A003159 -id:A003159 - cp's OEIS frontend

A352272 Numbers whose squarefree part is congruent to 1 modulo 6.

1, 4, 7, 9, 13, 16, 19, 25, 28, 31, 36, 37, 43, 49, 52, 55, 61, 63, 64, 67, 73, 76, 79, 81, 85, 91, 97, 100, 103, 109, 112, 115, 117, 121, 124, 127, 133, 139, 144, 145, 148, 151, 157, 163, 169, 171, 172, 175, 181, 187, 193, 196, 199, 205, 208, 211, 217, 220, 223, 225, 229

Offset: 1

Peter Munn, Mar 10 2022

Numbers of the form 4^i * 9^j * (6k+1), i, j, k >= 0.
Closed under multiplication.
The sequence forms a subgroup of the positive integers under the commutative operation A059897(.,.), one of 8 subgroups of the form {k : A007913(k) == 1 (mod m)} - in each case m is a divisor of 24. A059897 has a relevance to squarefree parts that arises from the identity A007913(k*n) = A059897(A007913(k), A007913(n)), where A007913(n) is the squarefree part of n.
The subgroup has 8 cosets, which partition the positive integers as follows. For each i in {1, 5}, j in {1, 2, 3, 6} there is a coset {m^2 * (6k+i) * j : m >= 1, k >= 0}. See the table in the examples.
None of the 8 cosets have been entered into the database previously, but many subgroups of the quotient group (which are formed of certain combinations of cosets) are represented among earlier OEIS sequences, including 6 of the 7 subgroups of index 2 (which combine 4 cosets). This sequence can therefore be defined as the intersection of pairs or triples of these sequences in many combinations (see the cross-references). See also the table in the example section of A352273 (the coset that includes 5).

			The squarefree part of 9 is 1, which is congruent to 1 (mod 6), so 9 is in the sequence.
The squarefree part of 14 is 14, which is congruent to 2 (mod 6), so 14 is not in the sequence.
The squarefree part of 52 = 2^2 * 13 is 13, which is congruent to 1 (mod 6), so 52 is in the sequence.
The 8 cosets described in the initial comments (forming a partition of the positive integers) are shown as rows of the following table. The first half of the table corresponds to (6k+i) with i=1; the second half to i=5, with row 5 being A352273.
   1,  4,   7,   9,  13,  16,  19,  25,  28,  31,  36, ...
   2,  8,  14,  18,  26,  32,  38,  50,  56,  62,  72, ...
   3, 12,  21,  27,  39,  48,  57,  75,  84,  93, 108, ...
   6, 24,  42,  54,  78,  96, 114, 150, 168, 186, 216, ...
   5, 11,  17,  20,  23,  29,  35,  41,  44,  45,  47, ...
  10, 22,  34,  40,  46,  58,  70,  82,  88,  90,  94, ...
  15, 33,  51,  60,  69,  87, 105, 123, 132, 135, 141, ...
  30, 66, 102, 120, 138, 174, 210, 246, 264, 270, 282, ...
The product of two positive integers is in this sequence if and only if they are in the same coset. The asymptotic density of cosets (containing) 1 and 5 is 1/4; of cosets 2 and 10 is 1/8; of cosets 3 and 15 is 1/12; of cosets 6 and 30 is 1/24.

Eric Weisstein's World of Mathematics, Group, Quotient Group, Squarefree Part.

Intersection of any 2 of A055047, A339690 and A352274.
Intersection of any 4 sets chosen from A003159, A007417, A026225, A036668, A189715 and A225837 (in most cases, only 3 sets are needed - specifically if the pairwise intersections of the 3 sets differ from each other).
Closure of A084089 under multiplication by 9.
Other subsequences: A000290\{0}, A016921, A229848 (apparently, with 55 the first difference).
A334832 lists equivalent sequences modulo other divisors of 24.
Cf. A007913, A059897, A307151, A352273.

PARI
```
isok(m) = core(m) % 6 == 1;
```

from itertools import count
def A352272(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 count(0):
            i2 = 9**i
            if i2>x:
                break
            for j in count(0,2):
                k = i2<x:
                    break
                c -= (x//k-1)//6+1
        return c
    return bisection(f,n,n) # Chai Wah Wu, Feb 14 2025

{a(n) : n >= 1} = {m >= 1 : A007913(m) == 1 (mod 6)}.
{a(n) : n >= 1} = A334832 U A334832/7 U A334832/13 U A334832/19 where A334832/k denotes {A334832(m)/k : m >= 1, k divides A334832(m)}.
Using the same denotation, {a(n) : n >= 1} = A352273/5 = {A307151(A352273(m)) : m >= 1}.

A368998 Numbers k such that A003415(k) and A276085(k) are both even, where A003415 is the arithmetic derivative and A276085 is the primorial base log-function; Numbers k for which A342001(k) is even.

Original entry on oeis.org

1, 4, 9, 12, 15, 16, 20, 21, 25, 28, 33, 35, 36, 39, 44, 48, 49, 51, 52, 55, 57, 60, 64, 65, 68, 69, 76, 77, 80, 81, 84, 85, 87, 91, 92, 93, 95, 100, 108, 111, 112, 115, 116, 119, 121, 123, 124, 129, 132, 133, 135, 140, 141, 143, 144, 145, 148, 155, 156, 159, 161, 164, 169, 172, 176, 177, 180, 183, 185, 187, 188, 189

Offset: 1

Antti Karttunen, Jan 14 2024

nonn

From Antti Karttunen, Jun 03 2024: (Start)
Also numbers k for which A373145(k) is even, or in other words, numbers k such that A003415(k) and A276085(k) are both even.
Because both A003159 and A235992 are multiplicative semigroups, this is also: if m and n are in the sequence then so is m*n.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000290, A003415, A276085, A368995 (subsequence), A368997 (characteristic function), A368999 (complement).
Positions of even terms in A342001 and in A373145.
Intersection of A003159 and A235992.
Disjoint union of A345452 and 2*A358776.

PARI
```
isA368998 = A368997;
```

Alternative definition added as a new primary definition by Antti Karttunen, Jun 04 2024

A372576 a(n) = A276085(n) mod 360, where A276085 is the primorial base log-function.

Original entry on oeis.org

0, 1, 2, 2, 6, 3, 30, 3, 4, 7, 210, 4, 150, 31, 8, 4, 150, 5, 30, 8, 32, 211, 210, 5, 12, 151, 6, 32, 150, 9, 30, 5, 212, 151, 36, 6, 210, 31, 152, 9, 210, 33, 330, 212, 10, 211, 150, 6, 60, 13, 152, 152, 210, 7, 216, 33, 32, 151, 330, 10, 30, 31, 34, 6, 156, 213, 30, 152, 212, 37, 210, 7, 150, 211, 14, 32, 240

Offset: 1

Antti Karttunen, May 25 2024

nonn

Completely additive modulo 360.

On average, every third term is a multiple of 4. See A369001.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to primorial numbers

Cf. A002110, A276085.

Cf. A003159 (positions of even terms), A036554 (of odd terms), A035263, A096268 (parity of terms), A369001, A369002 (positions of multiples of 4).

A002110(n) = prod(i=1,n,prime(i));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
A372576(n) = (A276085(n)%360);

A372587 Numbers k such that (sum of binary indices of k) + (sum of prime indices of k) is even.

Original entry on oeis.org

6, 7, 10, 11, 13, 14, 18, 19, 22, 23, 24, 25, 26, 27, 28, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 44, 49, 50, 52, 56, 57, 58, 62, 69, 70, 72, 74, 75, 76, 77, 82, 83, 85, 86, 87, 88, 90, 92, 96, 98, 100, 102, 103, 104, 106, 107, 108, 109, 112, 117, 120, 123

Offset: 1

Gus Wiseman, May 14 2024

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The odd version is A372586.

			The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
            {2,3}   6  (2,1)
          {1,2,3}   7  (4)
            {2,4}  10  (3,1)
          {1,2,4}  11  (5)
          {1,3,4}  13  (6)
          {2,3,4}  14  (4,1)
            {2,5}  18  (2,2,1)
          {1,2,5}  19  (8)
          {2,3,5}  22  (5,1)
        {1,2,3,5}  23  (9)
            {4,5}  24  (2,1,1,1)
          {1,4,5}  25  (3,3)
          {2,4,5}  26  (6,1)
        {1,2,4,5}  27  (2,2,2)
          {3,4,5}  28  (4,1,1)
        {2,3,4,5}  30  (3,2,1)
      {1,2,3,4,5}  31  (11)
            {1,6}  33  (5,2)
            {2,6}  34  (7,1)
          {1,2,6}  35  (4,3)
          {1,3,6}  37  (12)
          {2,3,6}  38  (8,1)

Positions of even terms in A372428, zeros A372427.
For minimum (A372437) we have A372440, complement A372439.
For length (A372441, zeros A071814) we have A372591, complement A372590.
For maximum (A372442, zeros A372436) we have A372589, complement A372588.
The complement is A372586.
For just binary indices:
- length: A001969, complement A000069
- sum: A158704, complement A158705
- minimum:  A036554, complement A003159
- maximum: A053754, complement A053738
For just prime indices:
- length: A026424 A028260 (count A027187), complement (count A027193)
- sum: A300061 (count A058696), complement A300063 (count A058695)
- minimum: A340933 (count A026805), complement A340932 (count A026804)
- maximum: A244990 (count A027187), complement A244991 (count A027193)
A005408 lists odd numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A031368 lists odd-indexed primes, even A031215.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A066207, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],EvenQ[Total[bix[#]]+Total[prix[#]]]&]

Numbers k such that A029931(k) + A056239(k) is even.

A372850 Numbers whose distinct prime indices are the binary indices of some prime number.

Original entry on oeis.org

3, 6, 9, 10, 12, 18, 20, 22, 24, 27, 30, 36, 40, 42, 44, 46, 48, 50, 54, 60, 66, 70, 72, 80, 81, 84, 88, 90, 92, 96, 100, 102, 108, 114, 118, 120, 126, 130, 132, 140, 144, 150, 160, 162, 168, 176, 180, 182, 184, 192, 198, 200, 204, 216, 228, 236, 238, 240, 242

Offset: 1

Gus Wiseman, May 16 2024

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Note the function taking a set s to its rank Sum_i 2^(s_i-1) is the inverse of A048793 (binary indices).

			The distinct prime indices of 45 are {2,3}, which are the binary indices of 6, which is not prime, so 45 is not in the sequence.
The distinct prime indices of 60 are {1,2,3}, which are the binary indices of 7, which is prime, so 60 is in the sequence.
The terms together with their prime indices begin:
    3: {2}
    6: {1,2}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   18: {1,2,2}
   20: {1,1,3}
   22: {1,5}
   24: {1,1,1,2}
   27: {2,2,2}
   30: {1,2,3}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   46: {1,9}
   48: {1,1,1,1,2}
   50: {1,3,3}
   54: {1,2,2,2}
   60: {1,1,2,3}
   66: {1,2,5}
   70: {1,3,4}

For even instead of prime we have A005408, with multiplicity A003159.
For odd instead of prime we have  A005843, with multiplicity A036554.
For prime indices with multiplicity we have A277319, counted by A372688.
The squarefree case is A372851, counted by A372687.
Partitions of this type are counted by A372887.
A014499 lists binary indices of prime numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A038499 counts partitions of prime length, strict A085756.
A048793 and A272020 (reverse) list binary indices:
- length A000120
- min A001511
- sum A029931
- max A070939
A058698 counts partitions of prime numbers, strict A064688.
A372885 lists primes whose binary indices sum to a prime, indices A372886.
Cf. A000040, A005940, A023506, A029837, A035100, A071814, A096111, A372429, A372436, A372441, A372471, A372689.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],PrimeQ[Total[2^(Union[prix[#]]-1)]]&]

Numbers k such that Sum_{i:prime(i)|k} 2^(i-1) is prime, where the sum is over the distinct prime indices of k.

A053661 For n > 1: if n is present, 2n is not.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 11, 12, 13, 15, 17, 19, 20, 21, 23, 25, 27, 28, 29, 31, 32, 33, 35, 36, 37, 39, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 57, 59, 60, 61, 63, 65, 67, 68, 69, 71, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 89, 91, 92, 93, 95, 97, 99, 100, 101, 103, 105

Offset: 1

Jeevan Chana Rai (Karanjit.Rai(AT)btinternet.com), Feb 16 2000

The Name line gives a property of the sequence, not a definition. The sequence can be defined simultaneously with b(n) := A171945(n) via a(n) = mex{a(i), b(i) : 0 <= i < n} (n >= 0}, b(n)=2a(n). The two sequences are complementary, hence A053661 is identical to A171944 (except for the first terms). Furthmore, A053661 is the same as A003159 except for the replacement of vile by dopey powers of 2. - Aviezri S. Fraenkel, Apr 28 2011
For n >= 2, either n = 2^k where k is odd or n = 2^k*m where m > 1 is odd and k is even (found by Kirk Bresniker and Stan Wagon). [Robert Israel, Oct 10 2010]
Subsequence of A175880; A000040, A001749, A002001, A002042, A002063, A002089, A003947, A004171 and A081294 are subsequences.

Robert Israel, Table of n, a(n) for n = 1..10000
Aviezri S. Fraenkel, The vile, dopey, evil and odious game players, Discrete Math. 312 (2012), no. 1, 42-46.

Essentially identical to A171944 and the complement of A171945.

a053661 n = a053661_list !! (n-1)
a053661_list = filter (> 0) a175880_list -- Reinhard Zumkeller, Feb 09 2011

N:= 1000: # to get all terms <= N
sort([1,seq(2^(2*i+1),i=0..(ilog2(N)-1)/2), seq(seq(2^(2*i)*(2*j+1),j=1..(N/2^(2*i)-1)/2),i=0..ilog2(N)/2)]); # Robert Israel, Jul 24 2015

Clear[T]; nn = 105; T[n_, k_] := T[n, k] = If[n < 1 || k < 1, 0, If[n == 1 || k == 1, 1, If[k > n, T[k, n], If[n > k, T[k, Mod[n, k, 1]], -Product[T[n, i], {i, n - 1}]]]]]; DeleteCases[Table[If[T[n, n] == -1, n, ""], {n, 1, nn}], ""] (* Mats Granvik, Aug 25 2012 *)

More terms from James Sellers, Feb 22 2000

A171946 N-positions for game of UpMark.

Original entry on oeis.org

0, 2, 4, 5, 6, 8, 10, 12, 13, 14, 16, 17, 18, 20, 21, 22, 24, 26, 28, 29, 30, 32, 34, 36, 37, 38, 40, 42, 44, 45, 46, 48, 49, 50, 52, 53, 54, 56, 58, 60, 61, 62, 64, 65, 66, 68, 69, 70, 72, 74, 76, 77, 78, 80, 81, 82, 84, 85, 86, 88, 90, 92, 93, 94, 96, 98

Offset: 1

N. J. A. Sloane, Oct 29 2010

It appears that this is the sequence of positions of 0 in the 1-limiting word of the morphism 0 -> 10, 1 -> 00; see A284948. - Clark Kimberling, Apr 18 2017
It appears that this sequence gives the positions of 1 in the limiting 0-word of the morphism 0->11, 1-> 01.  See A285383.  - Clark Kimberling, Apr 26 2017
Apparently a(n) = 1+A003159(n-1). - R. J. Mathar, Jun 24 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Aviezri S. Fraenkel, The vile, dopey, evil and odious game players, Discrete Math. 312 (2012), no. 1, 42-46.

Complement of A171947.

import Data.List (delete)
a171946 n = a171946_list !! (n-1)
a171946_list = 0 : f [2..] where
   f (w:ws) = w : f (delete (2 * w - 1) ws)
-- Reinhard Zumkeller, Oct 26 2014

def A171946(n):
    if n == 1: return 0
    def f(x):
        c, s = n, bin(x-1)[2:]
        l = len(s)
        for i in range(l&1,l,2):
            c += int(s[i])+int('0'+s[:i],2)
        return c
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Jan 29 2025

A171947 P-positions for game of UpMark.

Original entry on oeis.org

1, 3, 7, 9, 11, 15, 19, 23, 25, 27, 31, 33, 35, 39, 41, 43, 47, 51, 55, 57, 59, 63, 67, 71, 73, 75, 79, 83, 87, 89, 91, 95, 97, 99, 103, 105, 107, 111, 115, 119, 121, 123, 127, 129, 131, 135, 137, 139, 143, 147, 151, 153, 155, 159, 161, 163, 167, 169, 171, 175, 179

Offset: 1

N. J. A. Sloane, Oct 29 2010

The following description, due to D. R. Hofstadter, Email, Oct 23 2014, is presumably equivalent to Fraenkel's. Begin with 1, and then each new member is 2k-1, where k is the smallest unused non-member of the sequence. Thus k starts out as 2, so 2k-1 = 3, so 3 is the sequence's second member. The next value of k is 4, giving 2k-1 = 7, so 7 is the sequence's third member. Then k = 5, so 9 is the next member. Then k = 6, so 11 is the next member. Then k = 8, so 15 is the next member. Etc. - N. J. A. Sloane, Oct 26 2014
It appears that this is the sequence of positions of 1 in the 1-limiting word of the morphism 0 -> 10, 1 -> 00; see A284948. - Clark Kimberling, Apr 18 2017
It appears that this sequence gives the positions of 0 in the limiting 0-word of the morphism 0 -> 11, 1 -> 01. See A285383. - Clark Kimberling, Apr 26 2017
It appears that this sequence gives integers that are congruent to 2^k+1 (mod 2^(k+1)), where k is any odd integer >=1. - Jules Beauchamp, Dec 04 2023

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Aviezri S. Fraenkel, The vile, dopey, evil and odious game players, Discrete Math. 312 (2012), no. 1, 42-46.

Complement of A171946. Essentially identical to A072939.
A249034 gives missing odd numbers.
Cf. A003159.

import Data.List (delete)
a171947 n = a171947_list !! (n-1)
a171947_list = 1 : f [2..] where
   f (w:ws) = y : f (delete y ws) where y = 2 * w - 1
-- Reinhard Zumkeller, Oct 26 2014

# Maple code for M+1 terms of sequence, from N. J. A. Sloane, Oct 26 2014
m:=1; a:=[m]; M:=100;
for n from 1 to M do
m:=m+1; if m in a then m:=m+1; fi;
c:=2*m-1;
a:=[op(a),c];
od:
[seq(a[n],n=1..nops(a))];

f[n_] := Block[{a = {1}, b = {}, k}, Do[k = 2; While[MemberQ[a, k] || MemberQ[b, k], k++]; AppendTo[a, 2 k - 1]; AppendTo[b, k], {i, 2, n}]; a]; f@ 120 (* Michael De Vlieger, Jul 20 2015 *)

def A171947(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, s = n+x-1, bin(x-1)[2:]
        l = len(s)
        for i in range(l&1,l,2):
            c -= int(s[i])+int('0'+s[:i],2)
        return c
    return bisection(f,n,n) # Chai Wah Wu, Jan 29 2025

Presumably equal to 2*A003159 + 1. - Reinhard Zumkeller, Oct 26 2014

A336321 a(n) = A122111(A225546(n)).

Original entry on oeis.org

1, 2, 3, 4, 7, 5, 19, 6, 9, 11, 53, 10, 131, 23, 13, 8, 311, 15, 719, 22, 29, 59, 1619, 14, 49, 137, 21, 46, 3671, 17, 8161, 12, 61, 313, 37, 25, 17863, 727, 139, 26, 38873, 31, 84017, 118, 39, 1621, 180503, 20, 361, 77, 317, 274, 386093, 33, 71, 58, 733, 3673, 821641, 34, 1742537, 8167, 87, 18, 151, 67, 3681131, 626, 1627, 41, 7754077, 35, 16290047

Offset: 1

Antti Karttunen and Peter Munn, Jul 17 2020

nonn

A122111 and A225546 are both self-inverse permutations of the positive integers based on prime factorizations, and they share further common properties. For instance, they map the prime numbers to powers of 2: A122111 maps the k-th prime to 2^k, whereas A225546 maps it to 2^2^(k-1).

In composing these permutations, this sequence maps the squarefree numbers, as listed in A019565, to the prime numbers in increasing order; and the list of powers of 2 to the "normal" numbers (A055932), as listed in A057335.

			From _Peter Munn_, Jan 04 2021: (Start)
In this set of examples we consider [a(n)] as a function a(.) with an inverse, a^-1(.).
First, a table showing mapping of the powers of 2:
  n     a^-1(2^n) =    2^n =        a(2^n) =
        A001146(n-1)   A000079(n)   A057335(n)
  0             (1)         1            1
  1               2         2            2
  2               4         4            4
  3              16         8            6
  4             256        16            8
  5           65536        32           12
  6      4294967296        64           18
  ...
Next, a table showing mapping of the squarefree numbers, as listed in A019565 (a lexicographic ordering by prime factors):
  n   a^-1(A019565(n))   A019565(n)      a(A019565(n))   a^2(A019565(n))
      Cf. {A337533}      Cf. {A005117}   = prime(n)      = A033844(n-1)
  0              1               1             (1)               (1)
  1              2               2               2                 2
  2              3               3               3                 3
  3              8               6               5                 7
  4              6               5               7                19
  5             12              10              11                53
  6             18              15              13               131
  7            128              30              17               311
  8              5               7              19               719
  9             24              14              23              1619
  ...
As sets, the above columns are A337533, A005117, A008578, {1} U A033844.
Similarly, we get bijections between sets A000290\{0} -> {1} U A070003; and {1} U A335740 -> A005408 -> A066207.
(End)

A122111 composed with A225546.
Cf. A336322 (inverse permutation).
Other sequences used in a definition of this sequence: A000040, A000188, A019565, A248663, A253550, A253560.
Sequences used to express relationship between terms of this sequence: A003159, A003961, A297002, A334747.
Cf. A057335.
A mapping between the binary tree sequences A334866 and A253563.
Lists of sets (S_1, S_2, ... S_j) related by the bijection defined by the sequence: (A000290\{0}, {1} U A070003), ({1} U A001146, A000079, A055932), ({1} U A335740, A005408, A066207), (A337533, A005117, A008578, {1} U A033844).

a(n) = A122111(A225546(n)).
Alternative definition: (Start)
Write n = m^2 * A019565(j), where m = A000188(n), j = A248663(n).
a(1) = 1; otherwise for m = 1, a(n) = A000040(j), for m > 1, a(n) = A253550^j(A253560(a(m))).
(End)
a(A000040(m)) = A033844(m-1).
a(A001146(m)) = 2^(m+1).
a(2^n) = A057335(n).
a(n^2) = A253560(a(n)).
For n in A003159, a(2n) = b(a(n)), where b(1) = 2, b(n) = A253550(n), n >= 2.
More generally, a(A334747(n)) = b(a(n)).
a(A003961(n)) = A297002(a(n)).
a(A334866(m)) = A253563(m).

A374114 Numbers k such that A113177(k) and A276085(k) are both even, where A113177 and A276085 are fully additive with a(p) = Fibonacci(p) and a(p) = p#/p, respectively.

Original entry on oeis.org

1, 3, 4, 9, 12, 16, 25, 27, 35, 36, 48, 49, 55, 64, 65, 75, 77, 81, 85, 91, 95, 100, 105, 108, 115, 119, 121, 133, 140, 143, 144, 145, 147, 155, 161, 165, 169, 185, 187, 192, 195, 196, 203, 205, 209, 215, 217, 220, 221, 225, 231, 235, 243, 247, 253, 255, 256, 259, 260, 265, 273, 285, 287, 289, 295, 299, 300, 301, 305

Offset: 1

Antti Karttunen, Jun 29 2024

nonn

Numbers whose 2-adic valuation (A007814) is even, and the number of the prime factors (with multiplicity, A001222) and the 3-adic valuation (A007949) have the same parity.

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

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

Intersection of A003159 and A373586.
Indices of even terms in A374112.
Cf. A001222, A007814, A007949, A113177, A276085, A374113 (characteristic function), A374115 (complement).

PARI
```
isA374114 = A374113;
```

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