A073481 -id:A073481 - cp's OEIS frontend

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

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.

A073482 Largest prime factor of the n-th squarefree number.

Original entry on oeis.org

1, 2, 3, 5, 3, 7, 5, 11, 13, 7, 5, 17, 19, 7, 11, 23, 13, 29, 5, 31, 11, 17, 7, 37, 19, 13, 41, 7, 43, 23, 47, 17, 53, 11, 19, 29, 59, 61, 31, 13, 11, 67, 23, 7, 71, 73, 37, 11, 13, 79, 41, 83, 17, 43, 29, 89, 13, 31, 47, 19, 97, 101, 17, 103, 7, 53, 107, 109, 11, 37, 113, 19, 23, 59, 17, 61, 41

Offset: 1

Reinhard Zumkeller, Aug 03 2002

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck and Rafael Jakimczuk, Summing the largest prime factor over integer sequences, Revista de la Unión Matemática Argentina, Vol. 67, No. 1 (2024), pp. 27-35.

Cf. A073481, A001221, A005117, A006530, A265668, A323669.

a073482 = a006530 . a005117  -- Reinhard Zumkeller, Feb 04 2012

issquarefree := proc(n::integer) local nf, ifa, lar; nf := op(2,ifactors(n)); for ifa from 1 to nops(nf) do lar := op(1,op(ifa,nf)); if op(2,op(ifa,nf)) >= 2 then RETURN(0); fi; od : RETURN(lar); end: printf("1,"); for n from 2 to 100 do lfa := issquarefree(n); if lfa > 0 then printf("%a,",lfa); fi; od : # R. J. Mathar, Apr 02 2006

FactorInteger[#][[-1, 1]]& /@ Select[Range[100], SquareFreeQ] (* Jean-François Alcover, Feb 01 2018 *)
s[n_] := Module[{f = FactorInteger[n]}, If[AllTrue[f[[;; , 2]], # < 2 &], f[[-1, 1]], Nothing]]; Array[s, 200] (* Amiram Eldar, Mar 03 2024 *)

do(x)=my(v=List([1])); forfactored(n=2,x\1, if(vecmax(n[2][,2])==1, listput(v, vecmax(n[2][,1])))); Vec(v) \\ Charles R Greathouse IV, Nov 05 2017

from math import isqrt
from sympy import mobius, primefactors
def A073482(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    kmin, kmax = 0,1
    while f(kmax) > kmax:
        kmax <<= 1
    while kmax-kmin > 1:
        kmid = kmax+kmin>>1
        if f(kmid) <= kmid:
            kmax = kmid
        else:
            kmin = kmid
    return max(primefactors(kmax),default=1) # Chai Wah Wu, Aug 28 2024

a(n) = A006530(A005117(n)).
a(n) = A265668(n, A001221(n)). - Reinhard Zumkeller, Dec 13 2015
Sum_{A005117(n) <= x} a(n) = Sum_{i=1..k} d_i * x^2/log(x)^i + O(x^2/log(x)^(k+1)), for any given positive integer k, where d_i are constants, d_1 = 15/(2*Pi^2) = 0.759908... (A323669) (De Koninck and Jakimczuk, 2024). - Amiram Eldar, Mar 03 2024

More terms from Jason Earls, Aug 06 2002

A265668 Table read by rows: prime factors of squarefree numbers; a(1) = 1 by convention.

Original entry on oeis.org

1, 2, 3, 5, 2, 3, 7, 2, 5, 11, 13, 2, 7, 3, 5, 17, 19, 3, 7, 2, 11, 23, 2, 13, 29, 2, 3, 5, 31, 3, 11, 2, 17, 5, 7, 37, 2, 19, 3, 13, 41, 2, 3, 7, 43, 2, 23, 47, 3, 17, 53, 5, 11, 3, 19, 2, 29, 59, 61, 2, 31, 5, 13, 2, 3, 11, 67, 3, 23, 2, 5, 7, 71, 73, 2

Offset: 1

Reinhard Zumkeller, Dec 13 2015

For n > 1: A072047(n) = length of row n;
T(n,1) = A073481(n); T(n,A001221(n)) = A073482(n);
for n > 1: A111060(n) = sum of row n;
A005117(n) = product of row n.

			.   n | T(n,*)  A5117(n)    n | T(n,*)  A5117(n)    n | T(n,*)   A5117(n)
. ----+---------+------   ----+---------+------   ----+----------+------
.   1 | [1]     |  1       21 | [3,11]  | 33       41 | [2,3,11] | 66
.   2 | [2]     |  2       22 | [2,17]  | 34       42 | [67]     | 67
.   3 | [3]     |  3       23 | [5,7]   | 35       43 | [3,23]   | 69
.   4 | [5]     |  5       24 | [37]    | 37       44 | [2,5,7]  | 70
.   5 | [2,3]   |  6       25 | [2,19]  | 38       45 | [71]     | 71
.   6 | [7]     |  7       26 | [3,13]  | 39       46 | [73]     | 73
.   7 | [2,5]   | 10       27 | [41]    | 41       47 | [2,37]   | 74
.   8 | [11]    | 11       28 | [2,3,7] | 42       48 | [7,11]   | 77
.   9 | [13]    | 13       29 | [43]    | 43       49 | [2,3,13] | 78
.  10 | [2,7]   | 14       30 | [2,23]  | 46       50 | [79]     | 79
.  11 | [3,5]   | 15       31 | [47]    | 47       51 | [2,41]   | 82
.  12 | [17]    | 17       32 | [3,17]  | 51       52 | [83]     | 83
.  13 | [19]    | 19       33 | [53]    | 53       53 | [5,17]   | 85
.  14 | [3,7]   | 21       34 | [5,11]  | 55       54 | [2,43]   | 86
.  15 | [2,11]  | 22       35 | [3,19]  | 57       55 | [3,29]   | 87
.  16 | [23]    | 23       36 | [2,29]  | 58       56 | [89]     | 89
.  17 | [2,13]  | 26       37 | [59]    | 59       57 | [7,13]   | 91
.  18 | [29]    | 29       38 | [61]    | 61       58 | [3,31]   | 93
.  19 | [2,3,5] | 30       39 | [2,31]  | 62       59 | [2,47]   | 94
.  20 | [31]    | 31       40 | [5,13]  | 65       60 | [5,19]   | 95  .

Reinhard Zumkeller, Rows n = 1..1000 of triangle, flattened

Cf. A005117, A027748, A124010, A072047 (row lengths), A073481, A073482, A001221, A111060, A049200, A062822.

import Math.NumberTheory.Primes.Factorisation (factorise)
import Data.Maybe (mapMaybe)
a265668 n k = a265668_tabf !! (n-1) !! (k-1)
a265668_row n = a265668_tabf !! (n-1)
a265668_tabf = [1] : mapMaybe f [2..] where
   f x = if all (== 1) es then Just ps else Nothing
         where (ps, es) = unzip $ factorise x

FactorInteger[#][[All,1]]&/@Select[Range[100],SquareFreeQ]//Flatten (* Harvey P. Dale, Apr 27 2018 *)

A073483 For the n-th squarefree number: the product of all primes greater than its smallest factor and less than its largest factor and not dividing it.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 15, 1, 1, 1, 5, 105, 1, 1155, 1, 1, 1, 35, 15015, 1, 1, 255255, 385, 1, 5, 1, 4849845, 1, 5005, 1, 7, 85085, 111546435, 1, 1, 3234846615, 77, 35, 1, 1616615, 3, 1, 1, 100280245065, 1, 385, 1, 3710369067405, 1, 1001

Offset: 1

Reinhard Zumkeller, Aug 03 2002

a(n)=1 iff A073484(n)=0; a(A000040(n))=1, a(A006094(n))=1, a(A002110(n))=1.

			The 69th squarefree number is 110=2*5*11, primes between 2 and 11, not dividing 110, are 3 and 7, therefore a(69)=21.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A005117, A000040, A073484, A117214.

a073483 n = product $ filter ((> 0) . (mod m)) $
   dropWhile (<= a020639 m) $ takeWhile (<= a006530 m) a000040_list
   where m = a005117 n
-- Reinhard Zumkeller, Jan 15 2012

ppg[n_]:=Module[{f=Transpose[FactorInteger[n]][[1]]},Times@@Select[Prime[ Range[PrimePi[First[f]]+1,PrimePi[Last[f]]-1]],!MemberQ[f,#]&]]; ppg/@ Select[ Range[100],SquareFreeQ] (* Harvey P. Dale, Jan 16 2013 *)

a(n) = A002110(A073482(n))/(A005117(n)*A002110(A073481(n))).

a(44) and a(49) corrected by Reinhard Zumkeller, Jan 14 2012

Definition clarified by Harvey P. Dale, Jan 16 2013

A117183 a(n) = smallest prime dividing n-th nonsquarefree positive integer.

Original entry on oeis.org

2, 2, 3, 2, 2, 2, 2, 2, 5, 3, 2, 2, 2, 2, 2, 3, 2, 7, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 11, 2, 5, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 13, 3, 2, 5, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 5, 2, 2, 2, 2, 2, 3, 2, 2, 2

Offset: 1

Leroy Quet, Mar 01 2006

nonn

			12, the 4th nonsquarefree positive integer, is 2^2 * 3. 2 is the smallest prime dividing 12. So a(4) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A013929, A020639, A073481, A115074, A115090.

with(numtheory): a:=proc(n) if mobius(n)=0 then op(1,factorset(n)) fi end: seq(a(n),n=1..345); # Emeric Deutsch

FactorInteger[ # ][[1, 1]] & /@ Select[ Range@252, !SquareFreeQ@# &] (* Robert G. Wilson v, Mar 06 2006 *)
FactorInteger[#][[1,1]]&/@DeleteCases[Range[300],?SquareFreeQ] (* _Harvey P. Dale, Jun 02 2017 *)

list(lim) = apply(x -> factor(x)[1,1], select(x -> !issquarefree(x), vector(lim, i, i))); \\ Amiram Eldar, Jun 25 2025

a(n) = A020639(A013929(n)).

More terms from Emeric Deutsch and Robert G. Wilson v, Mar 06 2006

A304181 If n = Product (p_j^k_j) then a(n) = min{p_j}^min{k_j}.

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 7, 8, 9, 2, 11, 2, 13, 2, 3, 16, 17, 2, 19, 2, 3, 2, 23, 2, 25, 2, 27, 2, 29, 2, 31, 32, 3, 2, 5, 4, 37, 2, 3, 2, 41, 2, 43, 2, 3, 2, 47, 2, 49, 2, 3, 2, 53, 2, 5, 2, 3, 2, 59, 2, 61, 2, 3, 64, 5, 2, 67, 2, 3, 2, 71, 4, 73, 2, 3, 2, 7, 2, 79, 2, 81, 2, 83, 2, 5

Offset: 1

Ilya Gutkovskiy, May 07 2018

nonn

			a(72) = 4 because 72 = 2^3*3^2, min{2,3} = 2, min{3,2} = 2 and 2^2 = 4.

Eric Weisstein's World of Mathematics, Least Prime Factor
Index entries for sequences computed from exponents in factorization of n

Cf. A000040, A000961 (fixed points), A005117, A020639, A028233, A034684, A051904, A073481, A081811, A304180.

Table[(FactorInteger[n][[1, 1]])^(Min @@ Last /@ FactorInteger[n]), {n, 85}]

a(n) = A020639(n)^A051904(n).
a(p^k) = p^k where p is a prime.
a(A005117(k)) = A073481(k).

A087619 a(n) = 137*a(n-1) + a(n-2), with a(0) = 2 and a(1) = 137.

Original entry on oeis.org

2, 137, 18771, 2571764, 352350439, 48274581907, 6613970071698, 906162174404533, 124150831863492719, 17009570127472907036, 2330435258295651756651, 319286639956631763568223

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Oct 25 2003

a(n+1)/a(n) converges to (137+sqrt(18773))/2 = 137.00729888121410965...
a(0)/a(1) = 2/137;
a(1)/a(2) = 137/18771;
a(2)/a(3) = 18771/2571764;
a(3)/a(4) = 2571764/352350439; ... etc.
Lim_{n->infinity} a(n)/a(n+1) = 0.00729888121410965... = 2/(137+sqrt(18773)) = (sqrt(18773)-137)/2.

Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (137,1).

Cf. A037088, A073481.

a(n) = ((137+sqrt(18773))/2)^n + ((137-sqrt(18773))/2)^n.
(a(n))^2 = a(2*n)-2 if n = 1, 3, 5, ..., (a(n))^2 = a(2n) + 2 if n = 2, 4, 6, ...
G.f.: (2-137*x)/(1-137*x-x^2). - Philippe Deléham, Nov 23 2008

A115013 a(n) = difference between largest and smallest primes dividing the n-th squarefree integer (with a(1)=0).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 3, 0, 0, 5, 2, 0, 0, 4, 9, 0, 11, 0, 3, 0, 8, 15, 2, 0, 17, 10, 0, 5, 0, 21, 0, 14, 0, 6, 16, 27, 0, 0, 29, 8, 9, 0, 20, 5, 0, 0, 35, 4, 11, 0, 39, 0, 12, 41, 26, 0, 6, 28, 45, 14, 0, 0, 15, 0, 4, 51, 0, 0, 9, 34, 0, 17, 18, 57, 10, 59, 38, 0, 40, 11, 0, 12, 65, 0, 21, 0, 44, 69, 2

Offset: 1

Leroy Quet, Feb 28 2006

nonn

			a(7)=3 because the 7th squarefree number is 10 = 2*5 and 5 - 2 = 3.

with(numtheory): a:=proc(n) local FS: FS:=factorset(n): if abs(mobius(n))>0 then FS[nops(FS)]-FS[1] else fi end: seq(a(n),n=2..180); # Emeric Deutsch, Mar 07 2006

a(n) = A073482(n) - A073481(n).

More terms from Emeric Deutsch, Mar 07 2006

a(1) = 0 inserted by Georg Fischer, Dec 09 2022

cp's OEIS Frontend

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

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A073482 Largest prime factor of the n-th squarefree number.

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A265668 Table read by rows: prime factors of squarefree numbers; a(1) = 1 by convention.

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A073483 For the n-th squarefree number: the product of all primes greater than its smallest factor and less than its largest factor and not dividing it.

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A117183 a(n) = smallest prime dividing n-th nonsquarefree positive integer.

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A304181 If n = Product (p_j^k_j) then a(n) = min{p_j}^min{k_j}.

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A087619 a(n) = 137*a(n-1) + a(n-2), with a(0) = 2 and a(1) = 137.

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