A285109 -id:A285109 - cp's OEIS frontend

A285100 Numbers k for which A065642(k) = A285109(k).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 97, 99, 101

Offset: 1

Antti Karttunen, Apr 19 2017

nonn

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

Complement: A284342.
Positions of ones in A285337.
Subsequences: A000961, A005117.
Cf. A065642, A285109.

A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
A007947(n) = factorback(factorint(n)[, 1]); \\ From Andrew Lelechenko, May 09 2014
A065642(n) = { my(r=A007947(n)); if(1==n,n,n = n+r; while(A007947(n) <> r, n = n+r); n); };
isA285100(n) = (A065642(n) == n*A020639(n));
n=0; k=1; while(k <= 10000, n=n+1; if(isA285100(n),write("b285100.txt", k, " ", n);k=k+1));

from operator import mul
from sympy import primefactors
from functools import reduce
def a007947(n): return 1 if n<2 else reduce(mul, primefactors(n))
def a020639(n): return 1 if n==1 else primefactors(n)[0]
def a065642(n):
    if n==1: return 1
    r=a007947(n)
    n += r
    while a007947(n)!=r:
        n+=r
    return n
print([n for n in range(1, 102) if a065642(n) == n*a020639(n)]) # Indranil Ghosh, May 24 2017

;; With Antti Karttunen's IntSeq-library.
(define A285100 (MATCHING-POS 1 1 (lambda (n) (= (A065642 n) (A285109 n)))))

A284342 Numbers n such that A065642(n) < n*lpf(n), where lpf = least prime factor (A020639).

Original entry on oeis.org

12, 18, 24, 36, 40, 45, 48, 50, 54, 56, 60, 63, 72, 75, 80, 84, 90, 96, 98, 100, 108, 112, 120, 126, 132, 135, 144, 147, 150, 156, 160, 162, 168, 175, 176, 180, 189, 192, 196, 198, 200, 204, 208, 216, 224, 225, 228, 234, 240, 242, 245, 250, 252, 264, 270, 275, 276, 280, 288, 294, 297, 300

Offset: 1

Gionata Neri, Mar 25 2017

nonn

Numbers n for which A065642(n) < A285109(n). Positions of terms > 1 in A285337. - Antti Karttunen, Apr 19 2017

For any n in this sequence, k*n is also in this sequence. No term is squarefree. For any distinct primes p and q with p > q, p^2*q and p*q^(ceiling(log_q(p))) are in this sequence. - Charlie Neder, Oct 29 2018

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

Cf. A007947, A020639, A065642, A285100 (complement), A285109, A285337.

Select[Range[2, 300], Function[{n, c, lpf}, SelectFirst[Range[n + 1, n^2], Times @@ FactorInteger[#][[All, 1]] == c &] < n lpf] @@ {#1, Times @@ #2, #2[[1]]} & @@ {#, FactorInteger[#][[All, 1]]} &] (* Michael De Vlieger, Oct 31 2018 *)

for(n=1,300,for(k=1,n^2-n,a=factorback(factorint(n)[,1]); b=factorback(factorint(n+k)[,1]); c=vecmin(factor(n)[,1]); if(a==b&&n+k

A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
A007947(n) = factorback(factorint(n)[, 1]); \\ From Andrew Lelechenko, May 09 2014
A065642(n) = { my(r=A007947(n)); if(1==n,n,n = n+r; while(A007947(n) <> r, n = n+r); n); };
isA284342(n) = (A065642(n) < n*A020639(n));
n=0; k=1; while(k <= 10000, n=n+1; if(isA284342(n),write("b284342.txt", k, " ", n);k=k+1));
\\ Antti Karttunen, Apr 19 2017

from operator import mul
from sympy import primefactors
from functools import reduce
def a007947(n): return 1 if n<2 else reduce(mul, primefactors(n))
def a065642(n):
    if n==1: return 1
    r=a007947(n)
    n = n + r
    while a007947(n)!=r:
        n+=r
    return n
print([n for n in range(10, 301) if a065642(n)Indranil Ghosh, Apr 20 2017

A285326 a(0) = 0, for n > 0, a(n) = n + A006519(n).

Original entry on oeis.org

0, 2, 4, 4, 8, 6, 8, 8, 16, 10, 12, 12, 16, 14, 16, 16, 32, 18, 20, 20, 24, 22, 24, 24, 32, 26, 28, 28, 32, 30, 32, 32, 64, 34, 36, 36, 40, 38, 40, 40, 48, 42, 44, 44, 48, 46, 48, 48, 64, 50, 52, 52, 56, 54, 56, 56, 64, 58, 60, 60, 64, 62, 64, 64, 128, 66, 68, 68, 72, 70, 72, 72, 80, 74, 76, 76, 80, 78, 80, 80, 96, 82, 84, 84, 88, 86, 88, 88, 96, 90, 92, 92

Offset: 0

Antti Karttunen, Apr 19 2017

From M. F. Hasler, Oct 19 2019: (Start)
This sequence is equal to itself multiplied by 2 and interleaved with the positive even numbers: We have a(2n-1) = 2n (n >= 1) from the very definition, since A006519(m) = 1 for odd m. And a(2n) = 2n + A006519(2n) = 2*a(n), using A006519(2n) = 2*A006519(n).
The sequence repeats the pattern [A, B, C, C] where in the n-th occurrence C = 4n, B = C - 2, A = C if n is even, A = C + 4 if n = 3 (mod 4), and A = 16*a((n-1)/4) otherwise. (End)

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to binary expansion of n

Row 2 of A285325 (after the initial zero).
Cf. A006519, A019565, A048675, A065642, A129760, A285109.
Cf. A109168 (same terms divided by 2), also A140472.

Table[If[n>0, n + GCD[2^n, n], 0], {n, 0, 100}] (* Indranil Ghosh, Apr 20 2017 *)

a(n) = if(n>0, n + gcd(2^n, n), 0); \\ Indranil Ghosh, Apr 20 2017

A285326(n)=n+bitand(n,-n) \\ Or: {a(n)=-bitand(-n,bitneg(n))}, not faster. - M. F. Hasler, Oct 19 2019

from sympy import gcd
def a(n): return n + gcd(2**n, n) if n>0 else 0 # Indranil Ghosh, Apr 20 2017

(define (A285326 n) (if (zero? n) n (+ n (A006519 n))))
(define (A285326 n) (A048675 (A065642 (A019565 n))))

a(0) = 0; for n > 0, a(n) = n + A006519(n).
a(n) = A048675(A065642(A019565(n))) = A048675(A285109(A019565(n))).
For n >= 1, a(n) = 2*A109168(n).
a(n) = 2*A140472(n) and a(2n) = 2*a(n) and a(2^n) = 2^(n+1) for all n >= 0, a(2n-1) = 2n for all n >= 1. - M. F. Hasler, Oct 19 2019

A366807 a(n) = A020639(A120944(n))*A120944(n).

Original entry on oeis.org

12, 20, 28, 45, 63, 44, 52, 60, 99, 68, 175, 76, 117, 84, 92, 153, 275, 171, 116, 124, 325, 132, 207, 140, 148, 539, 156, 164, 425, 172, 261, 637, 279, 188, 475, 204, 315, 212, 220, 333, 228, 575, 236, 833, 244, 369, 387, 260, 931, 268, 276, 423, 284, 1573, 725

Offset: 1

Michael De Vlieger, Dec 16 2023

Define f(x) to be lpf(k)*k, where lpf(k) = A020639(k). This sequence contains the mappings of f(x) across composite squarefree numbers A120944.
Define sequence R_k = { m : rad(m) | k }, where rad(n) = A007947(n) and k is squarefree. Then the sequence k*R_k contains all numbers divisible by squarefree k that are also not divisible by any prime q coprime to k. It is plain to see that k is the first term in the sequence k*R_k. This sequence gives the second term in k*R_k since lpf(k) is the second term in R_k.
Permutation of A366825. Contains numbers whose prime signature has at least 2 terms, of which is 2, the rest of which are 1s.
Proper subset of A364996, which itself is contained in A126706.

			Let b(n) = A120944(n).
a(1) = 12 = 2^2*3^1 = b(1)*lpf(b(1)) = 6*lpf(6) = 6*2. In {6*A003586}, 12 is the second term.
a(2) = 20 = 2^2*5^1 = b(2)*lpf(b(2)) = 10*lpf(10) = 10*2. In {10*A003592}, 20 is the second term.
a(4) = 45 = 3^2*5^1 = b(4)*lpf(b(4)) = 15*lpf(15) = 15*3. In {15*A003593}, 45 is the second term, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000040, A007947, A020639, A065642, A120944, A126706, A285109, A364996, A366825.

nn = 150; s = Select[Range[nn], And[SquareFreeQ[#], CompositeQ[#]] &];
Array[#*FactorInteger[#][[1, 1]] &[s[[#]]] &, Length[s]]

from math import isqrt
from sympy import primepi, mobius, primefactors
def A366807(n):
    def f(x): return n+1+primepi(x)+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n+1, f(n+1)
    while m != k:
        m, k = k, f(k)
    return m*min(primefactors(m)) # Chai Wah Wu, Aug 02 2024

a(n) = A065642(A120944(n)), n > 1.

a(n) = A285109(A120944(n)).

A366786 a(n) = A073481(n)*A005117(n).

Original entry on oeis.org

1, 4, 9, 25, 12, 49, 20, 121, 169, 28, 45, 289, 361, 63, 44, 529, 52, 841, 60, 961, 99, 68, 175, 1369, 76, 117, 1681, 84, 1849, 92, 2209, 153, 2809, 275, 171, 116, 3481, 3721, 124, 325, 132, 4489, 207, 140, 5041, 5329, 148, 539, 156, 6241, 164, 6889, 425, 172

Offset: 1

Michael De Vlieger, Dec 16 2023

Define f(x) to be lpf(k)*k, where lpf(k) = A020639(k). This sequence contains the mappings of f(x) across squarefree numbers A005117.
a(1) = 1 by definition. 1 is the empty product and has no least prime factor.
Define sequence R_k = { m : rad(m) | k }, where rad(n) = A007947(n) and k > 1 is squarefree. Then the sequence k*R_k contains all numbers divisible by squarefree k that are also not divisible by any prime q coprime to k.
Plainly, k is the first term in the sequence k*R_k, because 1 is the first term in R_k. Hence a(n) is the second term in k*R_k for n > 1, since lpf(k) is the second term in R_k.

			Let b(n) = A005117(n).
a(2) = 4 = b(2)*lpf(b(2)) = 2*lpf(2) = 2*2. In {2*A000079}, 4 is the second term.
a(5) = 12 = b(5)*lpf(b(5)) = 6*lpf(6) = 6*2. In {6*A003586}, 12 is the second term..
a(11) = 45 = b(11)*lpf(b(11)) = 15*lpf(15) = 15*3. In {15*A003593}, 45 is the second term, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000040, A001248, A005117, A020639, A065642, A073481, A120944, A285109, A364996, A366807, A366825.

nn = 120; s = Select[Range[nn], SquareFreeQ];
Array[#*FactorInteger[#][[1, 1]] &[s[[#]]] &, Length[s]]

apply(x->(if (x==1,1, x*vecmin(factor(x)[,1]))), select(issquarefree, [1..150])) \\ Michel Marcus, Dec 17 2023

from math import isqrt
from sympy import mobius, primefactors
def A366786(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return (m:=bisection(f))*min(primefactors(m),default=1) # Chai Wah Wu, Aug 31 2024

a(n) = A065642(A005117(n)), n > 1.
a(n) = A285109(A005117(n)).
a(n) = A020639(A005117(n))*A005117(n).
For prime p, a(p) = p^2.
For composite squarefree k, a(k) = (p^2 * m) such that (p^2 * m) is in A364996.
Permutation of the union of {1}, A001248, and A366825.

A340896 Irregular triangle in which the n-th row consists of all multiples of n that have fewer than twice as many divisors as n.

Original entry on oeis.org

1, 2, 4, 3, 9, 4, 8, 16, 5, 25, 6, 12, 18, 7, 49, 8, 16, 32, 64, 9, 27, 81, 10, 20, 50, 11, 121, 12, 24, 36, 48, 13, 169, 14, 28, 98, 15, 45, 75, 16, 32, 64, 128, 256, 17, 289, 18, 36, 54, 162, 19, 361, 20, 40, 80, 100, 21, 63, 147, 22, 44, 242, 23, 529

Offset: 1

J. Lowell, Jan 25 2021

If n has d divisors, n has an infinite number of multiples with exactly 2d divisors, but only a finite number of multiples with fewer than 2d divisors.

Conjecture: row n includes n^2 if and only if n is a power of a prime number (A000961).

			Triangle begins:
   1;
   2,   4;
   3,   9;
   4,   8, 16;
   5,  25;
   6,  12, 18;
   7,  49;
   8,  16, 32,  64;
   9,  27, 81;
  10,  20, 50;
  11, 121;
  12,  24, 36,  48;
  13, 169;
  14,  28, 98;
  15,  45, 75;
  16,  32, 64, 128, 256;
  ...

Columns k=1..2 give: A000027, A285109 (for n>=2).
Last elements of rows give A225004.
Cf. A000005, A000961.

row(n) = select(x->((numdiv(x)<2*numdiv(n)) && !(x % n)), [1..n^2]); \\ Michel Marcus, Jan 26 2021

cp's OEIS Frontend

A285100 Numbers k for which A065642(k) = A285109(k).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Python

Scheme

A284342 Numbers n such that A065642(n) < n*lpf(n), where lpf = least prime factor (A020639).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

A285326 a(0) = 0, for n > 0, a(n) = n + A006519(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Scheme

Formula

A366807 a(n) = A020639(A120944(n))*A120944(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

A366786 a(n) = A073481(n)*A005117(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A340896 Irregular triangle in which the n-th row consists of all multiples of n that have fewer than twice as many divisors as n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI