A000290
    5|  1   5   7   25   11   35   13   125   49    55  -> A357852
    6|  1   6  15   36   35   90   77   216  225   210  -> A191002
    7|  1   7  11   49   13   77   17   343  121    91
    8|  1   8  27   64  125  216  343   512  729  1000  -> A000578
    9|  1   9  25   81   49  225  121   729  625   441
   10|  1  10  21  100   55  210   91  1000  441   550
From _Peter Munn_, Jun 24 2021: (Start)
The encoding, n, of polynomials, f(n), that is used for the table is further described in A206284. Examples of encoded polynomials:
   n      f(n)        n           f(n)
   1         0       16              4
   2         1       17            x^6
   3         x       21        x^3 + x
   4         2       25           2x^2
   5       x^2       27             3x
   6     x + 1       35      x^3 + x^2
   7       x^3       36         2x + 2
   8         3       49           2x^3
   9        2x       55      x^4 + x^2
  10   x^2 + 1       64              6
  11       x^4       77      x^4 + x^3
  12     x + 2       81             4x
  13       x^5       90   x^2 + 2x + 1
  15   x^2 + x       91      x^5 + x^3
(End)

Rémy Sigrist, Table of n, a(n) for n = 1..5050
Encyclopedia of Mathematics, Additive arithmetic function
Encyclopedia of Mathematics, Isomorphism
Eric Weisstein's World of Mathematics, Distributive
Eric Weisstein's World of Mathematics, Ring.
Wikipedia, Polynomial ring

Cf. A001221, A007913, A048720, A061395, A055396, A206284, A248601, A248663.
Row n: n=1: A000012, n=2: A000027, n=3: A003961, n=4: A000290, n=5: A357852, n=6: A191002, n=8: A000578.
Main diagonal: A297473.
Functions f satisfying f(T(n,k)) = f(n) * f(k): A001222, A048675 (and similarly, other rows of A104244), A195017.
Powers of k: k=3: A000040, k=4: A001146, k=5: A031368, k=6: A007188 (see also A066117), k=7: A031377, k=8: A023365, k=9: main diagonal of A329050.
Integers in the ideal of the related ring (see Jun 2021 comment) generated by S: S={3}: A005408, S={4}: A000290\{0}, S={4,3}: A003159, S={5}: A007310, S={5,4}: A339690, S={6}: A325698, S={6,4}: A028260, S={7}: A007775, S={8}: A000578\{0}, S={8,3}: A191257, S={8,6}: A332820, S={9}: A016754, S={10,4}: A340784, S={11}: A008364, S={12,8}: A145784, S={13}: A008365, S={15,4}: A345452, S={15,9}: A046337, S={16}: A000583\{0}, S={17}: A008366.
Equivalent sequence for polynomial composition: A326376.
Related binary operations: A003991, A306697/A059896, A329329/A059897.

T(n,k) = my (f=factor(n), p=apply(primepi, f[, 1]~), g=factor(k), q=apply(primepi, g[, 1]~)); prod (i=1, #p, prod(j=1, #q, prime(p[i]+q[j]-1)^(f[i, 2]*g[j, 2])))

T is completely multiplicative in both parameters:
- for any n > 0
- and k > 0 with prime factorization Prod_{i > 0} prime(i)^e_i:
- T(prime(n), k) = T(k, prime(n)) = Prod_{i > 0} prime(n + i - 1)^e_i.
For any m > 0, n > 0 and k > 0:
- T(n, k) = T(k, n) (T is commutative),
- T(m, T(n, k)) = T(T(m, n), k) (T is associative),
- T(n, 1) = 1 (1 is an absorbing element for T),
- T(n, 2) = n (2 is an identity element for T),
- T(n, 2^i) = n^i for any i >= 0,
- T(n, 4) = n^2 (A000290),
- T(n, 8) = n^3 (A000578),
- T(n, 3) = A003961(n),
- T(n, 3^i) = A003961(n)^i for any i >= 0,
- T(n, 6) = A191002(n),
- A001221(T(n, k)) <= A001221(n) * A001221(k),
- A001222(T(n, k)) = A001222(n) * A001222(k),
- A055396(T(n, k)) = A055396(n) + A055396(k) - 1 when n > 1 and k > 1,
- A061395(T(n, k)) = A061395(n) + A061395(k) - 1 when n > 1 and k > 1,
- T(A000040(n), A000040(k)) = A000040(n + k - 1),
- T(A000040(n)^i, A000040(k)^j) = A000040(n + k - 1)^(i * j) for any i >= 0 and j >= 0.
From Peter Munn, Mar 13 2020 and Apr 20 2021: (Start)
T(A329050(i_1, j_1), A329050(i_2, j_2)) = A329050(i_1+i_2, j_1+j_2).
T(n, m*k) = T(n, m) * T(n, k); T(n*m, k) = T(n, k) * T(m, k) (T distributes over multiplication).
A104244(m, T(n, k)) = A104244(m, n) * A104244(m, k).
For example, for m = 2, the above formula is equivalent to A048675(T(n, k)) = A048675(n) * A048675(k).
A195017(T(n, k)) = A195017(n) * A195017(k).
A248663(T(n, k)) = A048720(A248663(n), A248663(k)).
T(n, k) = A306697(n, k) if and only if T(n, k) = A329329(n, k).
A007913(T(n, k)) = A007913(T(A007913(n), A007913(k))) = A007913(A329329(n, k)).
(End)

New name from Peter Munn, Jul 17 2021

A339746 Positive integers of the form 2^i3^jk, gcd(k,6)=1, and i == j (mod 3).

Original entry on oeis.org

1, 5, 6, 7, 8, 11, 13, 17, 19, 23, 25, 27, 29, 30, 31, 35, 36, 37, 40, 41, 42, 43, 47, 48, 49, 53, 55, 56, 59, 61, 64, 65, 66, 67, 71, 73, 77, 78, 79, 83, 85, 88, 89, 91, 95, 97, 101, 102, 103, 104, 107, 109, 113, 114, 115, 119, 121, 125, 127, 131, 133, 135

Offset: 1

Griffin N. Macris, Dec 15 2020

nonn

From Peter Munn, Mar 16 2021: (Start)
The positive integers in the multiplicative subgroup of the positive rationals generated by 8, 6, and A215848 (primes greater than 3).
This subgroup, denoted H, has two cosets: 2H = (1/3)H and 3H = (1/2)H. It follows that the sequence is one part of a 3-part partition of the positive integers with the property that each part's terms are half the even terms of one of the other parts and also one third of the multiples of 3 in the remaining part.
(End)
Positions of multiples of 3 in A276085 (and in A276075). Because A276085 is completely additive, this is closed under multiplication: if m and n are in the sequence then so is m*n. - Antti Karttunen, May 27 2024
The coset sequences mentioned in Peter Munn's comment above are A373261 and A373262. - Antti Karttunen, Jun 04 2024

Robert Israel, Table of n, a(n) for n = 1..10000

Sequences of positive integers in a multiplicative subgroup of positive rationals generated by a set S and A215848: S={}: A007310, S={6}: A064615, S={3,4}: A003159, S={2,9}: A007417, S={4,6}: A036668, S={3,8}: A191257, S={4,9}: A339690, S={6,8}: this sequence.
Positions of 0's in A373153, positions of multiples of 3 in A276085 and in A372576.
Cf. A372573 (characteristic function), A373261, A373262.
Subsequences: A064615, A373144, A373373, A373484, A373837, A374042, A374044, A374120, A377872.
Sequences giving positions of multiples of k in A276085, for k=2, 3, 4, 5, 8, 27, 3125: A003159, this sequence, A369002, A373140, A373138, A377872, A377878.
Cf. also A332820, A373992, A383288.

N:= 1000: # for terms <= N
R:= {}:
for k1 from 0 to floor(N/6) do
  for k0 in [1,5] do
    k:= k0 + 6*k1;
    for j from 0 while 3^j*k <= N do
      for i from (j mod 3) by 3 do
        x:= 2^i * 3^j * k;
        if x > N then break fi;
        R:= R union {x}
od od od od:
sort(convert(R,list)); # Robert Israel, Apr 08 2021

Select[Range[130], Mod[IntegerExponent[#, 2] - IntegerExponent[#, 3], 3] == 0 &]

isA339746 = A372573; \\ Antti Karttunen, Jun 04 2024

from sympy import factorint
def ok(n):
  f = factorint(n, limit=4)
  i, j = 0 if 2 not in f else f[2], 0 if 3 not in f else f[3]
  return (i-j)%3 == 0
def aupto(limit): return [m for m in range(1, limit+1) if ok(m)]
print(aupto(200)) # Michael S. Branicky, Mar 26 2021

from itertools import count
def A339746(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c = n+x
        for i in range(x.bit_length()+1):
            i2 = 1<x:
                    break
                m = x//k
                c -= (m-1)//6+(m-5)//6+2
        return c
    return bisection(f,n,n) # Chai Wah Wu, Feb 12 2025

Formula

a(n) ~ (91/43)*n.

A067368 a(n) is the smallest positive even integer that cannot be expressed as the product of two or three previous terms (not necessarily distinct).

Original entry on oeis.org

2, 6, 10, 14, 16, 18, 22, 26, 30, 34, 38, 42, 46, 48, 50, 54, 58, 62, 66, 70, 74, 78, 80, 82, 86, 90, 94, 98, 102, 106, 110, 112, 114, 118, 122, 126, 128, 130, 134, 138, 142, 144, 146, 150, 154, 158, 162, 166, 170, 174, 176, 178, 182, 186, 190, 194, 198, 202, 206

Offset: 1

Jeremiah K. Hower (jhower(AT)vt.edu), Jan 20 2002

a(n+1) - a(n) = 2 or 4 for all n >= 1. See A067395 for the sequence of differences.
From Jianing Song, Sep 21 2018: (Start)
Numbers of the form 2^(3t+1)*s where s is an odd number.
Also positions of 1 in A191255. (End)
The asymptotic density of this sequence is 2/7. - Amiram Eldar, May 31 2024

			8 = 2*2*2, but 10 = 2*5 cannot be expressed with factors 2 and 6, so a(3) = 10.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A067395, A067396, A191255, A191257, A213258.

N:= 1000:
A:= {seq(seq(2^(3*k+1)*s,s=1..N/2^(3*k+1),2),k=0..floor(log[2](N/2)/3))}:
sort(convert(A,list)); # Robert Israel, Jul 23 2019

t = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0, 2}, 2 -> {0, 3}, 3 -> {0, 1}}] &, {0}, 9] (* A191255 *)
Flatten[Position[t, 0]] (* A005408, the odds *)
a = Flatten[Position[t, 1]] (* this sequence *)
b = Flatten[Position[t, 2]] (* A213258 *)
a/2  (* A191257 *)
b/4  (* a/2 *)
(* Clark Kimberling, May 28 2011 *)

isok(n) = valuation(n, 2)%3==1; \\ Altug Alkan, Sep 21 2018

def A067368(n):
    def f(x): return n+x-sum(((x>>i)-1>>1)+1 for i in range(0,x.bit_length(),3))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m<<1 # Chai Wah Wu, Feb 17 2025

Conjecture: a(n) = a(n-1) + 2 if (n = 2a(k) + k + 1) or (n = 2a(k) + k) for some k, otherwise a(n) = a(n-1) + 4. This has been confirmed for several hundred terms.
The above conjecture is correct because there are 2*(a(k+1)-a(k)) terms that are not divisible by 4 in the k-th interval which are determined by terms that are divisible by 4. For example, there are 2*(a(2)-a(1)) = 2*(6-2) = 8 terms between a(5) = 16 and a(14) = 48 because numbers of the form 2*s are always terms where s is an odd number. So first differences of a(n) determine the corresponding intervals and the formula above always holds. - Altug Alkan, Sep 24 2018
a(n) = 2*A191257(n) = A213258(n)/2. - Jianing Song, Sep 21 2018

Edited by John W. Layman, Jan 23 2002

A213258 Positive integers that are not in A213257.

Original entry on oeis.org

4, 12, 20, 28, 32, 36, 44, 52, 60, 68, 76, 84, 92, 96, 100, 108, 116, 124, 132, 140, 148, 156, 160, 164, 172, 180, 188, 196, 204, 212, 220, 224, 228, 236, 244, 252, 256, 260, 268, 276, 284, 288, 292, 300, 308, 316, 324, 332, 340, 348, 352, 356, 364, 372, 380, 388, 396, 404, 412, 416, 420, 428, 436, 444, 452, 460, 468, 476, 480, 484, 492, 500

Offset: 1

John W. Layman, Jun 07 2012

nonn

Conjecture. The terms of this sequence are given by the positions of 2 in the fixed-point of the morphism 0 -> 01, 1 -> 02, 2 -> 03, 3 -> 01 (see A191255). (This has been confirmed for over 5000 terms of A213257.) To illustrate, the fixed-point of the indicated morphism is {0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,1,0,1,0,2,0,...} and 2 occurs at positions {4,12,20,...}, integers in this sequence but missing from A213257.
It appears that the terms of this sequence are all of the form of 4 times an odd integer multiplied by a nonnegative power of 8.
The above two conjectures are correct. This is indeed positions of 2 in A191255, and numbers of the form 2^(3t+2)*s where s is an odd number. - Jianing Song, Sep 21 2018
The asymptotic density of this sequence is 1/7. - Amiram Eldar, May 31 2024

Cf. A067368, A191255, A191257, A213257.

Select[Range[500], Mod[IntegerExponent[#, 2], 3] == 2 &] (* Amiram Eldar, May 31 2024 *)

is(n) = valuation(n, 2) % 3 == 2; \\ Amiram Eldar, May 31 2024

def A213258(n):
    def f(x): return n+x-sum(((x>>i)-1>>1)+1 for i in range(0,x.bit_length(),3))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m<<2 # Chai Wah Wu, Feb 17 2025

a(n) = 2*A067368(n) = 4*A191257(n). - Amiram Eldar, May 31 2024

A191255 Fixed point of the morphism 0 -> 01, 1 -> 02, 2 -> 03, 3 -> 01.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 28 2011

nonn

See A191250.

The asymptotic density of the occurrences of k = 0, 1, 2 and 3 is 1/2, 2/7, 1/7 and 1/14, respectively. The asymptotic mean of this sequence is 11/14. - Amiram Eldar, May 31 2024

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A191250, A191254, A373157.

Positions of 0 or 3: A191257; positions of 0: A005408; positions of 1: A067368; positions of 2: A213258.

t = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0, 2}, 2 -> {0, 3}, 3 -> {0, 1}}] &, {0}, 9] (* this sequence *)
Flatten[Position[t, 0]] (* A005408, the odds *)
a = Flatten[Position[t, 1]] (* A067368 *)
b = Flatten[Position[t, 2]] (* A213258 *)
a/2  (* A191257 *)
b/4  (* a/2 *)

A191255(n) = if(n%2, 0, my(e=valuation(n, 2)%3); if(!e, 3, e)); \\ Antti Karttunen, May 28 2024, after Jianing Song's Sep 21 2018 formula

a(n) = 0 for odd n, otherwise a(n) is the unique number in {1,2,3} that is congruent to v2(n) modulo 3, where v2(n) = A007814(n) is the 2-adic valuation of n. - Jianing Song, Sep 21 2018 [Clarified by Jianing Song, May 30 2024]

Recurrence: a(2n-1) = 0, a(2n) = 1, 2, 3, 1 for a(n) = 0, 1, 2, 3 respectively. - Jianing Song, May 30 2024

A332821 One part of a 3-way classification of the positive integers. Numbers n for which A048675(n) == 1 (mod 3).

Original entry on oeis.org

2, 5, 9, 11, 12, 16, 17, 21, 23, 28, 30, 31, 39, 40, 41, 47, 49, 52, 54, 57, 59, 66, 67, 70, 72, 73, 75, 76, 83, 87, 88, 91, 96, 97, 100, 102, 103, 109, 111, 116, 126, 127, 128, 129, 130, 133, 135, 136, 137, 138, 148, 149, 154, 157, 159, 165, 167, 168, 169, 172, 175, 179, 180, 183, 184, 186, 190, 191, 197, 203, 211, 212

Offset: 1

Antti Karttunen and Peter Munn, Feb 25 2020

nonn

The positive integers are partitioned between A332820, this sequence and A332822.
For each prime p, the terms include exactly one of p and p^2. The primes alternate between this sequence and A332822. This sequence has the primes with odd indexes, those in A031368.
The terms are the even numbers in A332822 halved. The terms are also the numbers m such that 5m is in A332822, and so on for alternate primes: 11, 17, 23 etc. Likewise, the terms are the numbers m such that 3m is in A332820, and so on for alternate primes: 7, 13, 19 etc.
The numbers that are half of the even terms of this sequence are in A332820, which consists exactly of those numbers. The numbers that are one third of the terms that are multiples of 3 are in A332822, which consists exactly of those numbers. For larger primes, an alternating pattern applies as described in the previous paragraph.
If we take each odd term of this sequence and replace each prime in its factorization by the next smaller prime, the resulting number is in A332822, which consists entirely of those numbers.
The product of any 2 terms of this sequence is in A332822, the product of any 3 terms is in A332820, and the product of a term of A332820 and a term of this sequence is in this sequence. So if a number k is present, k^2 is in A332822, k^3 is in A332820, and k^4 is in this sequence.
If k is an even number, exactly one of {k/2, k, 2k} is in the sequence (cf. A191257 / A067368 / A213258); and generally if k is a multiple of a prime p, exactly one of {k/p, k, k*p} is in the sequence.

Positions of ones in A332823; equivalently, numbers in row 3k+1 of A277905 for some k >= 0.
Cf. A048675, A332820, A332822.
Comparable 2 or 3-way classifications: A000379/A000028, A001969/A000069, A003159/A036554, A005843/A005408, A028260/A026424, A191257/A067368/A213258, A325431/A325432, A329609/A329604/A332812.
Subsequences: intersection of A026478 and A066208, A031368 (prime terms), A033431\{0}, A052934\{1}, A069486, A099800, A167747\{1}, A244725\{0}, A244728\{0}, A338911 (semiprime terms).

Select[Range@ 212, Mod[Total@ #, 3] == 1 &@ Map[#[[-1]]*2^(PrimePi@ #[[1]] - 1) &, FactorInteger[#]] &] (* Michael De Vlieger, Mar 15 2020 *)

isA332821(n) =  { my(f = factor(n)); (1==((sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3)); };

{a(n) : n >= 1} = {2 * A332820(k) : k >= 1} U {A003961(A332822(k)) : k >= 1}.

{a(n) : n >= 1} = {A332822(k)^2 : k >= 1} U {A331590(2, A332820(k)) : k >= 1}.

A332822 One part of a 3-way classification of the positive integers. Numbers n for which A048675(n) == 2 (mod 3).

Original entry on oeis.org

3, 4, 7, 10, 13, 18, 19, 22, 24, 25, 29, 32, 34, 37, 42, 43, 45, 46, 53, 55, 56, 60, 61, 62, 71, 78, 79, 80, 81, 82, 85, 89, 94, 98, 99, 101, 104, 105, 107, 108, 113, 114, 115, 118, 121, 131, 132, 134, 139, 140, 144, 146, 150, 151, 152, 153, 155, 163, 166, 173, 174, 176, 181, 182, 187, 189, 192, 193, 194, 195, 199, 200, 204

Offset: 1

Antti Karttunen and Peter Munn, Feb 25 2020

nonn

The positive integers are partitioned between A332820, A332821 and this sequence.
For each prime p, the terms include exactly one of p and p^2. The primes alternate between this sequence and A332821. This sequence has the primes with even indexes, those in A031215.
The terms are the even numbers in A332820 halved. The terms are also the numbers m such that 5m is in A332820, and so on for alternate primes: 11, 17, 23 etc. Likewise, the terms are the numbers m such that 3m is in A332821, and so on for alternate primes: 7, 13, 19, 29 etc.
If we take each odd term of this sequence and replace each prime in its factorization by the next smaller prime, we get the same set of numbers as we get from halving the even terms of this sequence, and A332821 consists exactly of those numbers. The numbers that are one third of the terms that are multiples of 3 are in A332820, which consists exactly of those numbers. The numbers that are one fifth of the terms that are multiples of 5 constitute A332821, and for larger primes, an alternating pattern applies as described in the previous paragraph.
The product of any 2 terms of this sequence is in A332821, the product of any 3 terms is in A332820, and the product of a term of A332820 and a term of this sequence is in this sequence. So if a number k is present, k^2 is in A332821, k^3 is in A332820, and k^4 is in this sequence.
If k is an even number, exactly one of {k/2, k, 2k} is in the sequence (cf. A191257 / A067368 / A213258); and generally if k is a multiple of a prime p, exactly one of {k/p, k, k*p} is in the sequence.

Positions of terms valued -1 in A332823; equivalently, numbers in row 3k-1 of A277905 for some k >= 1.
Cf. A048675, A332820, A332821.
Comparable 2 or 3-way classifications: A000379/A000028, A001969/A000069, A003159/A036554, A005843/A005408, A028260/A026424, A191257/A067368/A213258, A325431/A325432, A329609/A329604/A332812.
Subsequences: intersection of A026478 and A066207, A031215 (prime terms), A033430\{0}, A117642\{0}, A169604, A244727\{0}, A244729\{0}, A338910 (semiprime terms).

Select[Range@ 204, Mod[Total@ #, 3] == 2 &@ Map[#[[-1]]*2^(PrimePi@ #[[1]] - 1) &, FactorInteger[#]] &] (* Michael De Vlieger, Mar 15 2020 *)

isA332822(n) =  { my(f = factor(n)); (2==((sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3)); };

{a(n) : n >= 1} = {2 * A332821(k) : k >= 1} U {A003961(A332821(k)) : k >= 1}.

{a(n) : n >= 1} = {A332821(k)^2 : k >= 1} U {A331590(2, A332821(k)) : k >= 1}.

A374044 Non-multiples of 3 whose 2-adic valuation is a multiple of 3.

Original entry on oeis.org

1, 5, 7, 8, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 40, 41, 43, 47, 49, 53, 55, 56, 59, 61, 64, 65, 67, 71, 73, 77, 79, 83, 85, 88, 89, 91, 95, 97, 101, 103, 104, 107, 109, 113, 115, 119, 121, 125, 127, 131, 133, 136, 137, 139, 143, 145, 149, 151, 152, 155, 157, 161, 163, 167, 169, 173, 175, 179, 181, 184, 185, 187

Offset: 1

Antti Karttunen, Jun 27 2024

A multiplicative semigroup: if m and n are in the sequence, then so is m*n.

The asymptotic density of this sequence is 8/21. - Amiram Eldar, Jun 28 2024

Antti Karttunen, Table of n, a(n) for n = 1..12000

Cf. A007814, A007949, A374043 (characteristic function).
Intersection of A001651 and A191257.
Subsequence of A339746, of A373992, and of A374042.

Select[Range[200], !Divisible[#, 3] && Divisible[IntegerExponent[#, 2], 3] &] (* Amiram Eldar, Jun 28 2024 *)

PARI
```
isA374044 = A374043;
```

{k | A007814(k) == 0 (mod 3), A007949(k) = 0}.

A319281 Numbers of the form 16^i(16j + 1).

Original entry on oeis.org

1, 16, 17, 33, 49, 65, 81, 97, 113, 129, 145, 161, 177, 193, 209, 225, 241, 256, 257, 272, 273, 289, 305, 321, 337, 353, 369, 385, 401, 417, 433, 449, 465, 481, 497, 513, 528, 529, 545, 561, 577, 593, 609, 625, 641, 657, 673, 689, 705, 721, 737, 753, 769, 784

Offset: 1

Jianing Song, Sep 16 2018

nonn

{a(n)} gives all positive fourth powers modulo all powers of 2, that is, positive fourth powers over 2-adic integers. So this sequence is closed under multiplication.

Jianing Song, Table of n, a(n) for n = 1..10002 (all terms <= 150000)

A158057 is a proper subsequence.
Perfect powers over 2-adic integers:
Squares: positive: A234000; negative: A004215 (negated);
Cubes: A191257;
Fourth powers: positive: this sequence; negative: A319282 (negated).

isA319281(n)= n\16^valuation(n, 16)%16==1

def A319281(n):
    if n<3: return 15*n-14
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n-1+x-sum((((x>>(i<<2))-1)>>4)+1 for i in range(x.bit_length()>>2))
    return bisection(f,n,n) # Chai Wah Wu, Feb 17 2025

a(n) = 15*n + O(log(n)).

cp's OEIS Frontend

A332820 Integers in the multiplicative subgroup of positive rationals generated by the products of two consecutive primes and the cubes of primes. Numbers k for which A048675(k) is a multiple of three.

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A297845 Encoded multiplication table for polynomials in one indeterminate with nonnegative integer coefficients. Symmetric square array T(n, k) read by antidiagonals, n > 0 and k > 0. See comment for details.

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A339746 Positive integers of the form 2^i*3^j*k, gcd(k,6)=1, and i == j (mod 3).

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A067368 a(n) is the smallest positive even integer that cannot be expressed as the product of two or three previous terms (not necessarily distinct).

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A213258 Positive integers that are not in A213257.

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A191255 Fixed point of the morphism 0 -> 01, 1 -> 02, 2 -> 03, 3 -> 01.

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A332821 One part of a 3-way classification of the positive integers. Numbers n for which A048675(n) == 1 (mod 3).

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A339746 Positive integers of the form 2^i3^jk, gcd(k,6)=1, and i == j (mod 3).

A319281 Numbers of the form 16^i(16j + 1).