A059897 -id:A059897 - cp's OEIS frontend

A366244 The largest infinitary divisor of n that is a term of A366242.

1, 2, 3, 1, 5, 6, 7, 2, 1, 10, 11, 3, 13, 14, 15, 16, 17, 2, 19, 5, 21, 22, 23, 6, 1, 26, 3, 7, 29, 30, 31, 32, 33, 34, 35, 1, 37, 38, 39, 10, 41, 42, 43, 11, 5, 46, 47, 48, 1, 2, 51, 13, 53, 6, 55, 14, 57, 58, 59, 15, 61, 62, 7, 16, 65, 66, 67, 17, 69, 70, 71

Offset: 1

Amiram Eldar, Oct 05 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A063694, A366242, A366243, A366245.

See the formula section for the relationships with A007913, A046100, A059895, A059896, A059897, A225546, A247503, A352780.

f[p_, e_] := p^BitAnd[e, Sum[2^k, {k, 0, Floor@ Log2[e], 2}]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(e) = sum(k = 0, e, (-2)^k*floor(e/2^k));
a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^s(f[i,2]));}

Multiplicative with a(p^e) = p^A063694(e).
a(n) = n / A366245(n).
a(n) >= 1, with equality if and only if n is a term of A366243.
a(n) <= n, with equality if and only if n is a term of A366242.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1-1/p)*(Sum_{k>=1} p^(A063694(k)-2*k)) = 0.35319488024808595542... .
From Peter Munn, Jan 09 2025: (Start)
a(n) = max({k in A366242 : A059895(k, n) = k}).
a(n) = Product_{k >= 0} A352780(n, 2k).
Also defined by:
- for n in A046100, a(n) = A007913(n);
- a(n^4) = (a(n))^4;
- a(A059896(n,k)) = A059896(a(n), a(k)).
Other identities:
a(n) = sqrt(A366245(n^2)).
a(A059897(n,k)) = A059897(a(n), a(k)).
a(A225546(n)) = A225546(A247503(n)).
(End)

A366245 The largest infinitary divisor of n that is a term of A366243.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 4, 1, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 4, 25, 1, 9, 4, 1, 1, 1, 1, 1, 1, 1, 36, 1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 1, 49, 25, 1, 4, 1, 9, 1, 4, 1, 1, 1, 4, 1, 1, 9, 4, 1, 1, 1, 4, 1, 1, 1, 36, 1, 1, 25, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1

Offset: 1

Amiram Eldar, Oct 05 2023

First differs from A335324 at n = 256.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A063695, A335324, A366242, A366243, A366244.

See the formula section for the relationships with A008833, A046100, A059895, A059896, A059897, A225546, A248101, A352780.

f[p_, e_] := p^BitAnd[e, Sum[2^k, {k, 1, Floor@ Log2[e], 2}]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(e) = -sum(k = 1, e, (-2)^k*floor(e/2^k));
a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^s(f[i,2]));}

Multiplicative with a(p^e) = p^A063695(e).
a(n) = n / A366244(n).
a(n) >= 1, with equality if and only if n is a term of A366242.
a(n) <= n, with equality if and only if n is a term of A366243.
From Peter Munn, Jan 09 2025: (Start)
a(n) = max({k in A366243 : A059895(k, n) = k}).
a(n) = Product_{k >= 0} A352780(n, 2k+1).
Also defined by:
- for n in A046100, a(n) = A008833(n);
- a(n^4) = (a(n))^4;
- a(A059896(n,k)) = A059896(a(n), a(k)).
Other identities:
a(n) = sqrt(A366244(n^2)).
a(A059897(n,k)) = A059897(a(n), a(k)).
a(A225546(n)) = A225546(A248101(n)).
(End)

A214682 Remove 2's that do not contribute to a factor of 4 from the prime factorization of n.

Original entry on oeis.org

1, 1, 3, 4, 5, 3, 7, 4, 9, 5, 11, 12, 13, 7, 15, 16, 17, 9, 19, 20, 21, 11, 23, 12, 25, 13, 27, 28, 29, 15, 31, 16, 33, 17, 35, 36, 37, 19, 39, 20, 41, 21, 43, 44, 45, 23, 47, 48, 49, 25, 51, 52, 53, 27, 55, 28, 57, 29

Offset: 1

Tyler Ball, Jul 25 2012

In this sequence, the number 4 exhibits some characteristics of a prime number since all extraneous 2's have been removed from the prime factorizations of all other numbers.

			For n=8, v_4(8)=1, v_2(8)=3, so a(8)=(8*4^1)/(2^3)=4.
For n=12, v_4(12)=1, v_2(12)=2, so a(12)=(12*4^1)/(2^2)=12.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A214681, A214685.
Range of values: A003159.
Missing values: A036554.
A056832, A059895, A073675 are used in a formula defining this sequence.
A059897 is used to express relationship between terms of this sequence.
Cf. A007814 (v_2(n)), A235127 (v_4(n)).

a[n_] := n/(2^Mod[IntegerExponent[n, 2], 2]); Array[a, 100] (* Amiram Eldar, Dec 09 2020 *)

a(n)=n>>(valuation(n,2)%2) \\ Charles R Greathouse IV, Jul 26 2012

def A214682(n): return n>>1 if (~n&n-1).bit_length()&1 else n # Chai Wah Wu, Jan 09 2023

C = []
for i in [1..n]:
    C.append(i*4^(Integer(i).valuation(4))/2^(Integer(i).valuation(2)))

a(n) = (n*4^(v_4(n)))/(2^(v_2(n))) where v_k(n) is the k-adic valuation of n. That is, v_k(n) is the largest power of k, a, such that k^a divides n.
For n odd, a(n)=n since n has no factors of 2 (or 4).
From Peter Munn, Nov 29 2020: (Start)
a(A003159(n)) = n.
a(A036554(n)) = n/2.
a(n) = n/A056832(n) = n/A059895(n, 2) = min(n, A073675(n)).
a(A059897(n, k)) = A059897(a(n), a(k)). (End)
Multiplicative with a(2^e) = 2^(2*floor(e/2)), and a(p^e) = p^e for odd primes p. - Amiram Eldar, Dec 09 2020
Sum_{k=1..n} a(k) ~ (5/12) * n^2. - Amiram Eldar, Nov 10 2022
Dirichlet g.f.: zeta(s-1)*(2^s+1)/(2^s+2). - Amiram Eldar, Dec 30 2022

A245630 Products of terms of A006094 (products of 2 successive primes).

Original entry on oeis.org

1, 6, 15, 35, 36, 77, 90, 143, 210, 216, 221, 225, 323, 437, 462, 525, 540, 667, 858, 899, 1147, 1155, 1225, 1260, 1296, 1326, 1350, 1517, 1763, 1938, 2021, 2145, 2491, 2622, 2695, 2772, 3127, 3150, 3240, 3315, 3375, 3599, 4002, 4087, 4757, 4845, 5005, 5148, 5183

Offset: 1

Robert Israel, Jul 27 2014

nonn

Multiplicative monoid generated by products of two successive primes.
All positive integers of the form Product_{i>=1} (prime(i)*prime(i+1))^m_i for integers m_i >= 0 (all but finitely many m_i = 0).
The smallest subset A of the positive integers such that
1) 1 is in A
2) if n is in A then so is n * prime(i) * prime(i+1) for all i.
Subsequence of A028260.
If A059897(.,.) is used as multiplicative operator in place of standard integer multiplication, A006094 generates A030229 (products of an even number of distinct primes). - Peter Munn, Oct 04 2019

			1 is in the sequence.
6 = 2*3 is in the sequence.
36 = (2*3)^2 is in the sequence.
90 = (2*3) * (3*5) is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..6742
Paul Erdős, Solution to Advanced Problem 4413, American Mathematical Monthly, 59 (1952) 259-261.

Cf. A006094, A030229, A059897, A245636, A267251.

Subsequence of: A028260, A325698.

N:= 10^6: # to get all terms <= N
PP:= [seq(ithprime(i)*ithprime(i+1),i=1.. numtheory[pi](floor(sqrt(N)))-1)]:
ext:= (x,p) -> seq(x*p^i,i=0..floor(log[p](N/x))):
S:= {1}:
for i from 1 to nops(PP) do S:= map(ext,S,PP[i]) od:
S;

M = 10^6;
T = Table[Prime[n] Prime[n + 1], {n, 1, PrimePi[Sqrt[M]]}];
T2 = Select[Join[T, T^2], # <= M &];
Join[{1}, T2 //. {a___, b_, c___, d_, e___} /; b*d <= M && FreeQ[{a, b, c, d, e}, b*d] :> Sort[{a, b, c, d, e, b*d}]] (* Jean-François Alcover, Apr 12 2019 *)

f(n) = prime(n)*prime(n+1); \\ A006094
mul(x,y) = x*y;
lista(nn) = {my(v = vector(nn, k, f(k)), lim = f(nn+1), ok = 0, nv); while (!ok,  nv = select(x->(xMichel Marcus, May 10 2021

Limit_{n->oo} a(n)/n^2 = Product_{i>=1} (1 - 1/sqrt(prime(i)*prime(i+1)))^2 / (1 - 1/prime(i))^2 = 1/A267251^2 (see Erdős reference).

A334832 Numbers whose squarefree part is congruent to 1 (mod 24).

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 49, 64, 73, 81, 97, 100, 121, 144, 145, 169, 193, 196, 217, 225, 241, 256, 265, 289, 292, 313, 324, 337, 361, 385, 388, 400, 409, 433, 441, 457, 481, 484, 505, 529, 553, 576, 577, 580, 601, 625, 649, 657, 673, 676, 697, 721, 729, 745, 769, 772, 784, 793, 817, 841

Offset: 1

Peter Munn, Jun 15 2020

Closed under multiplication.
The sequence forms a subgroup of the positive integers under the commutative operation A059897(.,.). A059897 has a relevance to squarefree parts that arises from the identity A007913(k*m) = A059897(A007913(k), A007913(m)), where A007913(n) is the squarefree part of n.
The subgroup is one of 8 A059897(.,.) subgroups of the form {k : A007913(k) == 1 (mod m)}. The list seems complete, in anticipation of proof that such sets form subgroups only when m is a divisor of 24 (based on the property described by A. G. Astudillo in A018253). This sequence might be viewed as primitive with respect to the other 7, as the latter correspond to subgroups of its quotient group, in the sense that each one (as a set) is also a union of cosets described below. The 7 include A003159 (m=2), A055047 (m=3), A277549 (m=4), A234000 (m=8) and the trivial A000027 (m=1).
The subgroup has 32 cosets. For each i in {1, 5, 7, 11, 13, 17, 19, 23}, j in {1, 2, 3, 6} there is a coset {n : n = k^2 * (24m + i) * j, k >= 1, m >= 0}. The divisors of 2730 = 2*3*5*7*13 form a transversal. (11, clearly not such a divisor, is in the same coset as 35 = 11 + 24; 17, 19, 23 are in the same cosets as 65, 91, 455 respectively.)
The asymptotic density of this sequence is 1/16. - Amiram Eldar, Mar 08 2021

			The squarefree part of 26 is 26, which is congruent to 2 (mod 24), so 26 is not in the sequence.
The squarefree part of 292 = 2^2 * 73 is 73, which is congruent to 1 (mod 24), so 292 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Quotient Group.
Eric Weisstein's World of Mathematics, Right Transversal.

A subgroup under A059897, defined using A007913.
Subsequences: A000290\{0}, A103214, A107008.
Equivalent sequences modulo other members of A018253: A000027 (1), A003159 (2), A055047 (3), A277549 (4), A352272(6), A234000 (8).

Select[Range[850], Mod[Sqrt[#] /. (c_ : 1)*a_^(b_ : 0) :> (c*a^b)^2, 24] == 1 &] (* Michael De Vlieger, Jun 24 2020 *)

isok(m) = core(m) % 24 == 1; \\ Peter Munn, Jun 21 2020

from sympy import integer_log
def A334832(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): return n+x-sum((x//(i<<(j<<1))-1)//24+1 for i in (9**k for k in range(integer_log(x,9)[0]+1)) for j in range((x//i>>1).bit_length()+1))
    return bisection(f,n,n) # Chai Wah Wu, Mar 21 2025

{a(n)} = {n : n = k^2 * (24m + 1), k >= 1, m >= 0}.

A352273 Numbers whose squarefree part is congruent to 5 modulo 6.

Original entry on oeis.org

5, 11, 17, 20, 23, 29, 35, 41, 44, 45, 47, 53, 59, 65, 68, 71, 77, 80, 83, 89, 92, 95, 99, 101, 107, 113, 116, 119, 125, 131, 137, 140, 143, 149, 153, 155, 161, 164, 167, 173, 176, 179, 180, 185, 188, 191, 197, 203, 207, 209, 212, 215, 221, 227, 233, 236, 239, 245, 251

Offset: 1

Peter Munn, Mar 10 2022

Numbers of the form 4^i * 9^j * (6k+5), i, j, k >= 0.
1/5 of each multiple of 5 in A352272.
The product of any two terms is in A352272.
The product of a term of this sequence and a term of A352272 is a term of this sequence.
The positive integers are usefully partitioned as {A352272, 2*A352272, 3*A352272, 6*A352272, {a(n)}, 2*{a(n)}, 3*{a(n)}, 6*{a(n)}}. There is a table in the example section giving sequences formed from unions of the parts.
The parts correspond to the cosets of A352272 considered as a subgroup of the positive integers under the operation A059897(.,.). Viewed another way, the parts correspond to the intersection of the integers with the cosets of the multiplicative subgroup of the positive rationals generated by the terms of A352272.
The asymptotic density of this sequence is 1/4. - Amiram Eldar, Apr 03 2022

			The squarefree part of 11 is 11, which is congruent to 5 (mod 6), so 11 is in the sequence.
The squarefree part of 15 is 15, which is congruent to 3 (mod 6), so 15 is not in the sequence.
The squarefree part of 20 = 2^2 * 5 is 5, which is congruent to 5 (mod 6), so 20 is in the sequence.
The table below lists OEIS sequences that are unions of the cosets described in the initial comments, and indicates the cosets included in each sequence. A352272 (as a subgroup) is denoted H, and this sequence (as a coset) is denoted H/5, in view of its terms being one fifth of the multiples of 5 in A352272.
             H    2H    3H    6H    H/5  2H/5  3H/5  6H/5
A003159      X           X           X           X
A036554            X           X           X           X
.
A007417      X     X                 X     X
A145204\{0}              X     X                 X     X
.
A026225      X           X                 X           X
A026179\{1}        X           X     X           X
.
A036668      X                 X     X                 X
A325424            X     X                 X     X
.
A055047      X                             X
A055048            X                 X
A055041                  X                             X
A055040                        X                 X
.
A189715      X                 X           X     X
A189716            X     X           X                 X
.
A225837      X     X     X     X
A225838                              X     X     X     X
.
A339690      X                       X
A329575                  X                       X
.
A352274      X           X
(The sequence groupings in the table start with the subgroup of the quotient group of H, followed by its cosets.)

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Squarefree Part.

Intersection of any three of A003159, A007417, A189716 and A225838.
Intersection of A036668 and A055048.
Complement within A339690 of A352272.
Cf. A007913, A059897, A307151, A334832.
Closure of A084088 under multiplication by 9.
Other subsequences: A033429\{0}, A016969.
Other sequences in the example table: A036554, A145204, A026179, A026225, A325424, A055040, A055041, A055047, A189715, A225837, A329575, A352274.

q[n_] := Module[{e2, e3}, {e2, e3} = IntegerExponent[n, {2, 3}]; EvenQ[e2] && EvenQ[e3] && Mod[n/2^e2/3^e3, 6] == 5]; Select[Range[250], q] (* Amiram Eldar, Apr 03 2022 *)

PARI
```
isok(m) = core(m) % 6 == 5;
```

from itertools import count
def A352273(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-5)//6+1
        return c
    return bisection(f,n,n) # Chai Wah Wu, Feb 14 2025

{a(n) : n >= 1} = {m >= 1 : A007913(m) == 5 (mod 6)}.
{a(n) : n >= 1} = A334832/5 U A334832/11 U A334832/17 U A334832/23 where A334832/k denotes {A334832(m)/k : m >= 1, k divides A334832(m)}.
Using the same notation, {a(n) : n >= 1} = A352272/5 = {A307151(A352272(m)) : m >= 1}.
{A225838(n) : n >= 1} = {m : m = a(j)*k, j >= 1, k divides 6}.

A268386 a(n) = A193231(A268387(n)).

Original entry on oeis.org

0, 1, 1, 3, 1, 0, 1, 2, 3, 0, 1, 2, 1, 0, 0, 5, 1, 2, 1, 2, 0, 0, 1, 3, 3, 0, 2, 2, 1, 1, 1, 4, 0, 0, 0, 0, 1, 0, 0, 3, 1, 1, 1, 2, 2, 0, 1, 4, 3, 2, 0, 2, 1, 3, 0, 3, 0, 0, 1, 3, 1, 0, 2, 6, 0, 1, 1, 2, 0, 1, 1, 1, 1, 0, 2, 2, 0, 1, 1, 4, 5, 0, 1, 3, 0, 0, 0, 3, 1, 3, 0, 2, 0, 0, 0, 5, 1, 2, 2, 0, 1, 1, 1, 3, 1, 0, 1, 1, 1, 1, 0, 4, 1, 1, 0, 2, 2, 0, 0, 2

Offset: 1

Antti Karttunen, Feb 10 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

A003987, A048720, A059897, A193231, A268385, A268387 are used in definitions of this sequence.

Cf. A000028 (indices of odd numbers), A000379 (indices of even numbers), A268390 (indices of zeros).

f[n_] := Which[0 <= # <= 1, #, EvenQ@ #, BitXor[2 #, #] &[f[#/2]], True, BitXor[#, 2 # + 1] &[f[(# - 1)/2]]] &@ Abs@ n; {0}~Join~Table[f[BitXor @@ Map[Last, FactorInteger@ n]], {n, 2, 120}] (* Michael De Vlieger, Feb 12 2016, after Robert G. Wilson v at A048724 and A065621 *)

a268387(n) = {my(f = factor(n), b = 0); for (k=1, #f~, b = bitxor(b, f[k, 2]); ); b; }
a193231(n) = {my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2)};
a(n) = a193231(a268387(n)); \\ Michel Marcus, May 09 2020

(define (A268386 n) (A193231 (A268387 n)))

The following two formulas are equivalent because A193231 distributes over bitwise XOR (A003987):
a(n) = A193231(A268387(n)) and
a(n) = A268387(A268385(n)).
a(2^k) = A193231(k). - Peter Munn, May 07 2020
From Peter Munn, Jun 02 2020: (Start)
Alternative definition, for n, k >= 1, where XOR denotes A003987:
a(prime(n)) = 1, where prime(n) = A000040(n);
a(n^2) = a(n) XOR (2 * a(n)) = A048720(a(n), 3);
a(A059897(n, k)) = a(n) XOR a(k).
(End)

A363340 a(n) is the smallest positive integer such that a(n) * n is the sum of two squares.

Original entry on oeis.org

1, 1, 3, 1, 1, 3, 7, 1, 1, 1, 11, 3, 1, 7, 3, 1, 1, 1, 19, 1, 21, 11, 23, 3, 1, 1, 3, 7, 1, 3, 31, 1, 33, 1, 7, 1, 1, 19, 3, 1, 1, 21, 43, 11, 1, 23, 47, 3, 1, 1, 3, 1, 1, 3, 11, 7, 57, 1, 59, 3, 1, 31, 7, 1, 1, 33, 67, 1, 69, 7, 71, 1, 1, 1, 3, 19, 77, 3, 79

Offset: 1

Peter Schorn, May 28 2023

Using Fermat's two-squares theorem it is easy to see that a(n) is the product of all prime factors of n that are congruent to 3 modulo 4 and have an odd exponent.
This implies that a(n) is also the smallest positive integer such that n / a(n) is the sum of two squares.
Equivalently, a(n) is the product of all primes of the form 4k+3 that divide the squarefree part of n. If we use the squarefree kernel instead, we get A170819. - Peter Munn, Aug 06 2023

			a(1) = a(2) = 1 since 1 and 2 are sums of two squares.
a(3) = 3 since 3 and 6 are not sums of two squares but 3*3 is.
a(6) = 3 since 6 and 12 are not sums of two squares but 3*6 = 3^2 + 3^2.

Cf. A001481 (positions of 1's), A167181 (range of values).
Fixed points: A167181.
Cf. A000265, A002145, A007913, A059897, A097706, A170819.

a(n) = my(r=1); foreach(mattranspose(factor(n)), f, if(f[1]%4==3&&f[2]%2==1, r*=f[1])); r

Multiplicative with a(p^e) = p if p^e == 3 (mod 4), otherwise 1. - Peter Munn, Jul 03 2023
From Peter Munn, Aug 06 2023: (Start)
a(n) = A007913(A097706(n)) = A097706(A007913(n)).
a(n) == A000265(n) (mod 4).
a(A059897(n, k)) = A059897(a(n), a(k)).
(End)

A002557 Odd squarefree numbers with an even number of prime factors that have no prime factors greater than 31.

Original entry on oeis.org

1, 15, 21, 33, 35, 39, 51, 55, 57, 65, 69, 77, 85, 87, 91, 93, 95, 115, 119, 133, 143, 145, 155, 161, 187, 203, 209, 217, 221, 247, 253, 299, 319, 323, 341, 377, 391, 403, 437, 493, 527, 551, 589, 667, 713, 899, 1155, 1365, 1785, 1995, 2145, 2415, 2805, 3003

Offset: 1

N. J. A. Sloane

Original name: A subset of A056913, definition unclear.
The definition is given on page 70 of Gupta (1943), but is hard to understand.
A variant of A056913, which has terms that also have prime factors > 31. - Arkadiusz Wesolowski, Jan 21 2016
The b-file contains the full sequence. - Robert Israel, Jan 21 2016
The sequence is closed under the commutative binary operation A059897(.,.). As integers are self-inverse under A059897, it forms a subgroup of the positive integers considered as a group under A059897. A subgroup of A056913. - Peter Munn, Jan 16 2020

H. Gupta, A formula for L(n), J. Indian Math. Soc., 7 (1943), 68-71.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert Israel, Table of n, a(n) for n = 1..512
H. Gupta, A formula for L(n), J. Indian Math. Soc., 7 (1943), 68-71. [Annotated scanned copy]

Cf. A002556, A046337, A059897. Subset of A056913.

a:= func< n | Factorization(n)>; [1] cat [n: n in [3..3003 by 2] | IsSquarefree(n) and (-1)^&+[p[2]: p in a(n)] eq 1 and f[#f][1] le 31 where f is a(n)]; // Arkadiusz Wesolowski, Jan 21 2016

S:= select(t -> (nops(t)::even), combinat:-powerset(select(isprime, [seq(i,i=3..31,2)]))):
sort(map(convert,S,`*`)); # Robert Israel, Jan 21 2016

npfQ[n_]:=With[{pf=FactorInteger[n][[;;,1]]},SquareFreeQ[n]&&EvenQ[PrimeOmega[n]]&&Max[pf]<32]; Select[Range[1,3003,2],npfQ] (* Harvey P. Dale, May 03 2025 *)

powerset = lambda lst: reduce(lambda result, x: result + [subset + [x] for subset in result], lst, [[]])
product = lambda lst: reduce(lambda x, y: x*y, lst, 1)
primes = [3, 5, 7, 11, 13, 17, 19, 23, 29, 31]
sequence = sorted(product(s) for s in powerset(primes) if len(s) % 2 == 0) # David Radcliffe, Jan 21 2016

Name changed and sequence extended by Arkadiusz Wesolowski, Jan 21 2016

A066616 a(1) = 1; a(n) = n*a(n-1) if n does not divide a(n-1), otherwise a(n) = a(n-1).

Original entry on oeis.org

1, 2, 6, 24, 120, 120, 840, 840, 7560, 7560, 83160, 83160, 1081080, 1081080, 1081080, 17297280, 294053760, 294053760, 5587021440, 5587021440, 5587021440, 5587021440, 128501493120, 128501493120, 3212537328000, 3212537328000

Offset: 1

Amarnath Murthy, Dec 24 2001

nonn

			a(5) = 120; as 6 divides a(5), we have a(6) = a(5) = 120. Though 9 is not coprime to a(8) but still 9 does not divide a(8) so a(9) = 9 * a(8).

Harry J. Smith, Table of n, a(n) for n = 1..200

Cf. A003418, A037992 (duplicates removed).

Replacing A059896 with A059897 in the formula gives A284567.

nxt[{n_,a_}]:={n+1,If[Mod[a,n+1]==0,a,a(n+1)]}; NestList[nxt,{1,1},30][[All,2]] (* Harvey P. Dale, Jul 01 2022 *)

{ for (n=1, 200, if (n==1, a=1, if (a%n, a=n*a)); write("b066616.txt", n, " ", a) ) } \\ Harry J. Smith, Mar 12 2010

a(1) = 1; for n > 1, a(n) = A059896(a(n-1), n). - Peter Munn, Jul 12 2022

More terms from Vladeta Jovovic, Dec 26 2001

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