A073482 -id:A073482 - cp's OEIS frontend

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

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 *)

A073481 Least prime factor of the n-th squarefree number.

Original entry on oeis.org

1, 2, 3, 5, 2, 7, 2, 11, 13, 2, 3, 17, 19, 3, 2, 23, 2, 29, 2, 31, 3, 2, 5, 37, 2, 3, 41, 2, 43, 2, 47, 3, 53, 5, 3, 2, 59, 61, 2, 5, 2, 67, 3, 2, 71, 73, 2, 7, 2, 79, 2, 83, 5, 2, 3, 89, 7, 3, 2, 5, 97, 101, 2, 103, 3, 2, 107, 109, 2, 3, 113, 2, 5, 2, 7, 2, 3, 127, 3, 2, 131, 7, 2, 137, 2, 139, 3, 2, 11, 5

Offset: 1

Reinhard Zumkeller, Aug 03 2002

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A073482.

Cf. A005117, A020639, A265668.

a073482 = a020639 . a005117  -- Reinhard Zumkeller, Feb 04 2012

a = Select[Range[300], SquareFreeQ[#]&]; Table[FactorInteger[a[[n]]][[1,1]], {n, Length[a]}] (* Vladimir Joseph Stephan Orlovsky, Jan 30 2012 *)

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

from math import isqrt
from sympy import mobius, primefactors
def A073481(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 min(primefactors(kmax),default=1) # Chai Wah Wu, Aug 28 2024

a(n) = A020639(A005117(n)).

a(n) = A265668(n,1). - Reinhard Zumkeller, Dec 13 2015

More terms from Jason Earls, Aug 06 2002

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

A117214 a(n) = (A117213(n))/(n-th squarefree positive integer).

Original entry on oeis.org

1, 1, 2, 6, 1, 30, 3, 210, 2310, 15, 2, 30030, 510510, 10, 105, 9699690, 1155, 223092870, 1, 6469693230, 70, 15015, 6, 200560490130, 255255, 770, 7420738134810, 5, 304250263527210, 4849845, 13082761331670030, 10010

Offset: 1

Leroy Quet, Mar 03 2006

nonn

Product of all primes up to greatest prime factor of n-th squarefree number that do not divide the n-th squarefree number. - Franklin T. Adams-Watters, Oct 09 2006

a(n) = least k such that k*A005117(n) is a primorial number. Every term is squarefree. Let m be any squarefree number, and let P be the smallest primorial such that m|P. Then a(P/m) = m, and for any primorial number Q > P, a(Q/m) = m. Since there are infinitely many Q > P it follows that every squarefree number appears in this sequence infinitely many times. - David James Sycamore, Jul 04 2024

			10 is the 7th squarefree integer. And 2*3*5 = 30 is the smallest primorial number divisible by 10 = 2*5. So a(7) = 30/10 = 3.

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

Cf. A117213.
Cf. A006530, A005117, A073482, A073483.
Cf.A000720, A002110, A006530,

a117214 n = product $
   filter ((> 0) . (mod m)) $ takeWhile (< a006530 m) a000040_list
   where m = a005117 n
-- Reinhard Zumkeller, Jan 14 2012

Product[Prime@ i, {i, PrimePi@ FactorInteger[#][[-1, 1]]}]/# & /@ Select[Range@ 52, SquareFreeQ] (* Michael De Vlieger, Sep 30 2017 *)

a(n) = A002110(A000720(A005117(n))))/A005117(n). a(A002110(n)) = 1 for all n >= 0. a(A000040(n) = A002110(n-1), n > 1. - David James Sycamore, Jul 04 2024

More terms from Franklin T. Adams-Watters, Oct 09 2006

A370833 a(n) is the greatest prime dividing the n-th cubefree number, for n >= 2; a(1)=1.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 3, 5, 11, 3, 13, 7, 5, 17, 3, 19, 5, 7, 11, 23, 5, 13, 7, 29, 5, 31, 11, 17, 7, 3, 37, 19, 13, 41, 7, 43, 11, 5, 23, 47, 7, 5, 17, 13, 53, 11, 19, 29, 59, 5, 61, 31, 7, 13, 11, 67, 17, 23, 7, 71, 73, 37, 5, 19, 11, 13, 79, 41, 83, 7, 17, 43

Offset: 1

Amiram Eldar, Mar 03 2024

Amiram Eldar, 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. A004709, A006530, A073482, A370834, A370835.

s[n_] := Module[{f = FactorInteger[n]}, If[AllTrue[f[[;; , 2]], # < 3 &], f[[-1, 1]], Nothing]]; Array[s, 200]

lista(kmax) = {my(f); print1(1, ", "); for(k = 2, kmax, f = factor(k); if(vecmax(f[, 2]) < 3, print1(f[#f~, 1], ", ")));}

from sympy import mobius, integer_nthroot, primefactors
def A370833(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return max(primefactors(m),default=1) # Chai Wah Wu, Aug 06 2024

a(n) = A006530(A004709(n)).

Sum_{A004709(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 = 315/(4*Pi^4) = 0.808446... (De Koninck and Jakimczuk, 2024).

A370834 a(n) is the greatest prime dividing the n-th powerful number, for n >= 2; a(1)=1.

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 3, 2, 3, 7, 2, 3, 3, 5, 3, 11, 5, 2, 3, 13, 7, 5, 3, 5, 3, 2, 3, 17, 3, 7, 19, 7, 5, 3, 7, 11, 5, 2, 23, 3, 5, 3, 5, 13, 3, 7, 5, 29, 3, 5, 31, 11, 3, 5, 2, 11, 5, 3, 17, 7, 3, 7, 11, 13, 37, 7, 19, 13, 7, 5, 41, 3, 7, 5, 43, 11, 3, 5, 5, 2, 23

Offset: 1

Amiram Eldar, Mar 03 2024

Amiram Eldar, 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.
Michael De Vlieger, Log log scatterplot of pi(a(n)), n = 2..2^20, highlighting in red pi(a(n)) such that b(n) is a prime power, where b(n) = A001694(n). The remaining terms are such that b(n) is in A286708. The topmost red line corresponds with b(n) in A001248 (prime squares), the row of red dots at the bottom corresponds with b(n) in A000079 (powers of 2), the topmost blue line corresponds with b(n) > 16 in A069262 (4*p^2, with prime p).

Cf. A001694, A006530, A073482, A082695, A370833, A370835.

s[n_] := Module[{f = FactorInteger[n]}, If[n == 1 || AllTrue[f[[;; , 2]], # > 1 &], f[[-1, 1]], Nothing]]; Array[s, 4000]

lista(kmax) = {my(f); print1(1, ", "); for(k = 2, kmax, f = factor(k); if(vecmin(f[, 2]) > 1, print1(f[#f~, 1], ", ")));}

a(n) = A006530(A001694(n)).

Sum_{A001694(n) <= x} a(n) = Sum_{i=1..k} e_i * x/log(x)^i + O(x/log(x)^(k+1)), for any given positive integer k, where e_i are constants, e_1 = zeta(2)*zeta(3)/zeta(6) = 1.943596... (A082695) (De Koninck and Jakimczuk, 2024).

A370835 a(n) is the greatest prime dividing the n-th cubefull number, for n >= 2; a(1)=1.

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 3, 5, 2, 3, 3, 2, 7, 3, 2, 5, 3, 3, 3, 5, 2, 3, 11, 3, 3, 5, 2, 3, 13, 7, 3, 7, 5, 5, 3, 3, 5, 2, 17, 5, 3, 7, 3, 3, 19, 3, 3, 5, 2, 7, 5, 5, 3, 11, 7, 3, 23, 3, 11, 3, 5, 5, 2, 7, 5, 3, 13, 7, 3, 5, 3, 11, 7, 3, 29, 5, 5, 3, 7, 13, 31, 5, 3, 5

Offset: 1

Amiram Eldar, Mar 03 2024

Amiram Eldar, 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. A036966, A006530, A073482, A082695, A370833, A370834.

s[n_] := Module[{f = FactorInteger[n]}, If[n == 1 || AllTrue[f[[;; , 2]], # > 2 &], f[[-1, 1]], Nothing]]; Array[s, 32000]

lista(kmax) = {my(f); print1(1, ", "); for(k = 2, kmax, f = factor(k); if(vecmin(f[, 2]) > 2, print1(f[#f~, 1], ", ")));}

a(n) = A006530(A036966(n)).

Sum_{A036966(n) <= x} a(n) = Sum_{i=1..k} e_i * x^(2/3)/log(x)^i + O(x^(2/3)/log(x)^(k+1)), for any given positive integer k, where e_i are constants, e_1 = (3/2) * Product_{p prime} (1 + Sum_{i>=3} 1/p^(2*i/3)) = 3.44968588450293915243... (De Koninck and Jakimczuk, 2024).

A117213 a(n) = smallest term of sequence A002110 divisible by n-th squarefree positive integer.

Original entry on oeis.org

1, 2, 6, 30, 6, 210, 30, 2310, 30030, 210, 30, 510510, 9699690, 210, 2310, 223092870, 30030, 6469693230, 30, 200560490130, 2310, 510510, 210, 7420738134810, 9699690, 30030, 304250263527210, 210, 13082761331670030, 223092870

Offset: 1

Leroy Quet, Mar 03 2006

nonn

			10 is the 7th squarefree integer. And 2*3*5 = 30 is the smallest primorial number divisible by 10 = 2*5. So a(7) = 30.

Michael De Vlieger, Table of n, a(n) for n = 1..1441

Cf. A002110, A073482, A117214.

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: primor := proc(n::integer) local resul, nepr ; resul :=2 ; nepr :=3 ; while nepr <= n do resul := resul*nepr ; nepr:=nextprime(nepr) ; od : RETURN(resul) ; end: printf("1,") ; for n from 2 to 100 do lfa := issquarefree(n) ; if lfa > 0 then printf("%a,",primor(lfa) ) ; fi ; od : # R. J. Mathar, Apr 02 2006

Select[Array[Which[# == 1, 1, SquareFreeQ@ #, Product[Prime@ i, {i, PrimePi@ FactorInteger[#][[-1, 1]]}], True, 0] &, 50], # > 0 & ] (* Michael De Vlieger, Sep 30 2017 *)

For n >= 2, a(n) = product of the primes <= A073482(n).

More terms from R. J. Mathar, Apr 02 2006

A304180 If n = Product (p_j^k_j) then a(n) = max{p_j}^max{k_j}.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 9, 13, 7, 5, 16, 17, 9, 19, 25, 7, 11, 23, 27, 25, 13, 27, 49, 29, 5, 31, 32, 11, 17, 7, 9, 37, 19, 13, 125, 41, 7, 43, 121, 25, 23, 47, 81, 49, 25, 17, 169, 53, 27, 11, 343, 19, 29, 59, 25, 61, 31, 49, 64, 13, 11, 67, 289, 23, 7, 71, 27, 73, 37, 25

Offset: 1

Ilya Gutkovskiy, May 07 2018

nonn

			a(40) = 125 because 40 = 2^3*5^1, max{2,5} = 5, max{3,1} = 3 and 5^3 = 125.

Ilya Gutkovskiy, Scatter plot of a(n) up to n=100000.
Eric Weisstein's World of Mathematics, Greatest Prime Factor.
Index entries for sequences computed from exponents in factorization of n.

Cf. A000040, A000961 (fixed points), A002110, A005117, A006530, A034699, A051903, A053585, A073482, A081810, A304181.

Table[(FactorInteger[n][[-1, 1]])^(Max @@ Last /@ FactorInteger[n]), {n, 75}]

a(n) = if(n == 1, 1, my(f = factor(n), p = f[, 1], e = f[, 2]); vecmax(p)^vecmax(e)); \\ Amiram Eldar, Sep 08 2024

a(n) = A006530(n)^A051903(n).
a(p^k) = p^k where p is a prime.
a(A005117(k)) = A073482(k).
a(A002110(k)) = A000040(k).

A318411 Least k (>1) such that m^k == m mod A005117(n) for 0 <= m <= A005117(n) - 1.

Original entry on oeis.org

2, 3, 5, 3, 7, 5, 11, 13, 7, 5, 17, 19, 7, 11, 23, 13, 29, 5, 31, 11, 17, 13, 37, 19, 13, 41, 7, 43, 23, 47, 17, 53, 21, 19, 29, 59, 61, 31, 13, 11, 67, 23, 13, 71, 73, 37, 31, 13, 79, 41, 83, 17, 43, 29, 89, 13, 31, 47, 37, 97, 101, 17, 103, 13, 53, 107, 109, 21, 37, 113, 19

Offset: 2

Seiichi Manyama, Aug 26 2018

nonn

This sequence is different from A073482.

			A005117(5) = 6.
    0^3  =   0 == 0 mod 6,
    1^3  =   1 == 1 mod 6,
    2^3  =   8 == 2 mod 6,
    3^3  =  27 == 3 mod 6,
    4^3  =  64 == 4 mod 6,
    5^3  = 125 == 5 mod 6.
------------------------------------------------
A005117(23) = 35.
    0^13 =                    0 ==  0 mod 35,
    1^13 =                    1 ==  1 mod 35,
    2^13 =                 8192 ==  2 mod 35,
   ...
   34^13 = 81138303245565435904 == 34 mod 35.
------------------------------------------------
------+------------+------
   n  | A005117(n) | a(n)
------+------------+------
    2 |          2 |    2
    3 |          3 |    3
    4 |          5 |    5
    5 |          6 |    3
    6 |          7 |    7
    7 |         10 |    5
    8 |         11 |   11
    9 |         13 |   13
   10 |         14 |    7
   11 |         15 |    5
   12 |         17 |   17
   13 |         19 |   19
   14 |         21 |    7
   15 |         22 |   11
   16 |         23 |   23
   17 |         26 |   13
   18 |         29 |   29
   19 |         30 |    5
   20 |         31 |   31
   21 |         33 |   11
   22 |         34 |   17
   23 |         35 |   13

Seiichi Manyama, Table of n, a(n) for n = 2..10000

Cf. A005117, A318572.

cp's OEIS Frontend

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

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A073481 Least prime factor of the n-th squarefree number.

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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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A117214 a(n) = (A117213(n))/(n-th squarefree positive integer).

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A370833 a(n) is the greatest prime dividing the n-th cubefree number, for n >= 2; a(1)=1.

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A370834 a(n) is the greatest prime dividing the n-th powerful number, for n >= 2; a(1)=1.

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A370835 a(n) is the greatest prime dividing the n-th cubefull number, for n >= 2; a(1)=1.

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