A034684 -id:A034684 - cp's OEIS frontend

A330755 Number of values of k, 1 <= k <= n, with A034684(k) = A034684(n), where A034684 gives the smallest unitary divisor of n that is larger than 1.

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 4, 3, 1, 1, 5, 1, 2, 4, 6, 1, 5, 1, 7, 1, 3, 1, 8, 1, 1, 6, 9, 2, 4, 1, 10, 7, 3, 1, 11, 1, 5, 4, 12, 1, 8, 1, 13, 9, 6, 1, 14, 5, 2, 10, 15, 1, 11, 1, 16, 3, 1, 6, 17, 1, 7, 12, 18, 1, 2, 1, 19, 13, 8, 4, 20, 1, 7, 1, 21, 1, 14, 8, 22, 15, 3, 1, 23, 5, 9, 16, 24, 9, 17, 1, 25, 2, 10, 1, 26, 1, 4, 18

Offset: 1

Antti Karttunen, Dec 30 2019

nonn

Ordinal transform of A034684.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A034684, A078898, A303759.

A034684[n_] := With[{f = FactorInteger[n]}, Min[f[[All, 1]]^f[[All, 2]]]];
Module[{b}, b[_] = 0;
a[n_] := With[{t = A034684[n]}, b[t] = b[t] + 1]];
Array[a, 105] (* Jean-François Alcover, Jan 10 2022 *)

up_to = 65537;
A034684(n) = if(1==n,n,my(f=factor(n)); vecmin(vector(#f[, 1], i, f[i, 1]^f[i, 2]))); \\ After code in A034699
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
v330755 = ordinal_transform(vector(up_to, n, A034684(n)));
A330755(n) = v330755[n];

A158377 a(1) = 0, a(n) = lcm(A034684(n), A034699(n)) for n >= 2.

Original entry on oeis.org

0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 10, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 14, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 15, 61, 62, 63, 64, 65, 22, 67, 68, 69, 14, 71, 72

Offset: 1

Jaroslav Krizek, Mar 17 2009

nonn

a(n) for n >= 2 equals LCM of minimal and maximal prime power factors in prime factorization of n. For n >= 2 holds: a(n)*A100994(n) = A034684(n)*A034699(n). a(n) for n >= 2 it deviates from A000027(n), first different term is a(30)=a(2*3*5), a(30)=lcm(2,5)=10, A000027(30)= 30. Sequence of deviations from A000027(n): 30,42,60,66,70,78,84,90,...

			For n = 30 = 2*3*5, a(30) = lcm(2,5) = 10.

Cf. A100994, A034684, A034699, A000027.

a(1) = 0, a(p) = p, a(pq) = pq, a(pq...z) = pz, a(p^k) = p^k, for p = primes (A000040), pq = product of two distinct primes (A006881), pq...z = product of k (k > 2) distinct primes p, q, ..., z (A120944), p^k = prime powers (A000961(n) for n > 1), k = natural numbers (A000027).

A006530 Gpf(n): greatest prime dividing n, for n >= 2; a(1)=1.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

The initial term a(1)=1 is purely conventional: The unit 1 is not a prime number, although it has been considered so in the past. 1 is the empty product of prime numbers, thus 1 has no largest prime factor. - Daniel Forgues, Jul 05 2011
Greatest noncomposite number dividing n, (cf. A008578). - Omar E. Pol, Aug 31 2013
Conjecture: Let a, b be nonzero integers and f(n) denote the maximum prime factor of a*n + b if a*n + b <> 0 and f(n)=0 if a*n + b=0 for any integer n. Then the set {n, f(n), f(f(n)), ...} is finite of bounded size. - M. Farrokhi D. G., Jan 10 2021

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section IV.1.
H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 210.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
K. Alladi and P. Erdős, On an additive arithmetic function, Pacific J. Math., Volume 71, Number 2 (1977), 275-294. MR 0447086 (56 #5401).
G. Back and M. Caragiu, The greatest prime factor and recurrent sequences, Fib. Q., 48 (2010), 358-362.
A. E. Brouwer, Two number theoretic sums, Stichting Mathematisch Centrum. Zuivere Wiskunde, Report ZW 19/74 (1974): 3 pages. [Cached copy, included with the permission of the author]
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
Mats Granvik, Mathematica program to check the conjectured formula, Feb 28 2021.
Nathan McNew, The Most Frequent Values of the Largest Prime Divisor Function, Exper. Math., 2017, Vol. 26, No. 2, 210-224.
OEIS Wiki, Greatest prime factor of n
H. P. Robinson, Letter to N. J. A. Sloane, Oct 1981
David Singmaster, Letter to N. J. A. Sloane, Oct 03 1982.
Eric Weisstein's World of Mathematics, Greatest Prime Factor
Index entries for "core" sequences

Cf. A000040, A020639 (smallest prime divisor), A034684, A028233, A034699, A053585.
See also A032742, A052126, A070087, A070089, A061395, A175723.
Cf. A046670 (partial sums), A104350 (partial products).
See A385503 for "popular" primes.

[ #f eq 0 select 1 else f[ #f][1] where f is Factorization(n): n in [1..86] ]; // Klaus Brockhaus, Oct 23 2008

with(numtheory,divisors); A006530 := proc(n) local i,t1,t2,t3,t4,t5; t1 := divisors(n); t2 := convert(t1,list); t3 := sort(t2); t4 := nops(t3); t5 := 1; for i from 1 to t4 do if isprime(t3[t4+1-i]) then return t3[t4+1-i]; fi; od; 1; end;
# alternative
A006530 := n->max(1,op(numtheory[factorset](n))); # Peter Luschny, Nov 02 2010

Table[ FactorInteger[n][[ -1, 1]], {n, 100}] (* Ray Chandler, Nov 12 2005 and modified by Robert G. Wilson v, Jul 16 2014 *)

A006530(n)=if(n>1,vecmax(factor(n)[,1]),1) \\ Edited to cover n=1. - M. F. Hasler, Jul 30 2015

from sympy import factorint
def a(n): return 1 if n == 1 else max(factorint(n))
print([a(n) for n in range(1, 87)]) # Michael S. Branicky, Aug 08 2022

def A006530(n): return list(factor(n))[-1][0] if n > 1 else 1
print([A006530(n) for n in range(1, 87)])  # Peter Luschny, Jan 07 2024

;; The following uses macro definec for the memoization (caching) of the results. A naive implementation of A020639 can be found under that entry. It could be also defined with definec to make it faster on the later calls. See http://oeis.org/wiki/Memoization#Scheme
(definec (A006530 n) (let ((spf (A020639 n))) (if (= spf n) spf (A006530 (/ n spf)))))
;; Antti Karttunen, Mar 12 2017

a(n) = A027748(n, A001221(n)) = A027746(n, A001222(n)); a(n)^A071178(n) = A053585(n). - Reinhard Zumkeller, Aug 27 2011
a(n) = A000040(A061395(n)). - M. F. Hasler, Jan 16 2015
a(n) = n + 1 - Sum_{k=1..n} (floor((k!^n)/n) - floor(((k!^n)-1)/n)). - Anthony Browne, May 11 2016
n/a(n) = A052126(n). - R. J. Mathar, Oct 03 2016
If A020639(n) = n [when n is 1 or a prime] then a(n) = n, otherwise a(n) = a(A032742(n)). - Antti Karttunen, Mar 12 2017
a(n) has average order Pi^2*n/(12 log n) [Brouwer]. See also A046670. - N. J. A. Sloane, Jun 26 2017

Edited by M. F. Hasler, Jan 16 2015

A020639 Lpf(n): least prime dividing n (when n > 1); a(1) = 1. Or, smallest prime factor of n, or smallest prime divisor of n.

Original entry on oeis.org

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

Offset: 1

David W. Wilson

Also, the largest number of distinct integers such that all their pairwise differences are coprime to n. - Max Alekseyev, Mar 17 2006
The unit 1 is not a prime number (although it has been considered so in the past). 1 is the empty product of prime numbers, thus 1 has no least prime factor. - Daniel Forgues, Jul 05 2011
a(n) = least m > 0 for which n! + m and n - m are not relatively prime. - Clark Kimberling, Jul 21 2012
For n > 1, a(n) = the smallest k > 1 that divides n. - Antti Karttunen, Feb 01 2014
For n > 1, records are at prime indices. - Zak Seidov, Apr 29 2015
The initials "lpf" might be mistaken for "largest prime factor" (A009190), using "spf" for "smallest prime factor" would avoid this. - M. F. Hasler, Jul 29 2015
n = 89 is the first index > 1 for which a(n) differs from the smallest k > 1 such that (2^k + n - 2)/k is an integer. - M. F. Hasler, Aug 11 2015
From Stanislav Sykora, Jul 29 2017: (Start)
For n > 1, a(n) is also the smallest k, 1 < k <= n, for which the binomial(n,k) is not divisible by n.
Proof: (A) When k and n are relatively prime then binomial(n,k) is divisible by n because k*binomial(n,k) = n*binomial(n-1,k-1). (B) When gcd(n,k) > 1, one of its prime factors is the smallest; let us denote it p, p <= k, and consider the binomial(n,p) = (1/p!)*Product_{i=0..p-1} (n-i). Since p is a divisor of n, it cannot be a divisor of any of the remaining numerator factors. It follows that, denoting as e the largest e > 0 such that p^e|n, the numerator is divisible by p^e but not by p^(e+1). Hence, the binomial is divisible by p^(e-1) but not by p^e and therefore not divisible by n. Applying (A), (B) to all considered values of k completes the proof. (End)
From Bob Selcoe, Oct 11 2017, edited by M. F. Hasler, Nov 06 2017: (Start)
a(n) = prime(j) when n == J (mod A002110(j)), n, j >= 1, where J is the set of numbers <= A002110(j) with smallest prime factor = prime(j). The number of terms in J is A005867(j-1). So:
a(n) = 2 when n == 0 (mod 2);
a(n) = 3 when n == 3 (mod 6);
a(n) = 5 when n == 5 or 25 (mod 30);
a(n) = 7 when n == 7, 49, 77, 91, 119, 133, 161 or 203 (mod 210);
etc. (End)
For n > 1, a(n) is the leftmost term, other than 0 or 1, in the n-th row of A127093. - Davis Smith, Mar 05 2019

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section IV.1.

Daniel Forgues, Table of n, a(n) for n = 1..100000 (terms 1..10000 from T. D. Noe)
A. E. Brouwer, Two number theoretic sums, Stichting Mathematisch Centrum. Zuivere Wiskunde, Report ZW 19/74 (1974): 3 pages. [Copy included with the permission of the author.]
OEIS Wiki, Least prime factor of n
David Singmaster, Letter to N. J. A. Sloane, Oct 3 1982.
Eric Weisstein's World of Mathematics, Least Prime Factor
Index entries for "core" sequences

Cf. A090368 (bisection).
Cf. A000040, A009190, A006530, A034684, A028233, A034699, A053585.
See also A032742, A055396, A068319, A088377, A007978, A053669, A117818.
Cf. A046669 (partial sums), A072486 (partial products).
Cf. A002110, A005867.
Cf. A127093.

a020639 n = spf a000040_list where
  spf (p:ps) | n < p^2      = n
             | mod n p == 0 = p
             | otherwise    = spf ps
-- Reinhard Zumkeller, Jul 13 2011

A020639 := proc(n) if n = 1 then 1; else min(op(numtheory[factorset](n))) ; end if; end proc: seq(A020639(n),n=1..20) ; # R. J. Mathar, Oct 25 2010

f[n_]:=FactorInteger[n][[1,1]]; Join[{1}, Array[f,120,2]]  (* Robert G. Wilson v, Apr 06 2011 *)
Join[{1}, Table[If[EvenQ[n], 2, FactorInteger[n][[1,1]]], {n, 2, 120}]] (* Zak Seidov, Nov 17 2013 *)
Riffle[Join[{1},Table[FactorInteger[n][[1,1]],{n,3,101,2}]],2] (* Harvey P. Dale, Dec 16 2021 *)

A020639(n) = { vecmin(factor(n)[,1]) } \\ [Will yield an error for n = 1.] - R. J. Mathar, Mar 02 2012

A020639(n)=if(n>1, if(n>n=factor(n,0)[1,1], n, factor(n)[1,1]), 1) \\ Avoids complete factorization if possible. Often the smallest prime factor can be found quickly even if it is larger than primelimit. If factoring takes too long for large n, use debugging level >= 3 (\g3) to display the smallest factor as soon as it is found. - M. F. Hasler, Jul 29 2015

from sympy import factorint
def a(n): return 1 if n == 1 else min(factorint(n))
print([a(n) for n in range(1, 98)]) # Michael S. Branicky, Dec 09 2021

def A020639_list(n) : return [1] + [prime_divisors(n)[0] for n in (2..n)]
A020639_list(97) # Peter Luschny, Jul 16 2012

[trial_division(n) for n in (1..100)] # Giuseppe Coppoletta, May 25 2016

(define (A020639 n) (if (< n 2) n (let loop ((k 2)) (cond ((zero? (modulo n k)) k) (else (loop (+ 1 k))))))) ;; Antti Karttunen, Feb 01 2014

A014673(n) = a(A032742(n)); A115561(n) = a(A054576(n)). - Reinhard Zumkeller, Mar 10 2006
A028233(n) = a(n)^A067029(n). - Reinhard Zumkeller, May 13 2006
a(n) = A027746(n,1) = A027748(n,1). - Reinhard Zumkeller, Aug 27 2011
For n > 1: a(n) = A240694(n,2). - Reinhard Zumkeller, Apr 10 2014
a(n) = A000040(A055396(n)) = n / A032742(n). - Antti Karttunen, Mar 07 2017
a(n) has average order n/(2 log n) [Brouwer] - N. J. A. Sloane, Sep 03 2017

Deleted wrong comment from M. Lagneau in 2012, following an observation by Gionata Neri. - M. F. Hasler, Aug 11 2015
Edited by M. F. Hasler, Nov 06 2017
Expanded definition to make this easier to find. - N. J. A. Sloane, Sep 21 2020

A067029 Exponent of least prime factor in prime factorization of n, a(1)=0.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 3, 1, 1

Offset: 1

Reinhard Zumkeller, Feb 17 2002

Even bisection is A001511: a(2n) = A007814(n) + 1. - Ralf Stephan, Jan 31 2004
Number of occurrences of the smallest part in the partition with Heinz number n. The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product_{j=1..r} (p_j-th prime) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). Example: a(24)=3 because the partition with Heinz number 24 = 3*2*2*2 is [2,1,1,1]. - Emeric Deutsch, Oct 02 2015
Together with A028234 is useful for defining sequences that are multiplicative with a(p^e) = f(e), as recurrences of the form: a(1) = 1 and for n > 1, a(n) = f(A067029(n)) * a(A028234(n)). - Antti Karttunen, May 29 2017

			a(18) = a(2^1 * 3^2) = 1.

T. D. Noe, Table of n, a(n) for n = 1..10000
Project Euler, Problem 779: Prime factor and exponent, (2022).

Cf. A051903, A020639, A028233, A034684, A071178, first column of A124010, A247180.

a067029 = head . a124010_row
-- Reinhard Zumkeller, Jul 05 2013, Jun 04 2012

A067029 := proc(n)
    local f,lp,a;
    a := 0 ;
    lp := n+1 ;
    for f in ifactors(n)[2] do
        p := op(1,f) ;
        if p < lp then
            a := op(2,f) ;
            lp := p;
        fi;
    end do:
    a ;
end proc: # R. J. Mathar, Jul 08 2015
seq(ifelse(n = 1, 0, ifactors(n)[2][1][2]), n = 1..90); # Peter Luschny, Jun 15 2025

Join[{0},Table[FactorInteger[n][[1,2]],{n,2,100}]] (* Harvey P. Dale, Oct 14 2011 *)

a(n) = if (n==1, 0, factor(n)[1,2]); \\ Michel Marcus, May 15 2017

from sympy import factorint
def a(n):
    f=factorint(n)
    return 0 if n==1 else f[min(f)] # Indranil Ghosh, May 15 2017

;; Naive implementation of A020639 is given under that entry. All of these functions could be also defined with definec to make them faster on the later calls. See http://oeis.org/wiki/Memoization#Scheme
(define (A067029 n) (if (< n 2) 0 (let ((mp (A020639 n))) (let loop ((e 0) (n (/ n mp))) (cond ((integer? n) (loop (+ e 1) (/ n mp))) (else e)))))) ;;  Antti Karttunen, May 29 2017

a(n) = A124010(n,1). - Reinhard Zumkeller, Aug 27 2011
A028233(n) = A020639(n)^a(n). - Reinhard Zumkeller, May 13 2006
a(A247180(n)) = 1. - Reinhard Zumkeller, Nov 23 2014
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} (Product_{i=1..k-1} (1 - 1/prime(i)))/(prime(k)-1) = 1/(prime(1)-1) + (1-1/prime(1))*(1/(prime(2)-1) + (1-1/prime(2))*(1/(prime(3)-1) + (1-1/prime(3))*( ... ))) = 1.6125177915... - Amiram Eldar, Oct 26 2021

A034699 Largest prime power factor of n.

Original entry on oeis.org

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

Offset: 1

David W. Wilson

n divides lcm(1, 2, ..., a(n)).
a(n) = A210208(n,A073093(n)) = largest term of n-th row in A210208. - Reinhard Zumkeller, Mar 18 2012
a(n) = smallest m > 0 such that n divides A003418(m). - Thomas Ordowski, Nov 15 2013
a(n) = n when n is a prime power (A000961). - Michel Marcus, Dec 03 2013
Conjecture: For all n between two consecutive prime numbers, all a(n) are different. - I. V. Serov, Jun 19 2019
Disproved with between p=prime(574) = 4177 and prime(575) = 4201, a(4180) = a(4199) = 19. See A308752. - Michel Marcus, Jun 19 2019
Conjecture: For any N > 0, there exist numbers n and m, N < n < n+a(n) <= m, such that all n..m are composite and a(n) = a(m). - I. V. Serov, Jun 21 2019
Conjecture: For all n between two consecutive prime numbers, all (-1)^n*a(n) are different. Checked up to 5*10^7. - I. V. Serov, Jun 23 2019
Disproved: between p = prime(460269635) = 10120168277 and p = prime(460269636) = 10120168507 the numbers n = 10120168284 and m = 10120168498 form a pair such that (-1)^n*a(n) = (-1)^m*a(m) = 107. - L. Joris Perrenet, Jan 05 2020
a(n) = cardinality of smallest set on which idempotence of order n+1 (f^{n+1} = f) differs from idempotence of order e for 2 <= e <= n (see von Eitzen link for proof); derivable from A245501. - Mark Bowron, May 22 2025

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from T. D. Noe)
Hagen von Eitzen, Least k to distinguish f^n = f on {1,...,k} from f^e = f for every 1 < e < n is largest prime power factor of n-1?, Math Stack Exchange.

Cf. A006530, A010055, A020639, A027750, A034684, A028233, A051283, A052128, A053585, A057110, A060818, A038610, A081805, A088387, A088388, A100574, A210208, A245501, A284600, A308752.

Numbers n ordered by a(n): A305325.

a034699 = last . a210208_row
-- Reinhard Zumkeller, Mar 18 2012, Feb 14 2012

f[n_] := If[n == 1, 1, Max[ #[[1]]^#[[2]] & /@ FactorInteger@n]]; Array[f, 79] (* Robert G. Wilson v, Sep 02 2006 *)
Array[Max[Power @@@ FactorInteger@ #] &, 79] (* Michael De Vlieger, Jul 26 2018 *)

a(n) = if(1==n,n,my(f=factor(n)); vecmax(vector(#f[, 1], i, f[i, 1]^f[i, 2]))); \\ Charles R Greathouse IV, Nov 20 2012, check for a(1) added by Antti Karttunen, Aug 06 2018

A034699(n) = if(1==n,n,fordiv(n, d, if(isprimepower(n/d), return(n/d)))); \\ Antti Karttunen, Aug 06 2018

from sympy import factorint
def A034699(n): return max((p**e for p, e in factorint(n).items()), default=1) # Chai Wah Wu, Apr 17 2023

If n = p_1^e_1 *...* p_k^e_k, p_1 < ... < p_k primes, then a(n) = Max_i p_i^e_i.
a(n) = A088387(n)^A088388(n). - Antti Karttunen, Jul 22 2018
a(n) = n/A284600(n) = n - A081805(n) = A034684(n) + A100574(n). - Antti Karttunen, Aug 06 2018
a(n) = a(m) iff m = d*a(n), where d is a divisor of A038610(a(n)). - I. V. Serov, Jun 19 2019

A008475 If n = Product (p_j^k_j) then a(n) = Sum (p_j^k_j) (a(1) = 0 by convention).

Original entry on oeis.org

0, 2, 3, 4, 5, 5, 7, 8, 9, 7, 11, 7, 13, 9, 8, 16, 17, 11, 19, 9, 10, 13, 23, 11, 25, 15, 27, 11, 29, 10, 31, 32, 14, 19, 12, 13, 37, 21, 16, 13, 41, 12, 43, 15, 14, 25, 47, 19, 49, 27, 20, 17, 53, 29, 16, 15, 22, 31, 59, 12, 61, 33, 16, 64, 18, 16, 67, 21, 26, 14, 71, 17, 73

Offset: 1

Olivier Gérard

For n>1, a(n) is the minimal number m such that the symmetric group S_m has an element of order n. - Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 26 2001

If gcd(u,w) = 1, then a(u*w) = a(u) + a(w); behaves like logarithm; compare A001414 or A056239. - Labos Elemer, Mar 31 2003

			a(180) = a(2^2 * 3^2 * 5) = 2^2 + 3^2 + 5 = 18.

F. J. Budden, The Fascination of Groups, Cambridge, 1972; pp. 322, 573.
József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter IV, p. 147.
T. Z. Xuan, On some sums of large additive number theoretic functions (in Chinese), Journal of Beijing normal university, No. 2 (1984), pp. 11-18.

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)
John Bamberg, Grant Cairns and Devin Kilminster, The crystallographic restriction, permutations and Goldbach's conjecture, Amer. Math. Monthly, Vol. 110, No. 3 (March 2003), pp. 202-209.
Roger B. Eggleton and William P. Galvin, Upper Bounds on the Sum of Principal Divisors of an Integer, Mathematics Magazine, Vol. 77, No. 3 (Jun., 2004), pp. 190-200.

Cf. A001414, A000961, A005117, A051613, A072691, A081402, A081403, A081404, A027748, A124010, A001221, A028233, A034684, A053585, A159077, A023888, A078771, A092509, A286875.

See A222416 for the variant with a(1)=1.

a008475 1 = 0
a008475 n = sum $ a141809_row n
-- Reinhard Zumkeller, Jan 29 2013, Oct 10 2011

A008475 := proc(n) local e,j; e := ifactors(n)[2]:
add(e[j][1]^e[j][2], j=1..nops(e)) end:
seq(A008475(n), n=1..60); # Peter Luschny, Jan 17 2010

f[n_] := Plus @@ Power @@@ FactorInteger@ n; f[1] = 0; Array[f, 73]

for(n=1,100,print1(sum(i=1,omega(n), component(component(factor(n),1),i)^component(component(factor(n),2),i)),","))

a(n)=local(t);if(n<1,0,t=factor(n);sum(k=1,matsize(t)[1],t[k,1]^t[k,2])) /* Michael Somos, Oct 20 2004 */

A008475(n) = { my(f=factor(n)); vecsum(vector(#f~,i,f[i,1]^f[i,2])); }; \\ Antti Karttunen, Nov 17 2017

from sympy import factorint
def a(n):
    f=factorint(n)
    return 0 if n==1 else sum([i**f[i] for i in f]) # Indranil Ghosh, May 20 2017

Additive with a(p^e) = p^e.
a(A000961(n)) = A000961(n); a(A005117(n)) = A001414(A005117(n)).
a(n) = Sum_{k=1..A001221(n)} A027748(n,k) ^ A124010(n,k) for n>1. - Reinhard Zumkeller, Oct 10 2011
a(n) = Sum_{k=1..A001221(n)} A141809(n,k) for n > 1. - Reinhard Zumkeller, Jan 29 2013
Sum_{k=1..n} a(k) ~ (Pi^2/12)* n^2/log(n) + O(n^2/log(n)^2) (Xuan, 1984). - Amiram Eldar, Mar 04 2021

A053585 If n = p_1^e_1 * ... * p_k^e_k, p_1 < ... < p_k primes, then a(n) = p_k^e_k.

Original entry on oeis.org

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

Offset: 1

Frederick Magata (frederick.magata(AT)uni-muenster.de), Jan 19 2000

Let p be the largest prime dividing n, a(n) is the largest power of p dividing n.

			a(42)=7 because 42=2*3*7, a(144)=9 because 144=16*9=2^4*3^2.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A020639, A006530, A034684, A028233, A051119, A008475.

Different from A034699.

a053585 = last . a141809_row  -- Reinhard Zumkeller, Jan 29 2013

a:= n-> `if`(n=1, 1, (i->i[1]^i[2])(sort(ifactors(n)[2])[-1])):
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 03 2023

Table[Power @@ Last @ FactorInteger @ n, {n, 79}] (* Jean-François Alcover, Apr 01 2011 *)

a(n)=if(n>1,my(f=factor(n)); f[#f~,1]^f[#f~,2],1) \\ Charles R Greathouse IV, Nov 10 2015

from sympy import factorint, primefactors
def a(n):
    if n==1: return 1
    p = primefactors(n)[-1]
    return p**factorint(n)[p] # Indranil Ghosh, May 19 2017

a(n) = A006530(n)^A071178(n). - Reinhard Zumkeller, Aug 27 2011

a(n) = A141809(n,A001221(n)). - Reinhard Zumkeller, Jan 29 2013

More terms from Andrew Gacek (andrew(AT)dgi.net), Apr 20 2000

A028233 If n = p_1^e_1 * ... * p_k^e_k, p_1 < ... < p_k primes, then a(n) = p_1^e_1, with a(1) = 1.

Original entry on oeis.org

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

Offset: 1

Marc LeBrun

Highest power of smallest prime dividing n. - Reinhard Zumkeller, Apr 09 2015

			From _Muniru A Asiru_, Jan 27 2018: (Start)
If n=10, then a(10) = 2 since 10 = 2^1*5^1.
If n=16, then a(16) = 16 since 16 = 2^4.
If n=29, then a(29) = 29 since 29 = 29^1.
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A020639, A006530, A034684, A034699, A053585.
See also A028234.
Cf. A008475.
Cf. A141809.

List(List(List(List([1..10^3], Factors), Collected), i -> i[1]), j -> j[1]^j[2]); # Muniru A Asiru, Jan 27 2018

a028233 = head . a141809_row
-- Reinhard Zumkeller, Jun 04 2012, Aug 17 2011

A028233 := proc(n)
    local spf,pf;
    if n = 1 then
        return 1 ;
    end if;
    spf := A020639(n) ;
    for pf in ifactors(n)[2] do
        if pf[1] = spf then
            return pf[1]^pf[2] ;
        end if;
    end do:
end proc: # R. J. Mathar, Jul 09 2016
# second Maple program:
a:= n-> `if`(n=1, 1, (i->i[1]^i[2])(sort(ifactors(n)[2])[1])):
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 29 2018

a[n_] := Power @@ First[ FactorInteger[n]]; Table[a[n], {n, 1, 86}] (* Jean-François Alcover, Dec 01 2011 *)

a(n)=if(n>1,n=factor(n);n[1,1]^n[1,2],1) \\ Charles R Greathouse IV, Apr 26 2012

from sympy import factorint
def a(n):
    f = factorint(n)
    return 1 if n==1 else min(f)**f[min(f)] # Indranil Ghosh, May 12 2017

;; Naive implementation of A020639 is given under that entry. All of these functions could be also defined with definec to make them faster on the later calls. See http://oeis.org/wiki/Memoization#Scheme
(define (A028233 n) (if (< n 2) n (let ((lpf (A020639 n))) (let loop ((m lpf) (n (/ n lpf))) (cond ((not (zero? (modulo n lpf))) m) (else (loop (* m lpf) (/ n lpf)))))))) ;; Antti Karttunen, May 29 2017

a(n) = A020639(n)^A067029(n). - Reinhard Zumkeller, May 13 2006
a(n) = A141809(n,1). - Reinhard Zumkeller, Jun 04 2012
a(n) = n / A028234(n). - Antti Karttunen, May 29 2017

Name edited to include a(1) = 1 by Franklin T. Adams-Watters, Jan 27 2018

cp's OEIS Frontend

A052125 a(n) = n/A034684(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A330755 Number of values of k, 1 <= k <= n, with A034684(k) = A034684(n), where A034684 gives the smallest unitary divisor of n that is larger than 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A158377 a(1) = 0, a(n) = lcm(A034684(n), A034699(n)) for n >= 2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A006530 Gpf(n): greatest prime dividing n, for n >= 2; a(1)=1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

SageMath

Scheme

Formula

Extensions

A020639 Lpf(n): least prime dividing n (when n > 1); a(1) = 1. Or, smallest prime factor of n, or smallest prime divisor of n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

Python

Sage

Sage

Scheme

Formula

Extensions

A067029 Exponent of least prime factor in prime factorization of n, a(1)=0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica