A210208 -id:A210208 - cp's OEIS frontend

A027750 Triangle read by rows in which row n lists the divisors of n.

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

Offset: 1

N. J. A. Sloane

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
Let there be an infinite number of tiles, each labeled with a positive integer m, initially placed on square m of an infinite 1D board. At step n, the leftmost unblocked tile (i.e., the top tile of the leftmost nonempty stack) moves forward exactly m squares, where m is its label. Tiles that land on the same square form a stack, and only the top tile of any stack may move. This sequence records the label m of the tile that moves at step n. - Ali Sada, May 23 2025
All divisors of a positive integer n form a finite set. Extending divisibility to n = 0 by using the definition (k|n <=> exists m such that m*k = n) makes the set of divisors infinite, suggesting the definition was not intended for zero, as arithmetic functions typically apply to n >= 1. So to preserve a core property when generalizing (cardinality), one can define divisors of n >= 0 as the fixed points of the greatest common divisor on the set [n] = {0, 1, 2, ..., n}. By this definition, the divisors of 0 are {0}, since 0|0 and gcd(0, 0) = 0. This definition is not circular because the gcd can be effectively calculated using the Euclidean algorithm. (Cf. links.) - Peter Luschny, Jun 02 2025

			Triangle begins:
  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;
  ...
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

Franklin T. Adams-Watters, Rows 1..1000, flattened
Franklin T. Adams-Watters, Rows 1..10000
Peter Luschny, A Python program using gcd for n >= 0.
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

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.

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

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

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

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

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

from sympy import divisors
for n in range(1, 16):
    print(divisors(n)) # Indranil Ghosh, Mar 30 2017

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

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

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

A073093 Number of prime power divisors of n.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Aug 24 2002

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 kBenoit 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

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

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

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

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

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

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 *)
PrimeOmega[Range[100]] + 1 (* Paolo Xausa, Nov 23 2024 *)

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

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

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

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 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

A023888 Sum of prime power divisors of n (1 included).

Original entry on oeis.org

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

Offset: 1

Olivier Gérard

nonn

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

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

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

Cf. A008475, A159077.

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

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

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

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

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

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

A183091 a(n) is the product of the divisors p^k of n where p is prime and k >= 1.

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Dec 25 2010

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

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

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

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

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

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

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 *)

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

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

A210241 Partial sums of A073093.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Mar 19 2012

nonn

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

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

Cf. A022559, A073093, A083342, A210208.

a210241 n = a210241_list !! (n-1)
a210241_list = scanl1 (+) a073093_list

Accumulate[Array[PrimeOmega[#] + 1 &, 100]] (* Amiram Eldar, Apr 05 2025 *)

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

a(n) = n * (log(log(n)) + B_2 + 1) + O(n/log(n)), where B_2 = A083342. - Amiram Eldar, Apr 05 2025

A248906 Binary representation of prime power divisors of n: Sum_{p^k | n} 2^(A065515(p^k)-1).

Original entry on oeis.org

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

Offset: 1

Franklin T. Adams-Watters, Mar 06 2015

nonn

			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.

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

Cf. A246655, A065515, A034729, A000961.

Cf. A095874, A210208, A073093, A000079.

a248906 = sum . map ((2 ^) . subtract 2 . a095874) . tail . a210208_row
-- Reinhard Zumkeller, Mar 07 2015

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

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

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A027750 Triangle read by rows in which row n lists the divisors of n.

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A034699 Largest prime power factor of n.

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A073093 Number of prime power divisors of n.

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A023888 Sum of prime power divisors of n (1 included).

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A183091 a(n) is the product of the divisors p^k of n where p is prime and k >= 1.

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A210241 Partial sums of A073093.

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