A325431 -id:A325431 - cp's OEIS frontend

A325432 Complement of A325431.

3, 4, 6, 8, 15, 20, 21, 27, 28, 30, 33, 36, 39, 40, 42, 44, 48, 51, 52, 54, 56, 57, 64, 66, 68, 69, 72, 75, 76, 78, 87, 88, 92, 93, 96, 100, 102, 104, 105, 111, 114, 116, 123, 124, 128, 129, 135, 136, 138, 140, 141, 147, 148, 150, 152, 159, 164, 165, 172

Offset: 1

Clark Kimberling, May 01 2019

These are the numbers 3x and 4x as x ranges through the numbers x > 1 in A325431.

Equivalently, numbers k whose exponent of the highest power of 3 dividing k and exponent of the highest power of 4 dividing k have an opposite parity. The asymptotic density of this sequence is 7/20. - Amiram Eldar, Sep 20 2020

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A325417, A325431.

a = {1}; Do[AppendTo[a, NestWhile[# + 1 &, Last[a] + 1, Apply[Or,
Map[MemberQ[a, #] &, Select[Flatten[{#/3, #/4}],
IntegerQ]]] &]], {150}]; a          (* A325431 *)
Complement[Range[Last[a]], a]       (* A325432 *)
(* Peter J. C. Moses, Apr 25 2019 *)
Select[Range[100], !Equal @@ Mod[IntegerExponent[#, {3, 4}], 2] &] (* Amiram Eldar, Sep 20 2020 *)

A325524 Difference sequence of A325431.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 06 2019

See A325417 for a guide to related sequences.

Conjecture: every term is in {1,2,3}.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A325417, A325431, A325525.

a = {1}; Do[AppendTo[a, NestWhile[# + 1 &, Last[a] + 1, Apply[Or,
Map[MemberQ[a, #] &, Select[Flatten[{#/3, #/4}],
IntegerQ]]] &]], {20000}]; a;         (* A315431 *)
c = Complement[Range[Last[a]], a] ;   (* A325432 *)
Differences[a]  (* A325524 *)
Differences[c]  (* A325525 *)
(* Peter J. C. Moses, Apr 23 2019 *)

Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = 20/13. - Amiram Eldar, Nov 26 2020

A325417 a(n) is the least number not 2a(m) or 3a(m)+1 for any m < n.

Original entry on oeis.org

1, 3, 5, 7, 8, 9, 11, 12, 13, 15, 17, 19, 20, 21, 23, 27, 29, 31, 32, 33, 35, 36, 39, 41, 43, 44, 45, 47, 48, 49, 50, 51, 53, 55, 56, 57, 59, 60, 63, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 89, 91, 92, 93, 95, 99, 101, 103, 104

Offset: 1

Clark Kimberling, Apr 24 2019

In column 1 of the following guide to related sequences, disallowed terms are indicated by the variable x representing a(m) for m < n.
Disallowed    Sequence(a)  Complement(c)  Differences(a)  Differences(c)
2x, 3x+1      A325417      A325418        A325444         A325445
3x, 2x+1      A325419      A325420        A325494         A325495
2x+1, 3x+1    A077477      A325422        A325496         A325497
2x, 3x        A036668      A325424        A325498         A325499
[3x/2], 2x    A325425      A325426        A325518         A325519
[3x/2], 2x+1  A325427      A325428        A325520         A325521
[3x/2], 3x    A325429      A325430        A325522         A325523
3x, 4x        A325431      A325432        A325525         A325526
2x, 3x-1      A325462      A325463        A325526         A325527
2x, 3x-2      A325464      A325465        A325528         A325529
2x-1, 3x-1    A325440      A325441        A325530         A325531
2x-1, 3x      A325442      A325443        A325532         A325533
2x+1, 3x+2    A325539      A325540        A325541         A325542

			The sequence necessarily starts with 1.  The next 2 terms are determined as follows:  because a(1) = 1, the numbers 2 and 4 are disallowed, so that a(2) = 3, whence the numbers 6 and 10 are disallowed, so that a(3) = 5.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A325418, A325444.

a = {1}; Do[AppendTo[a, NestWhile[# + 1 &, Last[a] + 1,
Apply[Or, Map[MemberQ[a, #] &, Select[Flatten[{#/2, (# - 1)/3}],
IntegerQ]]] &]], {150}]; a     (* A325417 *)
Complement[Range[Last[a]], a]  (* A325418 *)
(* Peter J. C. Moses, Apr 23 2019 *)

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.

Original entry on oeis.org

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

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}.

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A325432 Complement of A325431.

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A325524 Difference sequence of A325431.

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A325417 a(n) is the least number not 2*a(m) or 3*a(m)+1 for any m < n.

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

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A325417 a(n) is the least number not 2a(m) or 3a(m)+1 for any m < n.