A008475 -id:A008475 - cp's OEIS frontend

A056239 If n = Product_{k >= 1} (p_k)^(c_k) where p_k is k-th prime and c_k >= 0 then a(n) = Sum_{k >= 1} k*c_k.

0, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 6, 5, 5, 4, 7, 5, 8, 5, 6, 6, 9, 5, 6, 7, 6, 6, 10, 6, 11, 5, 7, 8, 7, 6, 12, 9, 8, 6, 13, 7, 14, 7, 7, 10, 15, 6, 8, 7, 9, 8, 16, 7, 8, 7, 10, 11, 17, 7, 18, 12, 8, 6, 9, 8, 19, 9, 11, 8, 20, 7, 21, 13, 8, 10, 9, 9, 22, 7, 8, 14, 23, 8, 10, 15, 12, 8, 24, 8, 10

Offset: 1

Leroy Quet, Aug 19 2000

A pseudo-logarithmic function in the sense that a(b*c) = a(b)+a(c) and so a(b^c) = c*a(b) and f(n) = k^a(n) is a multiplicative function. [Cf. A248692 for example.] Essentially a function from the positive integers onto the partitions of the nonnegative integers (1->0, 2->1, 3->2, 4->1+1, 5->3, 6->1+2, etc.) so each value a(n) appears A000041(a(n)) times, first with the a(n)-th prime and last with the a(n)-th power of 2. Produces triangular numbers from primorials. - Henry Bottomley, Nov 22 2001
Michael Nyvang writes (May 08 2006) that the Danish composer Karl Aage Rasmussen discovered this sequence in the 1990's: it has excellent musical properties.
All A000041(a(n)) different n's with the same value a(n) are listed in row a(n) of triangle A215366. - Alois P. Heinz, Aug 09 2012
a(n) is the sum of the parts of the partition having Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product_{j=1..r} (p_j-th prime) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: a(33) = 7 because the partition with Heinz number 33 = 3 * 11 is [2,5]. - Emeric Deutsch, May 19 2015

			a(12) = 1*2 + 2*1 = 4, since 12 = 2^2 *3^1 = (p_1)^2 *(p_2)^1.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Row sums of A112798.
Cf. A003963 (gives the corresponding products).
Cf. also A000041, A000217, A000720, A001221, A001222, A002110, A005940, A008475, A027748, A049084, A060437, A081401, A082090, A088314, A088318, A088850, A088880, A088881, A088887, A088902, A108951, A122111, A124010, A127668, A141128, A153734, A154351, A156552, A163517, A178503, A215366, A215369, A215501, A242422, A242423, A242424, A243070, A243353, A243354, A243503, A248692, A249336, A249337, A329605, A329902.

a056239 n = sum $ zipWith (*) (map a049084 $ a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Apr 27 2013

# To get 10000 terms. First make prime table: M:=10000; pl:=array(1..M); for i from 1 to M do pl[i]:=0; od: for i from 1 to M do if ithprime(i) > M then break; fi; pl[ithprime(i)]:=i; od:
# Decode Maple's amazing syntax for factoring integers: g:=proc(n) local e,p,t1,t2,t3,i,j,k; global pl; t1:=ifactor(n); t2:=nops(t1); if t2 = 2 and whattype(t1) <> `*` then p:=op(1,op(1,t1)); e:=op(2,t1); t3:=pl[p]*e; else
t3:=0; for i from 1 to t2 do j:=op(i,t1); if nops(j) = 1 then e:=1; p:=op(1,j); else e:=op(2,j); p:=op(1,op(1,j)); fi; t3:=t3+pl[p]*e; od: fi; t3; end; # N. J. A. Sloane, May 10 2006
A056239 := proc(n) add( numtheory[pi](op(1,p))*op(2,p), p = ifactors(n)[2]) ; end proc: # R. J. Mathar, Apr 20 2010
# alternative:
with(numtheory): a := proc (n) local B: B := proc (n) local nn, j, m: nn := op(2, ifactors(n)): for j to nops(nn) do m[j] := op(j, nn) end do: [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc: add(B(n)[i], i = 1 .. nops(B(n))) end proc: seq(a(n), n = 1 .. 130); # Emeric Deutsch, May 19 2015

a[1] = 0; a[2] = 1; a[p_?PrimeQ] := a[p] = PrimePi[p];
a[n_] := a[n] = Total[#[[2]]*a[#[[1]]] & /@ FactorInteger[n]]; a /@ Range[91] (* Jean-François Alcover, May 19 2011 *)
Table[Total[FactorInteger[n] /. {p_, c_} /; p > 0 :> PrimePi[p] c], {n, 91}] (* Michael De Vlieger, Jul 12 2017 *)

A056239(n) = if(1==n,0,my(f=factor(n)); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); \\ Antti Karttunen, Oct 26 2014, edited Jan 13 2020

from sympy import primepi, factorint
def A056239(n): return sum(primepi(p)*e for p, e in factorint(n).items()) # Chai Wah Wu, Jan 01 2023

(require 'factor) ;; Uses the function factor available in Aubrey Jaffer's SLIB Scheme library.
(define (A056239 n) (apply + (map A049084 (factor n))))
;; Antti Karttunen, Oct 26 2014

Totally additive with a(p) = PrimePi(p), where PrimePi(n) = A000720(n).
a(n) = Sum_{k=1..A001221(n)} A049084(A027748(k))*A124010(k). - Reinhard Zumkeller, Apr 27 2013
From Antti Karttunen, Oct 11 2014: (Start)
a(n) = n - A178503(n).
a(n) = A161511(A156552(n)).
a(n) = A227183(A243354(n)).
For all n >= 0:
a(A002110(n)) = A000217(n). [Cf. Henry Bottomley's comment above.]
a(A005940(n+1)) = A161511(n).
a(A243353(n)) = A227183(n).
Also, for all n >= 1:
a(A241909(n)) = A243503(n).
a(A122111(n)) = a(n).
a(A242424(n)) = a(n).
A248692(n) = 2^a(n). (End)
a(n) < A329605(n), a(n) = A001222(A108951(n)), a(A329902(n)) = A112778(n). - Antti Karttunen, Jan 14 2020

A000793 Landau's function g(n): largest order of permutation of n elements. Equivalently, largest LCM of partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 6, 12, 15, 20, 30, 30, 60, 60, 84, 105, 140, 210, 210, 420, 420, 420, 420, 840, 840, 1260, 1260, 1540, 2310, 2520, 4620, 4620, 5460, 5460, 9240, 9240, 13860, 13860, 16380, 16380, 27720, 30030, 32760, 60060, 60060, 60060, 60060, 120120

Offset: 0

N. J. A. Sloane

Also the largest orbit size (cycle length) for the permutation A057511 acting on Catalan objects (e.g., planar rooted trees, parenthesizations). - Antti Karttunen, Sep 07 2000
Grantham mentions that he computed a(n) for n <= 500000.
An easy lower bound is a(n) >= A002110(max{ m | A007504(m) <= n}), with strict inequality if n is not in A007504 (sum of the first m primes). Indeed, if A007504(m) <= n, the partition of n into the first m primes and maybe one additional term will have an LCM greater than or equal to primorial(m). If n > A007504(m) then a(n) >= (3/2)*A002110(m) by replacing the initial 2 by 3. But even for n = A007504(m), one has a(n) > A002110(m) for m > 8, since replacing 2+23 in 2+3+5+7+11+13+17+19+23 by 16+9, one has an LCM of 8*3*primorial(8) > primorial(9) because 24 > 23. - M. F. Hasler, Mar 29 2015
Maximum degree of the splitting field of a polynomial of degree n over a finite field, since over a finite field the degree of the splitting field is the least common multiple of the degrees of the irreducible polynomial factors of the polynomial. - Charles R Greathouse IV, Apr 27 2015
Maximum order of the elements in the symmetric group S_n. - Jianing Song, Dec 12 2021

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 6*x^6 + 12*x^7 + 15*x^8 + ...
From _Joerg Arndt_, Feb 15 2013: (Start)
The 15 partitions of 7 are the following:
[ #]  [ partition ]   lcm( parts )
[ 1]  [ 1 1 1 1 1 1 1 ]   1
[ 2]  [ 1 1 1 1 1 2 ]   2
[ 3]  [ 1 1 1 1 3 ]   3
[ 4]  [ 1 1 1 2 2 ]   2
[ 5]  [ 1 1 1 4 ]   4
[ 6]  [ 1 1 2 3 ]   6
[ 7]  [ 1 1 5 ]   5
[ 8]  [ 1 2 2 2 ]   2
[ 9]  [ 1 2 4 ]   4
[10]  [ 1 3 3 ]   3
[11]  [ 1 6 ]   6
[12]  [ 2 2 3 ]   6
[13]  [ 2 5 ]  10
[14]  [ 3 4 ]  12  (max)
[15]  [ 7 ]   7
The maximum (LCM) value attained is 12, so a(7) = 12.
(End)

J. Haack, "The Mathematics of Steve Reich's Clapping Music," in Bridges: Mathematical Connections in Art, Music and Science: Conference Proceedings, 1998, Reza Sarhangi (ed.), 87-92.
Edmund Georg Hermann Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea Publishing, NY 1953, p. 223.
J.-L. Nicolas, On Landau's function g(n), pp. 228-240 of R. L. Graham et al., eds., Mathematics of Paul Erdős I.
S. M. Shah, An inequality for the arithmetical function g(x), J. Indian Math. Soc., 3 (1939), 316-318. [See below for a scan of the first page.]
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 0..814 from David Wasserman)
Giedrius Alkauskas, Colouring monohedral tilings: defects and grain boundaries, 2024.
Joerg Arndt, Table of n, a(n) for n = 0..65536 (xz compressed).
Jan Brandts and A. Cihangir, Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group, arXiv preprint arXiv:1512.03044 [math.CO], 2015.
Marc Deléglise, Jean-Louis Nicolas, and Paul Zimmermann, Landau's function for one million billions, arXiv:0803.2160 [math.NT], 2008.
Marc Deléglise and Jean-Louis Nicolas, On the Largest Product of Primes with Bounded Sum, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.8.
Marc Deléglise and Jean-Louis Nicolas, The Landau function and the Riemann hypothesis, Univ. Lyon (France, 2019).
John D. Dixon and Daniel Panario, The Degree of the Splitting Field of a Random Polynomial over a Finite Field, The Electronic Journal of Combinatorics 11:1 (2004).
FindStat - Combinatorial Statistic Finder, The order of a permutation
Jon Grantham, The largest prime dividing the maximal order of an element of S_n, Math. Comput. 64, No. 209, 407-410 (1995).
Joel K. Haack, The Mathematics of Steve Reich's Clapping Music,in Bridges: Mathematical Connections in Art, Music and Science: Conference Proceedings, 1998, Reza Sarhangi (ed.), 87-92.
Jan Elmar Krauskopf, Andreas Rauh, and Andreas Hein, Discrete simulation of maypole braiding machines to create collision-free braiding programmes, Heliyon (2025), Art. No. e42917.
J. Kuzmanovich and A. Pavlichenkov, Finite groups of matrices whose entries are integers, Amer. Math. Monthly, 109 (2002), 173-186.
Jean-Pierre Massias, Majoration explicite de l'ordre maximum d'un élément du groupe symétrique, (French) [Explicit upper bound on the maximum order of an element of the symmetric group] Ann. Fac. Sci. Toulouse Math. (5) 6 (1984), no. 3-4, 269--281 (1985). MR0799599 (87a:11093).
Jean-Pierre Massias, Jean-Louis Nicolas, and Guy Robin, Effective bounds for the maximal order of an element in the symmetric group, Math. Comp. 53 (1989), no. 188, 665--678. MR0979940 (90e:11139).
W. Miller, The Maximum Order of an Element of Finite Symmetric Group, Am. Math. Monthly, Jun-Jul 1987, pp. 497-506.
Jean-Louis Nicolas, Sur l'ordre maximum d'un élément dans le groupe Sn des permutations, Acta Arith. 14, 315-332 (1968).
Jean-Louis Nicolas, Ordre maximal d'un élément du groupe S_n des permutations et 'highly composite numbers', Bull. Soc. Math. France 97 (1969), 129-191.
Jean-Louis Nicolas, Calcul de l'ordre maximum d'un élément du groupe symétrique S_n Revue française d'informatique et de recherche opérationnelle, série rouge 3.2 (1969): 43-50.
Jean-Louis Nicolas, Ordre maximal d'un e'le'ment du'un groupe de permutations, C. R. Acad. Sci. Paris, A, 270 (1970), 1-4. [Annotated scanned copy]
Roger D. Nussbaum, Lunel Verduyn, and M. Sjoerd, Asymptotic estimates for the periods of periodic points of non-expansive maps, Ergodic Theory Dynam. Systems 23 (2003), no. 4, pp. 1199-1226. MR1997973 (2004m:37033).
S. M. Shah, An inequality for the arithmetical function g(x) (scan of first page).
A. Wechsler, Re: Question (A000793(A007504(n)) =? A002110(n)), SeqFan mailing list, Mar 29 2015.
Eric Weisstein's World of Mathematics, Landau's Function
Herbert S. Wilf, The asymptotics of e^P(z) and the number of elements of each order in S_n, Bull. Amer. Math. Soc., 15.2 (1986), 225-232.
Index entries for sequences related to lcm's
Index entries for "core" sequences

Cf. A000792, A009490, A034891, A057731, A074859, A128305, A129759, A225655, A225648-A225650, A225651, A225636, A225558.

Row 1 of A225630.

a000793 = maximum . map (foldl lcm 1) . partitions where
   partitions n = ps 1 n where
      ps x 0 = [[]]
      ps x y = [t:ts | t <- [x..y], ts <- ps t (y - t)]
-- Reinhard Zumkeller, Mar 29 2015

A000793 := proc(n)
    l := 1:
    p := combinat[partition](n):
    for i from 1 to combinat[numbpart](n) do
        if ilcm( p[i][j] $ j=1..nops(p[i])) > l then
            l := ilcm( p[i][j] $ j=1..nops(p[i]))
        end if:
    end do:
    l ;
end proc:
seq(A000793(n),n=0..30) ; # James Sellers, Dec 07 2000
seq( max( op( map( x->ilcm(op(x)), combinat[partition](n)))), n=0..30); # David Radcliffe, Feb 28 2006
# third Maple program:
b:= proc(n, i) option remember; local p;
      p:= `if`(i<1, 1, ithprime(i));
      `if`(n=0 or i<1, 1, max(b(n, i-1),
           seq(p^j*b(n-p^j, i-1), j=1..ilog[p](n))))
    end:
a:=n->b(n, `if`(n<8, 3, numtheory[pi](ceil(1.328*isqrt(n*ilog(n)))))):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 16 2013

f[n_] := Max@ Apply[LCM, IntegerPartitions@ n, 1]; Array[f, 47] (* Robert G. Wilson v, Oct 23 2011 *)
b[n_, i_] := b[n, i] = Module[{p}, p = If[i<1, 1, Prime[i]]; If[n == 0 || i<1, 1, Max[b[n, i-1], Table[p^j*b[n-p^j, i-1], {j, 1, Log[p, n] // Floor}]]]]; a[n_] := b[n, If[n<8, 3, PrimePi[Ceiling[1.328*Sqrt[n*Log[n] // Floor]]]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 07 2014, after Alois P. Heinz *)

{a(n) = my(m, t, j, u); if( n<2, n>=0, m = ceil(n / exp(1)); t = ceil( (n/m)^m ); j=1; for( i=2, t, u = factor(i); u = sum( k=1, matsize(u)[1], u[k,1]^u[k,2]); if( u<=n, j=i)); j)}; /* Michael Somos, Oct 20 2004 */

c=0;A793=apply(t->eval(concat(Vec(t)[#Str(c++) .. -1])),select(t->#t,readstr("/tmp/b000793.txt")));A000793(n)=A793[n+1] \\ Assumes the b-file in the /tmp (or C:\tmp) folder. - M. F. Hasler, Mar 29 2015

A008475(n)=my(f=factor(n)); sum(i=1,#f~,f[i,1]^f[i,2]);
a(n)=
{
  if(n<2, return(1));
  forstep(i=ceil(exp(1.05315*sqrt(log(n)*n))), 2, -1,
    if(A008475(i)<=n, return(i))
  );
  1;
} \\ Charles R Greathouse IV, Apr 28 2015

{ \\ translated from code given by Tomas Rokicki
  my( N = 100 );
  my( V = vector(N,j,1) );
   forprime (i=2, N,  \\ primes i
      forstep (j=N, i,  -1,
         my( hi = V[j] );
         my( pp = i );  \\ powers of prime i
         while ( pp<=j,  \\ V[] is 1-based
             hi = max(if(j==pp, pp, V[j-pp]*pp), hi);
             pp *= i;
         );
         V[j] = hi;
      );
   );
   print( V );  \\ all values
\\   print( V[N] );  \\ just a(N)
\\  print("0 1");  for (n=1, N, print(n, " ", V[n]) );  \\ b-file
} \\ Joerg Arndt, Nov 14 2016

{a(n) = my(m=1); if( n<0, 0, forpart(v=n, m = max(m, lcm(Vec(v)))); m)}; /* Michael Somos, Sep 04 2017 */

from sympy import primerange
def aupton(N): # compute terms a(0)..a(N)
    V = [1 for j in range(N+1)]
    for i in primerange(2, N+1):
        for j in range(N, i-1, -1):
            hi = V[j]
            pp = i
            while pp <= j:
                hi = max((pp if j==pp else V[j-pp]*pp), hi)
                pp *= i
            V[j] = hi
    return V
print(aupton(47)) # Michael S. Branicky, Oct 09 2022 after Joerg Arndt

from sympy import primerange,sqrt,log,Rational
def f(N): # compute terms a(0)..a(N)
    V = [1 for j in range(N+1)]
    if N < 4:
        C = 2
    else:
        C = Rational(166,125)
    for i in primerange(C*sqrt(N*log(N))):
        for j in range(N, i-1, -1):
            hi = V[j]
            pp = i
            while pp <= j:
                hi = max(V[j-pp]*pp, hi)
                pp *= i
            V[j] = hi
    return V
# Philip Turecek, Mar 31 2023

def a(n):
  return max([lcm(l) for l in Partitions(n)])
# Philip Turecek, Mar 28 2023

;; A naive algorithm searching through all partitions of n:
(define (A000793 n) (let ((maxlcm (list 0))) (fold_over_partitions_of n 1 lcm (lambda (p) (set-car! maxlcm (max (car maxlcm) p)))) (car maxlcm)))
(define (fold_over_partitions_of m initval addpartfun colfun) (let recurse ((m m) (b m) (n 0) (partition initval)) (cond ((zero? m) (colfun partition)) (else (let loop ((i 1)) (recurse (- m i) i (+ 1 n) (addpartfun i partition)) (if (< i (min b m)) (loop (+ 1 i))))))))
;; From Antti Karttunen, May 17 2013.

Landau: lim_{n->oo} (log a(n)) / sqrt(n log n) = 1.
For bounds, see the Shah and Massias references.
For n >= 2, a(n) = max_{k} A008475(k) <= n. - Joerg Arndt, Nov 13 2016

More terms from David W. Wilson

Removed erroneous comment about a(16) which probably originated from misreading a(15)=105 as a(16) because of offset=0: a(16) = 4*5*7 = 140 is correct as it stands. - M. F. Hasler, Feb 02 2009

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

A034684 If n = p_1^e_1 * ... * p_k^e_k, p_1 < ... < p_k primes, then a(n) = min { p_i^e_i }.

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 7, 8, 9, 2, 11, 3, 13, 2, 3, 16, 17, 2, 19, 4, 3, 2, 23, 3, 25, 2, 27, 4, 29, 2, 31, 32, 3, 2, 5, 4, 37, 2, 3, 5, 41, 2, 43, 4, 5, 2, 47, 3, 49, 2, 3, 4, 53, 2, 5, 7, 3, 2, 59, 3, 61, 2, 7, 64, 5, 2, 67, 4, 3, 2, 71, 8, 73, 2, 3, 4, 7, 2, 79, 5, 81, 2, 83, 3, 5, 2, 3, 8, 89, 2, 7, 4

Offset: 1

Erich Friedman

a(1) = 1; for n > 1, smallest unitary divisor of n that is larger than 1.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A020639, A006530, A034684, A028233, A034699, A053585.
Cf. A034444, A034448, A052125.
Cf. A001221, A008475, A141809.

a034684 = minimum . a141809_row  -- Reinhard Zumkeller, Jan 29 2013

A034684[n_]:=Min[(#[[1]]^#[[2]])&/@FactorInteger[n]]; Array[A034684,100] (* Enrique Pérez Herrero, Nov 01 2011 *)

A034684(n) = {local(f,m);if(n==1,1,f=factor(n);m=f[1,1]^f[1,2];for(i=1,matsize(f)[1],m=min(m,f[i,1]^f[i,2]));m)} \\ Michael B. Porter, Jan 28 2010

a(n) = min{A141809(n,k): k=1..A001221(n)}. - Reinhard Zumkeller, Jan 29 2013

a(n) = n/A052125(n). - Amiram Eldar, Sep 16 2024

A141809 Irregular table: Row n (of A001221(n) terms, for n>=2) consists of the largest powers that divides n of each distinct prime that divides n. Terms are arranged by the size of the distinct primes. Row 1 = (1).

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 3, 7, 8, 9, 2, 5, 11, 4, 3, 13, 2, 7, 3, 5, 16, 17, 2, 9, 19, 4, 5, 3, 7, 2, 11, 23, 8, 3, 25, 2, 13, 27, 4, 7, 29, 2, 3, 5, 31, 32, 3, 11, 2, 17, 5, 7, 4, 9, 37, 2, 19, 3, 13, 8, 5, 41, 2, 3, 7, 43, 4, 11, 9, 5, 2, 23, 47, 16, 3, 49, 2, 25, 3, 17, 4, 13, 53, 2, 27, 5, 11, 8, 7, 3

Offset: 1

Leroy Quet, Jul 07 2008

In other words, except for row 1, row n contains the unitary prime power divisors of n, sorted by the prime. - Franklin T. Adams-Watters, May 05 2011

A034684(n) = smallest term of n-th row; A028233(n) = T(n,1); A053585(n) = T(n,A001221(n)); A008475(n) = sum of n-th row for n > 1. - Reinhard Zumkeller, Jan 29 2013

			60 has the prime factorization 2^2 * 3^1 * 5^1, so row 60 is (4,3,5).
From _M. F. Hasler_, Oct 12 2018: (Start)
The table starts:
    n : largest prime powers dividing n
    1 :  1
    2 :  2
    3 :  3
    4 :  4
    5 :  5
    6 :  2, 3
    7 :  7
    8 :  8
    9 :  9
   10 :  2, 5
   11 : 11
   12 :  4, 3
   etc. (End)

Reinhard Zumkeller, Rows n=1..10000 of triangle, flattened
Eric Weisstein's World of Mathematics, Prime Factorization

A027748, A124010 are used in a formula defining this sequence.
Cf. A001221 (row lengths), A008475 (row sums), A028233 (column 1), A034684 (row minima), A053585 (right edge).
Cf. A060175, A141810, A213925.

a141809 n k = a141809_row n !! (k-1)
a141809_row 1 = [1]
a141809_row n = zipWith (^) (a027748_row n) (a124010_row n)
a141809_tabf = map a141809_row [1..]
-- Reinhard Zumkeller, Mar 18 2012

f[{x_, y_}] := x^y; Table[Map[f, FactorInteger[n]], {n, 1, 50}] // Grid (* Geoffrey Critzer, Apr 03 2015 *)

A141809_row(n)=if(n>1, [f[1]^f[2]|f<-factor(n)~], [1]) \\ M. F. Hasler, Oct 12 2018, updated Aug 19 2022

T(n,k) = A027748(n,k)^A124010(n,k) for n > 1, k = 1..A001221(n). - Reinhard Zumkeller, Mar 15 2012

A046790 Positive numbers divisible by 8 or by the square of an odd prime.

Original entry on oeis.org

8, 9, 16, 18, 24, 25, 27, 32, 36, 40, 45, 48, 49, 50, 54, 56, 63, 64, 72, 75, 80, 81, 88, 90, 96, 98, 99, 100, 104, 108, 112, 117, 120, 121, 125, 126, 128, 135, 136, 144, 147, 150, 152, 153, 160, 162, 168, 169, 171, 175, 176, 180, 184, 189, 192, 196, 198, 200, 207, 208

Offset: 1

David W. Wilson, Dec 11 1999

This sequence has many equivalent definitions:
(D1) Positive numbers divisible by 8 or by the square of an odd prime. (We take this as the main definition, since it is the simplest.)
(D2) Moduli m for which there exist affine maps f:x->a*x + b modulo m, with a > 1, such that f has order m in the affine group. (For example, 8 is a term because f:x->(5x+1) mod 8 is a map with order 8 in the group of affine maps mod 8: the smallest power of f equal to identity is f^8. The maps x->x+1 always have this property, so are excluded from consideration.) - Emmanuel Amiot, Jul 28 2007
(D3) Numbers k such that A005361(k) < A003557(k). - Anthony Browne, Jun 03 2016
(D4) Numbers i such that there is a smaller positive number j such that (i+j)/2 and sqrt(i*j) are integers. (See A046791 for the smallest choice for j.) - David W. Wilson, Dec 11 1999
(D5) Numbers k such that A008475(k) is different from A001414(k). - Benoit Cloitre, Jan 11 2003
For a proof of the equivalence of definitions (D1)-(D5) see the Don Reble link.
(D6) Numbers m >= 8 having a divisor k^2 >= 4 such that m and m/k^2 are of the same parity. (See A046791 for the largest such k.) - Vladimir Shevelev, Jun 06 2006
(D7) Numbers that can be the semiperimeter of a isosceles triangle with integer sides and area. - Peter Kagey, May 17 2019
Closed under multiplication, which may be used to construct the sequence. - David A. Corneth, Jun 07 2016
Complement of A078779. - Omar E. Pol, Jun 11 2016
m is in this sequence if and only if m does not divide 2*radical(m). - Peter Luschny, Mar 05 2019
Verified up to a(290) = 1000, {a(n)} is identical to the sequence of group orders for which there exists at least one group G such that |Char(G)| is a nontrivial divisor of |Normal(G)|, where |Char(G)| is the number of characteristic subgroups of G and |Normal(G)| the number of normal subgroups of G. - Miles Englezou, Jul 20 2024

M. F. Hasler, Table of n, a(n) for n = 1..10000 (first 290 terms from N. J. A. Sloane).
Emmanuel Amiot, Autosimilar Melodies, J. Math. Music 2 (2008), no. 3, 157-180. DOI: 10.1080/17459730802598146.
Emmanuel Amiot, Mélodies autosimilaires (Self-Replicating Melodies) (in French).
Don Reble, Proof of equivalence of definitions (D1)-(D5), Jun 06 2016
Mathematics Stack Exchange user "George", Semiperimeter of isosceles Heronian triangles.

Cf. A001414, A003557, A005361, A008475, A046791.

ordreMax[a_, n_]:= Module[{mo, r, s, s0, gcd}, mo=MultiplicativeOrder[a,n]; s= s0=Mod[Sum[a^k,{k,0,mo-1}], n]; Max[Table[gcd=GCD[a-1,b];r=1; While[Mod[s *gcd, n]!=0, s=Mod[s0+a^mos,n];r++ ]; r*mo, {b,0,n-1} ]] ] ordreMax[n_] := Module[{candidats, m,t}, candidats = Select[Range[2,n-1], (GCD[n,# ]==1 && GCD[n, #-1]>1)&]; m=Max[t=Table[ordreMax[a,n], {a, candidats}] ]; {m,Part[candidats,Flatten@Position[t, m] ]}] Module[{resu}, Do[resu=ordreMax[n]; If[First[resu] >=n, Print[n ]], {n,4,200}]] (* This is for definition (D2). Emmanuel Amiot, Jul 28 2007 *)
Select[Range[210], Mod[#, 8] == 0 || AnyTrue[ Divisors[#], DivisorSigma[0, #] == 3 && Mod[#, 4] != 0 &] &] (* Carlos Eduardo Olivieri, Jun 07 2016 *)
Module[{upto=250,prs},prs=Prime[Range[2,PrimePi[Sqrt[upto]]]]^2;Join[ Range[ 8,upto,8],Select[Range[upto],AnyTrue[#/prs,IntegerQ]&]]] // Union (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 18 2018 *)

is(n)={n%8==0||!issquarefree(n>>!bittest(n,0))} \\ M. F. Hasler, Jun 07 2016

print([n for n in (1..208) if not ZZ(n).divides(2*radical(n))])  # Peter Luschny, Mar 05 2019

Let A(x) be the number of a(n) <= x. Then A(x) ~ (1 - 7/Pi^2)*x = 0.2907517...*x as x goes to infinity. - Vladimir Shevelev, Jun 07 2016

Entry revised by N. J. A. Sloane, with help from Don Reble and several OEIS editors, Jun 07 2016

A029908 Starting with n, repeatedly sum prime factors (with multiplicity) until reaching 0 or a fixed point. Then a(n) is the fixed point (or 0).

Original entry on oeis.org

0, 2, 3, 4, 5, 5, 7, 5, 5, 7, 11, 7, 13, 5, 5, 5, 17, 5, 19, 5, 7, 13, 23, 5, 7, 5, 5, 11, 29, 7, 31, 7, 5, 19, 7, 7, 37, 7, 5, 11, 41, 7, 43, 5, 11, 7, 47, 11, 5, 7, 5, 17, 53, 11, 5, 13, 13, 31, 59, 7, 61, 5, 13, 7, 5, 5, 67, 7, 5, 5, 71, 7, 73, 5, 13, 23, 5, 5, 79, 13, 7, 43, 83, 5, 13

Offset: 1

Dann Toliver

nonn

That is, the sopfr function (see A001414) applied repeatedly until reaching 0 or a fixed point.
For n > 1, the sequence reaches a fixed point which is either 4 or a prime.
A002217(n) is number of terms in sequence from n to a(n). - Reinhard Zumkeller, Apr 08 2003
Because sopfr(n) <= n (with equality at 4 and the primes), the first appearance of all primes is in the natural order: 2,3,5,7,11,... . - Zak Seidov, Mar 14 2011
The terms 0, 2, 3 and 4 occur exactly once, because no number > 5 can have factors that sum to be < 5, and therefore can never enter a trajectory that will drop below 5. - Christian N. K. Anderson, May 19 2013
For all primes p, where p is contained in A001359, then a(p^2) = p + 2. (A006512). Proof: p^2 has prime factors (p, p). This sums to 2p. 2p has factors (2, p). This sums to p + 2. Since p was the lesser of a twin prime, then p + 2 is the greater of a twin prime. - Ryan Bresler, Nov 04 2021

			20 -> 2+2+5 = 9 -> 3+3 = 6 -> 2+3 = 5, so a(20)=5.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Sum of Prime Factors.

Cf. A001414 (sum of prime factors of n).
Cf. A081758, A002217, A075860.
Cf. A001414, A056239, A008475, A082081, A082083.

f:= proc(n) option remember;
if isprime(n) then n
else `procname`(add(x[1]*x[2], x = ifactors(n)[2]))
fi
end proc:
f(1):= 0: f(4):= 4:
map(f, [$1..100]); # Robert Israel, Apr 27 2015

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] ep[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}] slog[x_] := slog[x_] := Apply[Plus, ba[x]*ep[x]] Table[FixedPoint[slog, w], {w, 1, 128}]
f[n_] := Plus @@ Flatten[ Table[ #[[1]], {#[[2]]}] & /@ FactorInteger@n]; Array[ FixedPoint[f, # ] &, 87] (* Robert G. Wilson v, Jan 18 2006 *)
fz[n_]:=Plus@@(#[[1]]*#[[2]]&/@FactorInteger@n); Array[FixedPoint[fz,#]&,1000] (* Zak Seidov, Mar 14 2011 *)

from sympy import factorint
def a(n, pn):
    if n == pn:
        return n
    else:
        return a(sum(p*e for p, e in factorint(n).items()), n)
print([a(i, None) for i in range(1, 100)]) # Gleb Ivanov, Nov 05 2021

A051613 a(n) = partitions of n into powers of distinct primes (1 not considered a power).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 0, 3, 2, 3, 2, 4, 3, 4, 4, 4, 8, 4, 8, 6, 9, 8, 10, 10, 13, 12, 13, 16, 16, 19, 17, 21, 23, 23, 25, 29, 31, 31, 31, 37, 40, 42, 44, 48, 49, 54, 55, 64, 67, 68, 70, 77, 84, 90, 92, 99, 102, 108, 115, 127, 133, 135, 138, 150, 165, 171, 183, 186, 198, 201, 220

Offset: 0

Vladeta Jovovic

			a(16) = 8 because we can write 16 = 2^4 = 3+13 = 5+11 = 3^2+7 = 2+3+11 = 2+3^2+5 = 2^3+3+5 = 2^2+5+7.

T. D. Noe, Table of n, a(n) for n = 0..1000
J. Bamberg, G. Cairns and D. Kilminster, The crystallographic restriction, permutations and Goldbach's conjecture, Amer. Math. Monthly, 110 (March 2003), 202-209.

Cf. A023894, A009490, A054685, A008475, A106245, A000961, A025473, A106244.

import Data.MemoCombinators (memo3, integral)
a051613' = p 1 2 where
   p x _ 0 = 1
   p x k m | m < qq       = 0
           | mod x q == 0 = p x (k + 1) m
           | otherwise    = p (q * x) (k + 1) (m - qq) + p x (k + 1) m
           where q = a025473 k; qq = a000961 k
-- Reinhard Zumkeller, Nov 23 2015

b:= proc(n,i) option remember; local p;
      p:= `if`(i<1, 1, ithprime(i));
      `if`(n=0, 1, `if`(i<1, 0, b(n,i-1)+
      add(b(n-p^j, i-1), j=1..ilog[p](n))))
    end:
a:= n-> b(n, numtheory[pi](n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Feb 15 2013

max = 70; f[x_] := Product[ 1 + Sum[x^(Prime[n]^k), {k, 1, If[n > 4, 1, 6]}], {n, 1, PrimePi[max]}]; CoefficientList[ Series[f[x], {x, 0, max}] , x](* Jean-François Alcover, Sep 12 2012 *)

first(n)=my(x='x,pr=O(x^(n+1))+1); forprime(p=sqrtint(n)+1,n, pr*=1+x^p); forprime(p=2,sqrtint(n), pr*=1+sum(e=1,logint(n,2), x^p^e)); Vec(pr) \\ Charles R Greathouse IV, Jun 25 2017

a(n) = number of m such that A008475(m) = n.

G.f.: Product_{p prime} (1 + Sum_{k >= 1} x^(p^k)).

Better description from David W. Wilson, Apr 19 2000

cp's OEIS Frontend

A056239 If n = Product_{k >= 1} (p_k)^(c_k) where p_k is k-th prime and c_k >= 0 then a(n) = Sum_{k >= 1} k*c_k.

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A000793 Landau's function g(n): largest order of permutation of n elements. Equivalently, largest LCM of partitions of n.

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

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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.

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A034684 If n = p_1^e_1 * ... * p_k^e_k, p_1 < ... < p_k primes, then a(n) = min { p_i^e_i }.

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