A210208 -id:A210208

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A027750

Triangle read by rows in which row n lists the divisors of n.

+10
505

1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 6, 1, 7, 1, 2, 4, 8, 1, 3, 9, 1, 2, 5, 10, 1, 11, 1, 2, 3, 4, 6, 12, 1, 13, 1, 2, 7, 14, 1, 3, 5, 15, 1, 2, 4, 8, 16, 1, 17, 1, 2, 3, 6, 9, 18, 1, 19, 1, 2, 4, 5, 10, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 2, 3, 4, 6, 8, 12, 24, 1, 5, 25, 1, 2, 13, 26, 1, 3, 9, 27, 1, 2, 4, 7, 14, 28, 1, 29

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Or, in the list of natural numbers (A000027), replace n with its divisors.

This gives the first elements of the ordered pairs (a,b) a >= 1, b >= 1 ordered by their product ab.

Also, row n lists the largest parts of the partitions of n whose parts are not distinct. - Omar E. Pol, Sep 17 2008

Concatenation of n-th row gives A037278(n). - Reinhard Zumkeller, Aug 07 2011

{A210208(n,k): k=1..A073093(n)} subset of {T(n,k): k=1..A000005(n)} for all n. - Reinhard Zumkeller, Mar 18 2012

Row sums give A000203. Right border gives A000027. - Omar E. Pol, Jul 29 2012

Indices of records are in A006218. - Irina Gerasimova, Feb 27 2013

The number of primes in the n-th row is omega(n) = A001221(n). - Michel Marcus, Oct 21 2015

The row polynomials P(n,x) = Sum_{k=1..A000005(n)} T(n,k)*x^k with composite n which are irreducible over the integers are given in A292226. - Wolfdieter Lang, Nov 09 2017

T(n,k) is also the number of parts in the k-th partition of n into equal parts (see example). - Omar E. Pol, Nov 20 2019

LINKS

Franklin T. Adams-Watters, Rows 1..1000, flattened

Franklin T. Adams-Watters, Rows 1..10000

Omar E. Pol, Illustration of initial terms, (2009).

Eric Weisstein's World of Mathematics, Divisor

Wikipedia, Table of divisors

Index entries for sequences related to divisors of numbers

FORMULA

a(A006218(n-1) + k) = k-divisor of n, 1 <= k <= A000005(n). - Reinhard Zumkeller, May 10 2006

T(n,k) = n / A056538(n,k) = A056538(n,n-k+1), 1 <= k <= A000005(n). - Reinhard Zumkeller, Sep 28 2014

EXAMPLE

Triangle begins:

1, 2;

1, 3;

1, 2, 4;

1, 5;

1, 2, 3, 6;

1, 7;

1, 2, 4, 8;

1, 3, 9;

1, 2, 5, 10;

1, 11;

1, 2, 3, 4, 6, 12;

...

For n = 6 the partitions of 6 into equal parts are [6], [3,3], [2,2,2], [1,1,1,1,1,1], so the number of parts are [1, 2, 3, 6] respectively, the same as the divisors of 6. - Omar E. Pol, Nov 20 2019

MAPLE

seq(op(numtheory:-divisors(a)), a = 1 .. 20) # Matt C. Anderson, May 15 2017

MATHEMATICA

Flatten[ Table[ Flatten [ Divisors[ n ] ], {n, 1, 30} ] ]

PROG

(Magma) [Divisors(n) : n in [1..20]];

(Haskell)

a027750 n k = a027750_row n !! (k-1)

a027750_row n = filter ((== 0) . (mod n)) [1..n]

a027750_tabf = map a027750_row [1..]

-- Reinhard Zumkeller, Jan 15 2011, Oct 21 2010

(PARI) v=List(); for(n=1, 20, fordiv(n, d, listput(v, d))); Vec(v) \\ Charles R Greathouse IV, Apr 28 2011

(Python)

from sympy import divisors

for n in range(1, 16):

print(divisors(n)) # Indranil Ghosh, Mar 30 2017

CROSSREFS

Cf. A000005 (row length), A001221, A027749, A027751, A056534, A056538, A127093, A135010, A161700, A163280, A240698 (partial sums of rows), A240694 (partial products of rows), A247795 (parities), A292226, A244051.

KEYWORD

nonn,easy,tabf,look

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

A034699

Largest prime power factor of n.

+10
67

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from T. D. Noe)

FORMULA

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

MATHEMATICA

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 *)

PROG

(Haskell)

a034699 = last . a210208_row

-- Reinhard Zumkeller, Mar 18 2012, Feb 14 2012

(PARI) 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

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

(Python)

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

CROSSREFS

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

Numbers n ordered by a(n): A305325.

KEYWORD

nonn,easy,nice

AUTHOR

David W. Wilson

STATUS

approved

A073093

Number of prime power divisors of n.

+10
42

1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 4, 2, 3, 3, 5, 2, 4, 2, 4, 3, 3, 2, 5, 3, 3, 4, 4, 2, 4, 2, 6, 3, 3, 3, 5, 2, 3, 3, 5, 2, 4, 2, 4, 4, 3, 2, 6, 3, 4, 3, 4, 2, 5, 3, 5, 3, 3, 2, 5, 2, 3, 4, 7, 3, 4, 2, 4, 3, 4, 2, 6, 2, 3, 4, 4, 3, 4, 2, 6, 5, 3, 2, 5, 3, 3, 3, 5, 2, 5, 3, 4, 3, 3, 3, 7, 2, 4, 4, 5, 2, 4, 2, 5, 4

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Also, number of prime divisors of 2n (counted with multiplicity).

A001221(n) < a(n) <= A000005(n) for all n; a(n)=A001221(n)+1 iff n is squarefree (A005117); a(n)=A000005(n) iff n is a prime power (A000961).

a(n) is also the number of k<n such that the resultant of the k-th cyclotomic polynomial and the n-th cyclotomic polynomial is not 1. It is well known that if (k,n)=1, res(polcyclo(n),polcyclo(k))=1. - Benoit Cloitre, Oct 13 2002

a(n) is also 1 + the number of divisors of n with omega(d)=1, where omega is A001221. - Enrique Pérez Herrero, Nov 05 2009

Length of n-th row of triangle A210208. - Reinhard Zumkeller, Mar 18 2012

a(n) depends only on the prime signature of n with a(A025487(n)) = 1, 2, 3, 3, 4, 4, 5, 5, 4, 6, 5, 6, 5, 7, 6, 7 ,.. = A036041(n)+1; (n>=1). - R. J. Mathar, May 28 2017

LINKS

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

T. M. Apostol, Resultants of Cyclotomic Polynomials, Proc. Amer. Math. Soc. 24, 457-462, 1970.

T. M. Apostol, The Resultant of the Cyclotomic Polynomials Fm(ax) and Fn(bx), Math. Comput. 29, 1-6, 1975.

Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

FORMULA

If n = Product (p_j^k_j), a(n) = 1 + Sum (k_j).

a(n) = bigomega(n)+1 = A001222(n)+1 = A001222(2*n).

a(n) = if n=1 then 1 else a(A032742(n)) + 1. - Reinhard Zumkeller, Sep 24 2009

a(n) = max { a(d) ; d<n and d|n } + 1, if n > 1. - David W. Wilson, Dec 08 2010

a(n) = Sum_{k = 1 .. A001221(n)} A010055(A027750(n,k)). - Reinhard Zumkeller, Mar 18 2012

G.f.: x/(1 - x) + Sum_{k>=2} floor(1/omega(k))*x^k/(1 - x^k), where omega(k) is the number of distinct prime factors (A001221). - Ilya Gutkovskiy, Jan 04 2017

MAPLE

seq(numtheory:-bigomega(n)+1, n=1..1000); # Robert Israel, Sep 06 2015

MATHEMATICA

f[n_] := Plus @@ Flatten[ Table[1, {#[[2]]}] & /@ FactorInteger[n]]; Table[ f[2n], {n, 105}] (* Robert G. Wilson v, Dec 23 2004 *)

A001221[n_] := (Length[ FactorInteger[n]]); SetAttributes[A001221, Listable]; A073093[n_]:=Length[Select[A001221[Divisors[n]], # == 1 &]]; (* Enrique Pérez Herrero, Nov 05 2009 *)

PROG

(PARI) a(n)=sum(k=1, n, if(1-polresultant(polcyclo(n), polcyclo(k)), 1, 0))

(PARI) A073093(n)=bigomega(n)+1 \\ M. F. Hasler, Dec 08 2010

(MuPAD) numlib::Omega (2*n)$ n=1..105 // Zerinvary Lajos, May 13 2008

(Haskell)

a073093 = length . a210208_row -- Reinhard Zumkeller, Mar 18 2012

(Magma) [n eq 1 select 1 else &+[p[2]: p in Factorization(n)]+1: n in [1..100]]; // Vincenzo Librandi, Jan 06 2017

CROSSREFS

Cf. A000961, A023888, A054372. Bisection of A001222.

KEYWORD

nonn,easy

AUTHOR

Reinhard Zumkeller, Aug 24 2002

STATUS

approved

A023888

Sum of prime power divisors of n (1 included).

+10
11

1, 3, 4, 7, 6, 6, 8, 15, 13, 8, 12, 10, 14, 10, 9, 31, 18, 15, 20, 12, 11, 14, 24, 18, 31, 16, 40, 14, 30, 11, 32, 63, 15, 20, 13, 19, 38, 22, 17, 20, 42, 13, 44, 18, 18, 26, 48, 34, 57, 33, 21, 20, 54, 42, 17, 22, 23, 32, 60, 15, 62, 34, 20, 127, 19, 17, 68, 24, 27

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Sum of n-th row of triangle A210208. [Reinhard Zumkeller, Mar 18 2012]

LINKS

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

FORMULA

a(n) = A000203(n) - A035321(n) = A023889(n) + 1.

a(1) = 1, a(p) = p+1, a(pq) = p+q+1, a(pq...z) = (p+q+...+z) + 1, a(p^k) = (p^(k+1)-1) / (p-1), for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.

G.f.: x/(1 - x) + Sum_{k>=2} floor(1/omega(k))*k*x^k/(1 - x^k), where omega(k) is the number of distinct prime factors (A001221). - Ilya Gutkovskiy, Jan 04 2017

EXAMPLE

For n = 12, set of such divisors is {1, 2, 3, 4}; a(12) = 1+2+3+4 = 10. From

MAPLE

f:= n -> 1 + add((t[1]^(t[2]+1)-t[1])/(t[1]-1), t=ifactors(n)[2]):

map(f, [$1..100]); # Robert Israel, Jan 04 2017

MATHEMATICA

Array[ Plus @@ (Select[ Divisors[ # ], (Length[ FactorInteger[ # ] ]<=1)& ])&, 70 ]

PROG

(PARI) for(n=1, 100, s=1; fordiv(n, d, if((ispower(d, , &z)&&isprime(z)) || isprime(d), s+=d)); print1(s, ", "))

(Haskell)

a023888 = sum . a210208_row -- Reinhard Zumkeller, Mar 18 2012

(PARI)

a(n) = {

my(f = factor(n), fsz = matsize(f)[1]);

1 + sum(k = 1, fsz, f[k, 1]*(f[k, 1]^f[k, 2] - 1)\(f[k, 1]-1));

};

vector(100, n, a(n)) \\ Gheorghe Coserea, Jan 04 2017

CROSSREFS

Cf. A008475, A159077.

KEYWORD

nonn

AUTHOR

Olivier Gérard

STATUS

approved

A183091

a(n) is the product of the divisors p^k of n where p is prime and k >= 1.

+10
5

1, 2, 3, 8, 5, 6, 7, 64, 27, 10, 11, 24, 13, 14, 15, 1024, 17, 54, 19, 40, 21, 22, 23, 192, 125, 26, 729, 56, 29, 30, 31, 32768, 33, 34, 35, 216, 37, 38, 39, 320, 41, 42, 43, 88, 135, 46, 47, 3072, 343, 250, 51, 104, 53, 1458

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Product of n-th row of triangle A210208. - Reinhard Zumkeller, Mar 18 2012

LINKS

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

FORMULA

a(n) = A007955(n) / A183092(n).

Multiplicative with a(p^k) = p^(k*(k+1)/2).

The Dirichlet g.f. of a(n) / abs(A153038(n)) is Product_{k >= 0} zeta(s+k). - Álvar Ibeas, Nov 10 2014

EXAMPLE

For n = 12, set of such divisors is {1, 2, 3, 4}; a(12) = 1*2*3*4 = 24.

MAPLE

A183091 := proc(n) local a, d; a := 1 ; for d in numtheory[divisors](n) minus {1} do if nops( numtheory[factorset](d)) = 1 then a := a*d; end if; end do: a ; end proc: # R. J. Mathar, Apr 14 2011

MATHEMATICA

Table[Product[d, {d, Select[Divisors[n], Length[FactorInteger[#]] == 1 &]}], {n, 1, 54}] (* Geoffrey Critzer, Mar 18 2015 *)

f[p_, e_] := p^(e*(e+1)/2); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 31 2023 *)

PROG

(Haskell)

a183091 = product . a210208_row -- Reinhard Zumkeller, Mar 18 2012

(PARI) a(n)=my(f=factor(n)); prod(i=1, #f~, f[i, 1]^binomial(f[i, 2]+1, 2)) \\ Charles R Greathouse IV, Nov 11 2014

CROSSREFS

Cf. A007955, A023888, A153038, A183092, A210208.

KEYWORD

nonn,easy,mult

AUTHOR

Jaroslav Krizek, Dec 25 2010

STATUS

approved

A210241

Partial sums of A073093.

+10
2

1, 3, 5, 8, 10, 13, 15, 19, 22, 25, 27, 31, 33, 36, 39, 44, 46, 50, 52, 56, 59, 62, 64, 69, 72, 75, 79, 83, 85, 89, 91, 97, 100, 103, 106, 111, 113, 116, 119, 124, 126, 130, 132, 136, 140, 143, 145, 151, 154, 158, 161, 165, 167, 172, 175, 180, 183, 186, 188

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Number of terms in the first n rows of triangle A210208;

a(n) = A022559(n) + n.

LINKS

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

PROG

(Haskell)

a210241 n = a210241_list !! (n-1)

a210241_list = scanl1 (+) a073093_list

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Mar 19 2012

STATUS

approved

A248906

Binary representation of prime power divisors of n: Sum_{p^k | n} 2^(A065515(p^k)-1).

+10
2

0, 1, 2, 5, 8, 3, 16, 37, 66, 9, 128, 7, 256, 17, 10, 549, 1024, 67, 2048, 13, 18, 129, 4096, 39, 8200, 257, 16450, 21, 32768, 11, 65536, 131621, 130, 1025, 24, 71, 262144, 2049, 258, 45, 524288, 19, 1048576, 133, 74, 4097, 2097152, 551, 4194320, 8201

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

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

FORMULA

Additive with a(p^k) = Sum_{j=1..k} 2^(A065515(p^j)-1).

a(A051451(k)) = 2^k - 1.

a(n) = Sum_{k=1..A073093(n)} 2^(A095874(A210208(n,k))-2). - Reinhard Zumkeller, Mar 07 2015

EXAMPLE

The prime power divisors of 12 are 2, 3, and 4. These are indices 1, 2, and 3 in the list of prime powers, so a(12) = 2^(1-1) + 2^(2-1) + 2^(3-1) = 7.

PROG

(PARI) al(n) = my(r=vector(n), pps=[p| p <- [1..n], isprimepower(p)], p2); for(k=1, #pps, p2=2^(k-1); forstep(j=pps[k], n, pps[k], r[j]+=p2)); r

(Haskell)

a248906 = sum . map ((2 ^) . subtract 2 . a095874) . tail . a210208_row

-- Reinhard Zumkeller, Mar 07 2015

CROSSREFS

Cf. A246655, A065515, A034729, A000961.

Cf. A095874, A210208, A073093, A000079.

KEYWORD

nonn

AUTHOR

Franklin T. Adams-Watters, Mar 06 2015

STATUS

approved

A330043

Product of largest prime power factors of numbers <= n.

+10
0

1, 1, 2, 6, 24, 120, 360, 2520, 20160, 181440, 907200, 9979200, 39916800, 518918400, 3632428800, 18162144000, 290594304000, 4940103168000, 44460928512000, 844757641728000, 4223788208640000, 29566517460480000, 325231692065280000, 7480328917501440000, 59842631340011520000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Partial products of A034699.

LINKS

Table of n, a(n) for n=0..24.

FORMULA

A001221(a(n)) = A000720(n).

MATHEMATICA

a[n_] := Product[Max[#[[1]]^#[[2]] & /@ FactorInteger@k], {k, 1, n}]; Table[a[n], {n, 0, 24}]

CROSSREFS