A225546 -id:A225546 - cp's OEIS frontend

A352780 Square array A(n,k), n >= 1, k >= 0, read by descending antidiagonals, such that the row product is n and column k contains only (2^k)-th powers of squarefree numbers.

1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 4, 5, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 14

Offset: 1

Antti Karttunen and Peter Munn, Apr 02 2022

This is well-defined because positive integers have a unique factorization into powers of nonunit squarefree numbers with distinct exponents that are powers of 2.
Each (infinite) row is the lexicographically earliest with product n and terms that are a (2^k)-th power for all k.
For all k, column k is column k+1 of A060176 conjugated by A225546.

			The top left corner of the array:
  n/k |   0   1   2   3   4   5   6
------+------------------------------
    1 |   1,  1,  1,  1,  1,  1,  1,
    2 |   2,  1,  1,  1,  1,  1,  1,
    3 |   3,  1,  1,  1,  1,  1,  1,
    4 |   1,  4,  1,  1,  1,  1,  1,
    5 |   5,  1,  1,  1,  1,  1,  1,
    6 |   6,  1,  1,  1,  1,  1,  1,
    7 |   7,  1,  1,  1,  1,  1,  1,
    8 |   2,  4,  1,  1,  1,  1,  1,
    9 |   1,  9,  1,  1,  1,  1,  1,
   10 |  10,  1,  1,  1,  1,  1,  1,
   11 |  11,  1,  1,  1,  1,  1,  1,
   12 |   3,  4,  1,  1,  1,  1,  1,
   13 |  13,  1,  1,  1,  1,  1,  1,
   14 |  14,  1,  1,  1,  1,  1,  1,
   15 |  15,  1,  1,  1,  1,  1,  1,
   16 |   1,  1, 16,  1,  1,  1,  1,
   17 |  17,  1,  1,  1,  1,  1,  1,
   18 |   2,  9,  1,  1,  1,  1,  1,
   19 |  19,  1,  1,  1,  1,  1,  1,
   20 |   5,  4,  1,  1,  1,  1,  1,

Antti Karttunen, Table of n, a(n) for n = 1..33153; the first 257 antidiagonals

Sequences used in a formula defining this sequence: A000188, A007913, A060176, A225546.
Cf. A007913 (column 0), A335324 (column 1).
Range of values: {1} U A340682 (whole table), A005117 (column 0), A062503 (column 1), {1} U A113849 (column 2).
Row numbers of rows:
- with a 1 in column 0: A000290\{0};
- with a 1 in column 1: A252895;
- with a 1 in column 0, but not in column 1: A030140;
- where every 1 is followed by another 1: A337533;
- with 1's in all even columns: A366243;
- with 1's in all odd columns: A366242;
- where every term has an even number of distinct prime factors: A268390;
- where every term is a power of a prime: A268375;
- where the terms are pairwise coprime: A138302;
- where the last nonunit term is coprime to the earlier terms: A369938;
- where the last nonunit term is a power of 2: A335738.
Number of nonunit terms in row n is A331591(n); their positions are given (in reversed binary) by A267116(n); the first nonunit is in column A352080(n)-1 and the infinite run of 1's starts in column A299090(n).

up_to = 105;
A352780sq(n, k) = if(k==0, core(n), A352780sq(core(n, 1)[2], k-1)^2);
A352780list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, forstep(col=a-1,0,-1, i++; if(i > up_to, return(v)); v[i] = A352780sq(a-col,col))); (v); };
v352780 = A352780list(up_to);
A352780(n) = v352780[n];

A(n,0) = A007913(n); for k > 0, A(n,k) = A(A000188(n), k-1)^2.
A(n,k) = A225546(A060176(A225546(n), k+1)).
A331591(A(n,k)) <= 1.

A030140 The nonsquares squared.

Original entry on oeis.org

4, 9, 25, 36, 49, 64, 100, 121, 144, 169, 196, 225, 289, 324, 361, 400, 441, 484, 529, 576, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2500, 2601, 2704, 2809, 2916, 3025

Offset: 1

N. J. A. Sloane

The complement of the fourth powers A000583 within the squares A000290. - Peter Munn, Aug 20 2019

			a(1)=2^2, a(2)=3^2, a(3)=5^2, a(4)=6^2, a(5)=7^2, ..., a(n)=(integer which is not a perfect square)^2.

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

Cf. A000037, A000290, A000583, A013661, A013662, A062503.
Positions of 2's in A352080.
Related to A016945 via A225546.

[(n + Floor(1/2 + Sqrt(n)))^2: n in [1..60]]; // Vincenzo Librandi, Apr 06 2020

a:=proc(n) if type(sqrt(n),integer)=false then n^2 else fi end: seq(a(n),n=1..70); # Emeric Deutsch, Apr 11 2007

a[n_] := (n + Floor[1/2 + Sqrt[n]])^2;
Array[a, 50] (* Jean-François Alcover, Apr 05 2020 *)

from math import isqrt
def A030140(n): return (n+(k:=isqrt(n))+int(n>=k*(k+1)+1))**2 # Chai Wah Wu, Jun 17 2024

a(n) = A000037(n)^2.
Sum_{n>=1} 1/a(n) = zeta(2) - zeta(4) = A013661 - A013662 = 0.5626108331... - Amiram Eldar, Nov 14 2020
{a(n) : n >= 1} = {A225546(6m+3) : m >= 0}. - Peter Munn, Nov 17 2022

Edited by N. J. A. Sloane, Jul 02 2008 at the suggestion of R. J. Mathar

A334110 The squares of squarefree numbers (A062503), ordered lexicographically according to their prime factors. a(n) = Product_{k in I} prime(k+1)^2, where I are the indices of nonzero binary digits in n = Sum_{k in I} 2^k.

Original entry on oeis.org

1, 4, 9, 36, 25, 100, 225, 900, 49, 196, 441, 1764, 1225, 4900, 11025, 44100, 121, 484, 1089, 4356, 3025, 12100, 27225, 108900, 5929, 23716, 53361, 213444, 148225, 592900, 1334025, 5336100, 169, 676, 1521, 6084, 4225, 16900, 38025, 152100, 8281, 33124, 74529, 298116, 207025, 828100, 1863225, 7452900, 20449, 81796, 184041

Offset: 0

Antti Karttunen and Peter Munn, May 01 2020

nonn

For the lexicographic ordering, the prime factors must be written in nonincreasing order. If we write the factors in nondecreasing order, we get a lexicographically ordered set with an order type that is greater than a natural number index - the resulting sequence does not include all qualifying numbers. (Note also that the symbols used for the lexicographic order are the prime numbers, not their digits.)
a(n) is the n-th power of 4 in the monoid defined in A331590.
Conjecture: a(n) is the position of the first occurrence of n in A334109.

			The initial terms are shown below, equated with the product of their prime factors to exhibit the lexicographic ordering. The list starts with 1, since 1 is factored as the empty product and the empty list is first in lexicographic order.
    1 = .
    4 = 2*2.
    9 = 3*3.
   36 = 3*3*2*2.
   25 = 5*5.
  100 = 5*5*2*2.
  225 = 5*5*3*3.
  900 = 5*5*3*3*2*2.
   49 = 7*7.
  196 = 7*7*2*2.
  441 = 7*7*3*3.

Cf. A000079, A019565 (square roots), A048675, A097248, A225546, A267116, A332382, A334109 (a left inverse).
Column 2 of A329332. Permutation of A062503.
After 1, the right children of the leftmost edge of A334860, or respectively, the left children of the rightmost edge of A334866.
Subsequences: A001248, A061742, A166329.
Subsequence of A052330.
A003961, A003987, A059897, A331590 are used to express relationship between terms of this sequence.

Array[If[# == 0, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[3^#]]] &, 51, 0] (* Michael De Vlieger, May 26 2020 *)

A334110(n) = { my(p=2,m=1); while(n, if(n%2, m *= p^2); n >>= 1; p = nextprime(1+p)); (m); };

a(n) = A019565(n)^2.
For n >= 1, a(A000079(n-1)) = A001248(n).
For all n >= 0, A334109(a(n)) = n.
a(n+k) = A331590(a(n), a(k)).
a(n XOR k) = A059897(a(n), a(k)), where XOR denotes bitwise exclusive-or, A003987.
a(n) = A225546(3^n).
a(2n) = A003961(a(n)).
a(2n+1) = 4 * a(2n).
a(2^k-1) = A061742(k).
A267116(a(n)) = 2.
A048675(a(n)) = 2n.
A097248(a(n)) = A332382(n) = A019565(2n).

A051144 Nonsquarefree nonsquares: each term has a square factor but is not a perfect square itself.

Original entry on oeis.org

8, 12, 18, 20, 24, 27, 28, 32, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 125, 126, 128, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180

Offset: 1

Michael Minic (Rassilon6(AT)aol.com)

At least one exponent in the canonical prime factorization (cf. A124010) of n is odd, and at least one exponent is greater than 1. - Reinhard Zumkeller, Jan 24 2013

Compare this sequence, as a set, with A177712, numbers that have an odd factor, but are not odd. The self-inverse function defined by A225546, maps the members of either one of these sets 1:1 onto the other set. - Peter Munn, Jul 31 2020

			63 is included because 63 = 3^2 * 7.
64 is not included because it is a perfect square (8^2).
65 is not included because it is squarefree (5 * 13).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Square Number.
Eric Weisstein's World of Mathematics, Squarefree.
Wikipedia, Square number.
Wikipedia, Squarefree integer.

Cf. A210490 (complement), intersection of A013929 and A000037.

Related to A177712 via A225546.

a051144 n = a051144_list !! (n-1)
a051144_list = filter ((== 0) . a008966) a000037_list
-- Reinhard Zumkeller, Sep 02 2013, Jan 24 2013

[k:k in [1..200]| not IsSquare(k) and not IsSquarefree(k)]; // Marius A. Burtea, Dec 29 2019

N:= 10000;  # to get all entries up to N
A051144:= remove(numtheory:-issqrfree,{$1..N}) minus {seq(i^2,i=1..floor(sqrt(N)))}:
# Robert Israel, Mar 30 2014

searchMax = 32; Complement[Select[Range[searchMax^2], MoebiusMu[#] == 0 &], Range[searchMax]^2] (* Alonso del Arte, Dec 20 2019 *)

is(n)=!issquare(n) && !issquarefree(n) \\ Charles R Greathouse IV, Sep 18 2015

from math import isqrt
from sympy import mobius
def A051144(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 int(n-1+(y:=isqrt(x))+sum(mobius(k)*(x//k**2) for k in range(1, y+1)))
    return bisection(f,n,n) # Chai Wah Wu, Mar 23 2025

(1 - A008966(a(n)))*(1 - A010052(a(n))) = 1; A008966(a(n)) + A010052(a(n)) = 0. - Reinhard Zumkeller, Jan 24 2013

Sum_{n>=1} 1/a(n)^s = 1 + zeta(s) - zeta(2*s) - zeta(s)/zeta(2*s), for s > 1. - Amiram Eldar, Dec 03 2022

Incorrect comment removed by Charles R Greathouse IV, Mar 19 2010

Offset corrected by Reinhard Zumkeller, Jan 24 2013

A177712 Even numbers that have a nontrivial odd divisor.

Original entry on oeis.org

6, 10, 12, 14, 18, 20, 22, 24, 26, 28, 30, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 130, 132, 134, 136, 138, 140

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 11 2010

Numbers which can be expressed as a sum of a set of positive consecutive even numbers: sum_{i=m..m+k} A005843(i), m>=1, k>=1.
Differs from A054741, which contains 105 for example.
These are the numbers that are not free of odd prime factors, but are not odd. Compare with A051144, nonsquarefree nonsquares. The self-inverse function defined by A225546 maps the members of either set 1:1 onto the other set. - Peter Munn, Jul 31 2020 with edit Feb 14 2022

			6=2+4. 10=4+6. 12=2+4+6. 14=6+8. 18=4+6+8. 20=2+4+6+8. 22=10+12. 24=6+8+10.

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

Cf. A000217, A054741.
Intersection of A057716 and A299174.
Related to A051144 via A225546.

z=200;lst={};Do[c=a;Do[c+=b;If[c<=2*z,AppendTo[lst,c]],{b,a-2,1,-2}], {a,2,z,2}];Union@lst

isA177712(n) = (!(n%2)&&(0<#select(x -> x%2,factor(n)[,1]))); \\ Antti Karttunen, Jul 31 2020

isA177712(n) = (!(n%2)&&bitand(n,n-1)); \\ Antti Karttunen, Jul 31 2020

def A177712(n): return n+(m:=n.bit_length())+(n>=(1<Chai Wah Wu, Jun 30 2024

a(n) = 2 * A057716(n).

Definition moved into a comment by R. J. Mathar, Aug 15 2010

New name from Peter Munn, Jul 31 2020

A217319 Numbers with binary representation ending in 4k+2 or 4k+3 zeros.

Original entry on oeis.org

4, 8, 12, 20, 24, 28, 36, 40, 44, 52, 56, 60, 64, 68, 72, 76, 84, 88, 92, 100, 104, 108, 116, 120, 124, 128, 132, 136, 140, 148, 152, 156, 164, 168, 172, 180, 184, 188, 192, 196, 200, 204, 212, 216, 220, 228, 232, 236, 244, 248, 252, 260, 264, 268, 276, 280

Offset: 1

Vladimir Shevelev, Mar 18 2013

Or numbers having infinitary divisor 4, or the same, having factor 4 in Fermi-Dirac representation as a product of distinct terms of A050376.
From Peter Munn, Aug 25 2020: (Start)
Compare the terms, as a set, with A145204\{0} (numbers having 3 as a Fermi-Dirac factor). The self-inverse function defined by A225546 maps the members of either one of these sets 1:1 onto the other set.
Numbers whose 4th-power-free part is divisible by 4.
(End)
Numbers k such that the exponent of the highest power of 4 dividing k, A235127(k), is odd. The asymptotic density of this sequence is 1/5. - Amiram Eldar, Sep 20 2020

Peter J. C. Moses, Table of n, a(n) for n = 1..10000

Cf. A050376, A171949, A235127.

Related to A145204\{0} via A225546.

isA007814 := proc(n)
    if modp( A007814(n),4) in {2,3} then
        true ;
    else
        false ;
    end if;
end proc:
for n from 1 to 1000 do
    if isA007814(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Nov 22 2023

okQ[n_] := (cnt = Count[ Split[ IntegerDigits[n, 2]] // Last, 0]; k0 = k /. ToRules@ Reduce[ (cnt == 2*k || cnt == 2*k+1), k, Integers]; OddQ[k0]); Select[ Range[312], okQ] (* Jean-François Alcover, Mar 18 2013 *)
Select[Map[# Boole[IntegerQ[(1/4 (1+#))]||IntegerQ[(1/4 (2+#))]&[Length[Last[Split[IntegerDigits[#,2]]]]]]&,Range[2,500,2]],#>0&]
Select[Range[280], OddQ @ IntegerExponent[#, 4] &] (* Amiram Eldar, Sep 20 2020 *)

def A217319(n):
    def f(x): return n+x-sum((k:=x>>(m<<1))-(k>>2) for m in range(0,(x.bit_length()+1>>1),2))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m<<2 # Chai Wah Wu, May 25 2025

Conjecture. For n>=1, a(n) = A171949(n+1).

Named edited by David A. Corneth, Sep 22 2020

A334871 Number of steps needed to reach 1 when starting from n and iterating with A334870.

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 8, 3, 3, 5, 16, 4, 32, 9, 6, 3, 64, 4, 128, 6, 10, 17, 256, 5, 5, 33, 5, 10, 512, 7, 1024, 4, 18, 65, 12, 4, 2048, 129, 34, 7, 4096, 11, 8192, 18, 7, 257, 16384, 5, 9, 6, 66, 34, 32768, 6, 20, 11, 130, 513, 65536, 8, 131072, 1025, 11, 4, 36, 19, 262144, 66, 258, 13, 524288, 5, 1048576, 2049, 7, 130

Offset: 1

Antti Karttunen, Jun 08 2020

nonn

Distance of n from the root (1) in binary trees like A334860 and A334866.
Each n > 0 occurs 2^(n-1) times.
a(n) is the size of the inner lining of the integer partition with Heinz number A225546(n), which is also the size of the largest hook of the same partition. (After Gus Wiseman's Apr 02 2019 comment in A252464).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A007913, A008833, A070939, A225546, A252464, A299090, A334859, A334860, A334865, A334866, A334869, A334870, A334872.

A334870(n) = if(issquare(n),sqrtint(n),my(c=core(n), m=n); forprime(p=2, , if(!(c % p), m/=p; break, m*=p)); (m));
A334871(n) = { my(s=0); while(n>1,s++; n = A334870(n)); (s); };

\\ Much faster:
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A334871(n) = { my(s=0); while(n>1, if(issquare(n), s++; n = sqrtint(n), s += A048675(core(n)); n /= core(n))); (s); };

a(1) = 0; for n > 1, a(n) = 1 + a(A334870(n)).
a(n) = A252464(A225546(n)).
a(n) = A048675(A007913(n)) + a(A008833(n)).
For n > 1, a(n) = 1 + A048675(A007913(n)) + a(A000188(n)).
For n > 1, a(n) = A070939(A334859(n)) = A070939(A334865(n)).
For all n >= 1, a(n) >= A299090(n).
For all n >= 1, a(n) >= A334872(n).

A335324 Square part of 4th-power-free part of n.

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

Peter Munn, May 31 2020

Equivalently, biquadratefree (4th-power-free) part of square part of n.
Multiplicative. The terms are squares of squarefree numbers (A062503).
Every positive integer n is the product of a unique subset S_n of the terms of A050376 (sometimes called Fermi-Dirac primes). a(n) is the product of the members of S_n that are squares of prime numbers (A001248).

			Removing the 4th powers from 192 = 2^6 * 3^1 gives 2^(6 - 4) * 3^1 = 2^2 * 3 = 12. So the 4th-power-free part of 192 is 12. The square part of 12 (largest square dividing 12) is 4. So a(192) = 4.

Michel Marcus, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Biquadratefree.
Eric Weisstein's World of Mathematics, Square part.

A007913, A008833, A008835, A053165 are used in formulas defining the sequence.
Column 1 of A352780.
Range of values is A062503.
Positions of 1's: A252895.
Related to A038500 by A225546.
The formula section details how the sequence maps the terms of A003961, A331590.
Cf. A001248, A050376, A083730.
Cf. A002117, A013662, A078434.

f[p_, e_] := p^(2*Floor[e/2] - 4*Floor[e/4]); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jun 01 2020 *)

A053165(n)=my(f=factor(n)); f[, 2]=f[, 2]%4; factorback(f);
a(n) = my(m=A053165(n)); m/core(m); \\ Michel Marcus, Jun 01 2020

from math import prod
from sympy import factorint
def A335324(n): return prod(p**(e&2) for p, e in factorint(n).items()) # Chai Wah Wu, Aug 07 2024

a(n) = A053165(A008833(n)) = A008833(A053165(n)).
a(n) = A053165(n) / A007913(n).
a(n) = A008833(n) / A008835(n).
n = A007913(n) * a(n) * A008835(n).
a(n) = A225546(A038500(A225546(n))).
a(n^2) = A007913(n)^2.
a(A003961(n)) = A003961(a(n)).
a(A331590(n, k)) = A331590(a(n), a(k)).
a(p^e) = p^(2*floor(e/2) - 4*floor(e/4)). - Amiram Eldar, Jun 01 2020
From Amiram Eldar, Sep 21 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * zeta(4*s)/(zeta(2*s) * zeta(4*s-4)).
Sum_{k=1..n} a(k) ~ (4*zeta(3/2)*zeta(4))/(21*zeta(3)) * n^(3/2). (End)

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

Original entry on oeis.org

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)

cp's OEIS Frontend

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A217319 Numbers with binary representation ending in 4k+2 or 4k+3 zeros.