A141809 -id:A141809 - cp's OEIS frontend

A027748 Irregular triangle in which first row is 1, n-th row (n > 1) lists distinct prime factors of n.

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

Offset: 1

N. J. A. Sloane

Number of terms in n-th row is A001221(n) for n > 1.
From Reinhard Zumkeller, Aug 27 2011: (Start)
A008472(n) = Sum_{k=1..A001221(n)} T(n,k), n>1;
A007947(n) = Product_{k=1..A001221(n)} T(n,k);
A006530(n) = Max_{k=1..A001221(n)} T(n,k).
A020639(n) = Min_{k=1..A001221(n)} T(n,k).
(End)
Subsequence of A027750 that lists the divisors of n. - Michel Marcus, Oct 17 2015

			Triangle begins:
   1;
   2;
   3;
   2;
   5;
   2, 3;
   7;
   2;
   3;
   2, 5;
  11;
   2, 3;
  13;
   2, 7;
  ...

T. D. Noe, Rows n=1..2048 of triangle, flattened
Eric Weisstein's World of Mathematics, Distinct Prime Factors.

Cf. A000027, A001221, A001222 (with repetition), A027746, A141809, A141810.
a(A013939(A000040(n))+1) = A000040(n).
Cf. A020639, A027750.
A284411 gives column medians.

import Data.List (unfoldr)
a027748 n k = a027748_tabl !! (n-1) !! (k-1)
a027748_tabl = map a027748_row [1..]
a027748_row 1 = [1]
a027748_row n = unfoldr fact n where
   fact 1 = Nothing
   fact x = Just (p, until ((> 0) . (`mod` p)) (`div` p) x)
            where p = a020639 x  -- smallest prime factor of x
-- Reinhard Zumkeller, Aug 27 2011

with(numtheory): [ seq(factorset(n), n=1..100) ];

Flatten[ Table[ FactorInteger[n][[All, 1]], {n, 1, 62}]](* Jean-François Alcover, Oct 10 2011 *)

print1(1);for(n=2,20,f=factor(n)[,1];for(i=1,#f,print1(", "f[i]))) \\ Charles R Greathouse IV, Mar 20 2013

from sympy import primefactors
for n in range(2, 101):
    print([i for i in primefactors(n)]) # Indranil Ghosh, Mar 31 2017

More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)

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

A115627 Irregular triangle read by rows: T(n,k) = multiplicity of prime(k) as a divisor of n!.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 1, 4, 2, 1, 4, 2, 1, 1, 7, 2, 1, 1, 7, 4, 1, 1, 8, 4, 2, 1, 8, 4, 2, 1, 1, 10, 5, 2, 1, 1, 10, 5, 2, 1, 1, 1, 11, 5, 2, 2, 1, 1, 11, 6, 3, 2, 1, 1, 15, 6, 3, 2, 1, 1, 15, 6, 3, 2, 1, 1, 1, 16, 8, 3, 2, 1, 1, 1, 16, 8, 3, 2, 1, 1, 1, 1

Offset: 2

Franklin T. Adams-Watters, Jan 26 2006

The factorization of n! is n! = 2^T(n,1)*3^T(n,2)*...*p_(pi(n))^T(n,pi(n)) where p_k = k-th prime, pi(n) = A000720(n).
Nonzero terms of A085604; T(n,k) = A085604(n,k), k = 1..A000720(n). - Reinhard Zumkeller, Nov 01 2013
For n=2, 3, 4 and 5, all terms of the n-th row are odd. Are there other such rows? - Michel Marcus, Nov 11 2018
From Gus Wiseman, May 15 2019: (Start)
Differences between successive rows are A067255, so row n is the sum of the first n row-vectors of A067255 (padded with zeros on the right so that all n row-vectors have length A000720(n)). For example, the first 10 rows of A067255 are
  {}
  1
  0 1
  2 0
  0 0 1
  1 1 0
  0 0 0 1
  3 0 0 0
  0 2 0 0
  1 0 1 0
with column sums (8,4,2,1), which is row 10.
(End)
For all prime p > 7, 3*p > 2*nextprime(p), so for any n > 21 there will always be a prime p dividing n! with exponent 2 and there are no further rows with all entries odd. - Charlie Neder, Jun 03 2019

			From _Gus Wiseman_, May 09 2019: (Start)
Triangle begins:
   1
   1  1
   3  1
   3  1  1
   4  2  1
   4  2  1  1
   7  2  1  1
   7  4  1  1
   8  4  2  1
   8  4  2  1  1
  10  5  2  1  1
  10  5  2  1  1  1
  11  5  2  2  1  1
  11  6  3  2  1  1
  15  6  3  2  1  1
  15  6  3  2  1  1  1
  16  8  3  2  1  1  1
  16  8  3  2  1  1  1  1
  18  8  4  2  1  1  1  1
(End)
m such that 5^m||101!: floor(log(101)/log(5)) = 2 terms. floor(101/5) = 20. floor(20/5) = 4. So m = u_1 + u_2 = 20 + 4 = 24. - _David A. Corneth_, Jun 22 2014

T. D. Noe, Rows n = 2..300, flattened
H. T. Davis, Tables of the Mathematical Functions, Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX. [Annotated scan of pages 204-208 of Volume 2.] See Table 2 on page 206.
Wenguang Zhai, On the prime power factorization of n!, Journal of Number Theory, Volume 129, Issue 8, August 2009, pages 1820-1836.
Index entries for sequences related to factorial numbers

Row lengths are A000720.
Row-sums are A022559.
Row-products are A135291.
Row maxima are A011371.
Columns include A011371, A054861, A027868, A054896, A090617, A064458, A090620.
Cf. A090622, A090623, A000142, A115628.
Cf. A085604, A141809.
Cf. A034876, A067255, A071626, A076934, A322583, A325272, A325273, A325276, A325508, A325509.

a115627 n k = a115627_tabf !! (n-2) !! (k-1)
a115627_row = map a100995 . a141809_row . a000142
a115627_tabf = map a115627_row [2..]
-- Reinhard Zumkeller, Nov 01 2013

A115627 := proc(n,k) local d,p; p := ithprime(k) ; n-add(d,d=convert(n,base,p)) ; %/(p-1) ; end proc: # R. J. Mathar, Oct 29 2010

Flatten[Table[Transpose[FactorInteger[n!]][[2]], {n, 2, 20}]] (* T. D. Noe, Apr 10 2012 *)
T[n_, k_] := Module[{p, jm}, p = Prime[k]; jm = Floor[Log[p, n]]; Sum[Floor[n/p^j], {j, 1, jm}]]; Table[Table[T[n, k], {k, 1, PrimePi[n]}], {n, 2, 20}] // Flatten (* Jean-François Alcover, Feb 23 2015 *)

a(n)=my(i=2);while(n-primepi(i)>1,n-=primepi(i);i++);p=prime(n-1);sum(j=1,log(i)\log(p),i\=p) \\ David A. Corneth, Jun 21 2014

T(n,k) = Sum_{i=1..inf} floor(n/(p_k)^i). (Although stated as an infinite sum, only finitely many terms are nonzero.)

T(n,k) = Sum_{i=1..floor(log(n)/log(p_k))} floor(u_i) where u_0 = n and u_(i+1) = floor((u_i)/p_k). - David A. Corneth, Jun 22 2014

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

A287352 Irregular triangle T(n,k) = A112798(n,1) followed by first differences of A112798(n).

Original entry on oeis.org

0, 1, 2, 1, 0, 3, 1, 1, 4, 1, 0, 0, 2, 0, 1, 2, 5, 1, 0, 1, 6, 1, 3, 2, 1, 1, 0, 0, 0, 7, 1, 1, 0, 8, 1, 0, 2, 2, 2, 1, 4, 9, 1, 0, 0, 1, 3, 0, 1, 5, 2, 0, 0, 1, 0, 3, 10, 1, 1, 1, 11, 1, 0, 0, 0, 0, 2, 3, 1, 6, 3, 1, 1, 0, 1, 0, 12, 1, 7, 2, 4, 1, 0, 0, 2, 13

Offset: 1

Michael De Vlieger, May 23 2017

Irregular triangle T(n,k) = first differences of indices of prime divisors p of n.
Row lengths = (big) Omega(n) = A001222(n).
Row sums = A061395(n).
Row maxima = A286469(n).
We can concatenate the rows 1 <= n <= 28 as none of the values of k in this range exceed 9: {0, 1, 2, 10, 3, 11, 4, 100, 20, 12, 5, 101, 6, 13, 21, 1000, 7, 110, 8, 102, 22, 14, 9, 1001, 30, 15, 200, 103}; a(29) = {10}, which would require a digit greater than 9.
a(1) = 0 by convention.
a(0) is not defined (i.e., null set). a(n) is defined for positive nonzero n.
a(p) = A000720(p) for p prime.
a(p^e) = A000720(p) followed by (e - 1) zeros.
a(Product(p^e)) is the concatenation of the a(p^e) of the unitary prime power divisors p^e of n, sorted by the prime p (i.e. the function a(n) mapped across the terms of row n of A141809).
a(A002110(n)) = an array of n 1s.
T(n,k) could be used to furnish A054841(n). We read data in row n of T(n,k). If T(n,1) = 0, then write 0. If T(n,1) > 0, then increment the k-th place from the right. For k > 1, increment the k-th place to the right of the last-incremented place.
T(n,k) can be used to render n in decimal. If T(n,1) = 0, then write 1. If T(n,1) > 0, then multiply 1 by A000720(T(n,1)). For k > 1, multiply the previous product by pi(x) = A000720(x) of the running total of T(n,k) for each k.
Ignoring zeros in row n > 1 and decoding the remaining values of T(n,k) as immediately above yields the squarefree kernel of n = A007947(n).
Leading zeros of a(n) are trimmed, but as in decimal notation numbers that include leading zeros symbolize the same n as without them. Zeros that precede nonzero values merely multiply implicit 1 by itself until we encounter nonzero values. Thus, {0,0,2} = 1*1*pi(2) = 3, as {2} = pi(2) = 3. Because of this no row n > 1 has 0 for k = 1 of T(n,k).
Interpreting n written in binary as a row of a(n) yields A057335(n).

			a(1) = {0} by convention.
a(2) = {pi(2)} = {1}.
a(4) = {pi(2), pi(2) - pi(2)}, = {1, 0} since 4 = 2 * 2.
a(6) = {pi(2), pi(3) - pi(2)} = {1, 1} since 6 = 2 * 3.
a(12) = {pi(2), pi(2) - pi(2), pi(3) - pi(2) - pi(2)} = {1, 0, 1}, since 12 = 2 * 2 * 3.
The triangle starts:
   1:  0;
   2:  1;
   3:  2;
   4:  1, 0;
   5:  3;
   6:  1, 1;
   7:  4;
   8:  1, 0, 0;
   9:  2, 0;
  10:  1, 2;
  11:  5;
  12:  1, 0, 1;
  13:  6;
  14:  1, 3;
  15:  2, 1;
  16:  1, 0, 0, 0;
  17:  7;
  18:  1, 1, 0;
  19:  8;
  20:  1, 0, 2;
       ...

Michael De Vlieger, Table of n, a(n) for n = 1..15568 (rows 1 <= n <= 5000).

Cf. A000720, A001222, A007947, A054841, A057335, A061395, A112798, A141809, A286469.

Table[Prepend[Differences@ #, First@ #] & Flatten[FactorInteger[n] /. {p_, e_} /; p > 0 :> ConstantArray[PrimePi@ p, e]], {n, 41}] // Flatten (* Michael De Vlieger, May 23 2017 *)

T(n,1) = A117798(n,1); T(n,k) = A117798(n,k) - A117798(n, k - 1) for 2 <= k <= A001222(n).

A375136 Number of maximal strictly increasing runs in the weakly increasing prime factors of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 04 2024

nonn

For n > 1, this is one more than the number of adjacent equal terms in the multiset of prime factors of n.

			The prime factors of 540 are {2,2,3,3,3,5}, with maximal strictly increasing runs ({2},{2,3},{3},{3,5}), so a(540) = 4.

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

For compositions we have A124768, row-lengths of A374683, sum A374684.
For sum of prime indices we have A374706.
Row-lengths of A375128.
A112798 lists prime indices:
- distinct A001221
- length A001222
- leader A055396
- sum A056239
- reverse A296150
Cf. A034296, A046660, A066328, A141199, A141809, A279790, A320324, A333213, A358836, A374700.

Table[Length[Split[Flatten[ConstantArray@@@FactorInteger[n]],Less]],{n,100}]

For n > 1, a(n) = A046660(n) + 1 = A001222(n) - A001221(n) + 1.

A028236 If n = Product (p_j^k_j), a(n) = numerator of Sum 1/p_j^k_j.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 1, 1, 7, 1, 7, 1, 9, 8, 1, 1, 11, 1, 9, 10, 13, 1, 11, 1, 15, 1, 11, 1, 31, 1, 1, 14, 19, 12, 13, 1, 21, 16, 13, 1, 41, 1, 15, 14, 25, 1, 19, 1, 27, 20, 17, 1, 29, 16, 15, 22, 31, 1, 47, 1, 33, 16, 1, 18, 61, 1, 21, 26, 59, 1, 17, 1, 39, 28, 23, 18, 71, 1, 21, 1, 43

Offset: 1

N. J. A. Sloane

			Fractions begin with 1, 1/2, 1/3, 1/4, 1/5, 5/6, 1/7, 1/8, 1/9, 7/10, 1/11, 7/12, ...

Klaus Brockhaus, Table of n, a(n) for n = 1..10000

Denominator is n (A000027).

Cf. A001221, A141809, A179119.

a028236 n = sum $ map (div n) $ a141809_row n
-- Reinhard Zumkeller, Nov 10 2013

a028236:=func< k | k eq 1 select 1 else Numerator(&+[ f[i, 1]^-f[i,2]: i in [1..#f] ]) where f is Factorization(k) >; [ a028236(n):n in [1..82] ]; // Klaus Brockhaus, Nov 06 2010

a[n_] := n * Total[1/Power @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Sep 29 2023 *)

Fraction is additive with a(p^e) = 1/p^e.
a(n) = Sum_{k=1..A001221(n)} n/A141809(n,k). - Reinhard Zumkeller, Nov 10 2013
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/k = Sum_{p prime} 1/(p*(p+1)) = 0.330229... (A179119). - Amiram Eldar, Sep 29 2023

More terms from Erich Friedman

A080737 a(1) = a(2) = 0; for n > 2, the least dimension of a lattice possessing a symmetry of order n.

Original entry on oeis.org

0, 0, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 6, 8, 16, 6, 18, 6, 8, 10, 22, 6, 20, 12, 18, 8, 28, 6, 30, 16, 12, 16, 10, 8, 36, 18, 14, 8, 40, 8, 42, 12, 10, 22, 46, 10, 42, 20, 18, 14, 52, 18, 14, 10, 20, 28, 58, 8, 60, 30, 12, 32, 16, 12, 66, 18, 24, 10, 70, 10, 72, 36, 22, 20, 16, 14

Offset: 1

N. J. A. Sloane, Mar 08 2003

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J. Bamberg, G. Cairns and D. Kilminster, The crystallographic restriction, permutations and Goldbach's conjecture, Amer. Math. Monthly, 110 (March 2003), 202-209.
Savinien Kreczman, Luca Prigioniero, Eric Rowland, and Manon Stipulanti, Magic numbers in periodic sequences, Univ. Liège (Belgium, 2023). See p. 7.

Cf. A080736, A080738, A080739, A080740, A067240, A000010, A141809.

See A152455 for another version.

a080737 n = a080737_list !! (n-1)
a080737_list = 0 : (map f [2..]) where
f n | mod n 4 == 2 = a080737 $ div n 2
| otherwise = a067240 n
-- Reinhard Zumkeller, Jun 13 2012, Jun 11 2012

a[1] = a[2] = 0; a[p_?PrimeQ] := a[p] = p-1; a[n_] := a[n] = If[Length[fi = FactorInteger[n]] == 1, EulerPhi[n], Total[a /@ (fi[[All, 1]]^fi[[All, 2]])]]; Table[a[n], {n, 1, 78}] (* Jean-François Alcover, Jun 20 2012 *)

for(n=1,78,k=0; if(n>1,f=factor(n); k=sum(j=1,matsize(f)[1],eulerphi(f[j,1]^f[j,2])); if(f[1,1]==2&&f[1,2]==1,k--)); print1(k,",")) \\ Klaus Brockhaus, Mar 10 2003

For n > 2, a(2^r) = 2^(r-1) with r>1, a(p^r) = phi(p^r) with p > 2 prime, r >= 1, where phi is Euler's function A000010; in general if a(Product p_i^e_i) = Sum a(p_i^e_i).

More terms from Klaus Brockhaus, Mar 10 2003

cp's OEIS Frontend

A027748 Irregular triangle in which first row is 1, n-th row (n > 1) lists distinct prime factors of n.

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

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A115627 Irregular triangle read by rows: T(n,k) = multiplicity of prime(k) as a divisor 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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