A285100 -id:A285100 - cp's OEIS frontend

A285109 a(n) = n multiplied by its smallest prime factor; a(1) = 1.

1, 4, 9, 8, 25, 12, 49, 16, 27, 20, 121, 24, 169, 28, 45, 32, 289, 36, 361, 40, 63, 44, 529, 48, 125, 52, 81, 56, 841, 60, 961, 64, 99, 68, 175, 72, 1369, 76, 117, 80, 1681, 84, 1849, 88, 135, 92, 2209, 96, 343, 100, 153, 104, 2809, 108, 275, 112, 171, 116, 3481, 120, 3721, 124, 189, 128, 325, 132, 4489, 136, 207

Offset: 1

Antti Karttunen, Apr 19 2017

nonn

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

Cf. A005117, A020639, A032742, A253560, A285100.

Differs from A065642 for the first time at n=12. See A284342 for all the differing points.

a[n_] := n * FactorInteger[n][[1, 1]]; Array[a, 100] (* Amiram Eldar, Jun 30 2022 *)

a(n)=if(n==1, 1, n*factor(n)[1,1]); \\ Joerg Arndt, Oct 27 2021

Scheme
```
(define (A285109 n) (* (A020639 n) n))
```

a(n) = A020639(n) * n.
Other identities. For all n >= 1:
a(A285100(n)) = A065642(A285100(n)). [Agrees with A065642 on all terms of A285100, but not on any other points.]

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

A285337 a(n) = denominator of A065642(n)/n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 1, 3, 1, 1, 8, 1, 5, 1, 1, 1, 3, 1, 4, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 5, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 8, 1, 7, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4

Offset: 1

Antti Karttunen, Apr 19 2017

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

Cf. A065642, A285100 (positions of ones), A284342 (positions of terms > 1).

Cf. A285336 for the numerator.

from sympy import primefactors, prod, Integer
def a007947(n): return 1 if n<2 else prod(primefactors(n))
def a065642(n):
    if n==1: return 1
    r=a007947(n)
    n += r
    while a007947(n)!=r:
        n+=r
    return n
def a(n): return (a065642(n)/Integer(n)).denominator # Indranil Ghosh, Apr 20 2017

(define (A285337 n) (denominator (/ (A065642 n) n)))

cp's OEIS Frontend

A285109 a(n) = n multiplied by its smallest prime factor; a(1) = 1.

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

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A285337 a(n) = denominator of A065642(n)/n.

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