A000188 -id:A000188 - cp's OEIS frontend

A336321 a(n) = A122111(A225546(n)).

1, 2, 3, 4, 7, 5, 19, 6, 9, 11, 53, 10, 131, 23, 13, 8, 311, 15, 719, 22, 29, 59, 1619, 14, 49, 137, 21, 46, 3671, 17, 8161, 12, 61, 313, 37, 25, 17863, 727, 139, 26, 38873, 31, 84017, 118, 39, 1621, 180503, 20, 361, 77, 317, 274, 386093, 33, 71, 58, 733, 3673, 821641, 34, 1742537, 8167, 87, 18, 151, 67, 3681131, 626, 1627, 41, 7754077, 35, 16290047

Offset: 1

Antti Karttunen and Peter Munn, Jul 17 2020

nonn

A122111 and A225546 are both self-inverse permutations of the positive integers based on prime factorizations, and they share further common properties. For instance, they map the prime numbers to powers of 2: A122111 maps the k-th prime to 2^k, whereas A225546 maps it to 2^2^(k-1).

In composing these permutations, this sequence maps the squarefree numbers, as listed in A019565, to the prime numbers in increasing order; and the list of powers of 2 to the "normal" numbers (A055932), as listed in A057335.

			From _Peter Munn_, Jan 04 2021: (Start)
In this set of examples we consider [a(n)] as a function a(.) with an inverse, a^-1(.).
First, a table showing mapping of the powers of 2:
  n     a^-1(2^n) =    2^n =        a(2^n) =
        A001146(n-1)   A000079(n)   A057335(n)
  0             (1)         1            1
  1               2         2            2
  2               4         4            4
  3              16         8            6
  4             256        16            8
  5           65536        32           12
  6      4294967296        64           18
  ...
Next, a table showing mapping of the squarefree numbers, as listed in A019565 (a lexicographic ordering by prime factors):
  n   a^-1(A019565(n))   A019565(n)      a(A019565(n))   a^2(A019565(n))
      Cf. {A337533}      Cf. {A005117}   = prime(n)      = A033844(n-1)
  0              1               1             (1)               (1)
  1              2               2               2                 2
  2              3               3               3                 3
  3              8               6               5                 7
  4              6               5               7                19
  5             12              10              11                53
  6             18              15              13               131
  7            128              30              17               311
  8              5               7              19               719
  9             24              14              23              1619
  ...
As sets, the above columns are A337533, A005117, A008578, {1} U A033844.
Similarly, we get bijections between sets A000290\{0} -> {1} U A070003; and {1} U A335740 -> A005408 -> A066207.
(End)

A122111 composed with A225546.
Cf. A336322 (inverse permutation).
Other sequences used in a definition of this sequence: A000040, A000188, A019565, A248663, A253550, A253560.
Sequences used to express relationship between terms of this sequence: A003159, A003961, A297002, A334747.
Cf. A057335.
A mapping between the binary tree sequences A334866 and A253563.
Lists of sets (S_1, S_2, ... S_j) related by the bijection defined by the sequence: (A000290\{0}, {1} U A070003), ({1} U A001146, A000079, A055932), ({1} U A335740, A005408, A066207), (A337533, A005117, A008578, {1} U A033844).

a(n) = A122111(A225546(n)).
Alternative definition: (Start)
Write n = m^2 * A019565(j), where m = A000188(n), j = A248663(n).
a(1) = 1; otherwise for m = 1, a(n) = A000040(j), for m > 1, a(n) = A253550^j(A253560(a(m))).
(End)
a(A000040(m)) = A033844(m-1).
a(A001146(m)) = 2^(m+1).
a(2^n) = A057335(n).
a(n^2) = A253560(a(n)).
For n in A003159, a(2n) = b(a(n)), where b(1) = 2, b(n) = A253550(n), n >= 2.
More generally, a(A334747(n)) = b(a(n)).
a(A003961(n)) = A297002(a(n)).
a(A334866(m)) = A253563(m).

A056042 a(n) = n!/(k!)^2, where k is the largest number such that (k!)^2 divides n!.

Original entry on oeis.org

1, 2, 6, 6, 30, 20, 140, 70, 630, 7, 77, 924, 12012, 3432, 51480, 12870, 218790, 48620, 923780, 184756, 3879876, 705432, 16224936, 2704156, 67603900, 10400600, 280816200, 178296, 5170584, 155117520, 4808643120, 601080390, 19835652870

Offset: 1

Labos Elemer, Jul 25 2000

nonn

Least integer of the form n!/{(n-k)!}^2.

Similar to but different from A001405.

			E.g. for n=9, 10, 11, 12, a(n)=630, 7, 77, 924 while the corresponding central binomial coefficients are 126, 252, 462, 924 respectively.

Ivan Neretin, Table of n, a(n) for n = 1..1000

Cf. A000142, A001405, A000188, A055772, A096123, A096125.

f[n_] := Min[ Select[ Table[ n!/(n - k)!^2, {k, n}], IntegerQ[ # ] &]]; Table[ f[n], {n, 33}] (Robert G. Wilson v)

A059591 Squarefree part of n^2+1.

Original entry on oeis.org

1, 2, 5, 10, 17, 26, 37, 2, 65, 82, 101, 122, 145, 170, 197, 226, 257, 290, 13, 362, 401, 442, 485, 530, 577, 626, 677, 730, 785, 842, 901, 962, 41, 1090, 1157, 1226, 1297, 1370, 5, 1522, 1601, 2, 1765, 74, 1937, 2026, 2117, 2210, 2305, 2402, 2501, 2602

Offset: 0

Marc LeBrun, Jan 25 2001

Related to period-1 continued fractions [z,z,z,...].

			a(7)=2 since 7^2+1 = 50 = 25*2 = (5^2)*2.

Enrique Pérez Herrero, Table of n, a(n) for n = 0..5000

Cf. A007913, A059592, A000188.

A:= proc(n)
local F;
F:= select(t -> t[2]::odd, ifactors(n^2+1)[2]);
mul(t[1],t=F)
end proc:
map(A, [$0..100]); # Robert Israel, Jun 18 2015

Table[Times @@ Flatten[Table[#1, {#2}] & @@@ Select[FactorInteger[n^2 + 1], OddQ@ Last@ # &]], {n, 120}] (* Michael De Vlieger, Jun 19 2015 *)

a(n) = core(n^2+1); \\ Michel Marcus, Jun 18 2015

a(n) = A007913(n^2+1).

a(n)*A059592(n)^2 = n^2+1.

A160696 Largest k such that k^2 divides prime(n)+1.

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 3, 2, 2, 1, 4, 1, 1, 2, 4, 3, 2, 1, 2, 6, 1, 4, 2, 3, 7, 1, 2, 6, 1, 1, 8, 2, 1, 2, 5, 2, 1, 2, 2, 1, 6, 1, 8, 1, 3, 10, 2, 4, 2, 1, 3, 4, 11, 6, 1, 2, 3, 4, 1, 1, 2, 7, 2, 2, 1, 1, 2, 13, 2, 5, 1, 6, 4, 1, 2, 8, 1, 1, 1, 1, 2, 1, 12, 1, 2, 2, 15, 1, 1, 4, 6, 4, 2, 2, 10, 6, 1, 3, 2, 1, 2, 3, 2

Offset: 1

Reinhard Zumkeller, May 24 2009

nonn

A160697 and A160698 give record values and where they occur.

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

Cf. A000188, A008864, A049097, A049098.

a(n) = core(prime(n)+1, 1)[2]; \\ Michel Marcus, Nov 06 2022

a(A049097(n)) = 1; a(A049098(n)) > 1;

a(n) = A000188(A008864(n)).

A283484 Odd bisection of A283983; square root of the largest square dividing A277324.

Original entry on oeis.org

1, 1, 3, 1, 3, 3, 15, 1, 3, 15, 45, 15, 15, 15, 105, 1, 3, 105, 225, 525, 1575, 1125, 1575, 105, 105, 525, 1575, 525, 105, 105, 1155, 1, 3, 1155, 1575, 3675, 7875, 275625, 55125, 5775, 17325, 275625, 4134375, 55125, 55125, 275625, 121275, 1155, 1155, 40425, 385875, 202125, 606375, 1929375, 606375, 5775, 8085, 40425, 121275, 40425, 1155, 1155, 15015, 1, 3

Offset: 0

Antti Karttunen, Mar 25 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..2048

Cf. A000188, A001222, A277324, A283983, A284265.

A003961[p_?PrimeQ] := A003961[p] = Prime[ PrimePi[p] + 1]; A003961[1] = 1; A003961[n_] := A003961[n] = Times @@ ( A003961[First[#]] ^ Last[#] & ) /@ FactorInteger[n]  (* after Jean-François Alcover, Dec 01 2011 *); A260443[n_]:= If[n<2, n + 1, If[EvenQ[n], A003961[A260443[n/2]], A260443[(n - 1)/2] * A260443[(n + 1)/2]]]; A275812[n_]:= PrimeOmega[n] - If[n<2, 0,Count[Transpose[FactorInteger[n]][[2]], 1]]; A277324[n_]:=A260443[2n + 1];  A000188[n_]:=  Sum[Boole[Mod[i^2, n] == 0], {i, n}]; Table[A000188[A277324[n]], {n, 0, 50}] (* Indranil Ghosh, Mar 28 2017 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From Michel Marcus
A260443(n) = if(n<2, n+1, if(n%2, A260443(n\2)*A260443(n\2+1),  A003961(A260443(n\2)))); \\ Cf. Charles R Greathouse IV's code for "ps" in A186891 and A277013.
A277324(n) = A260443((2*n)+1);
A000188(n) = core(n, 1)[2]; \\ This function from Michel Marcus, Feb 27 2013
A283484(n) = A000188(A277324(n));

(define (A283484 n) (A000188 (A277324 n)))
(define (A283484 n) (A283983 (+ n n 1)))

a(n) = A283983((2*n)+1).
a(n) = A000188(A277324(n)).
A001222(a(n)) = A284265(n).

A284264 a(n) = A001222(A283983(n)).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Mar 25 2017

nonn

a(n) = Sum_{c} floor(c/2), where c ranges over each coefficient of terms c * x^k in the Stern polynomial B(n,x), thus sum of the halved terms (for odd terms floored down) on row n of table A125184.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Cf. A000188, A002487, A001222, A125184, A260443, A277700, A283983, A284265 (odd bisection), A284272.

Cf. A023758 (gives the positions of zeros).

A003961[p_?PrimeQ] := A003961[p] = Prime[ PrimePi[p] + 1]; A003961[1] = 1; A003961[n_] := A003961[n] = Times @@ ( A003961[First[#]] ^ Last[#] & ) /@ FactorInteger[n]  (* after Jean-François Alcover, Dec 01 2011 *); A260443[n_]:= If[n<2, n + 1, If[EvenQ[n], A003961[A260443[n/2]], A260443[(n - 1)/2] * A260443[(n + 1)/2]]]; A000188[n_]:= Sum[Boole[Mod[i^2, n] == 0], {i, n}]; Table[PrimeOmega[A000188[A260443[n]]], {n, 0, 120}] (* Indranil Ghosh, Mar 28 2017 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From Michel Marcus
A260443(n) = if(n<2, n+1, if(n%2, A260443(n\2)*A260443(n\2+1), A003961(A260443(n\2)))); \\ Cf. Charles R Greathouse IV's code for "ps" in A186891 and A277013.
A000188(n) = core(n, 1)[2]; \\ This function from Michel Marcus, Feb 27 2013
A283983(n) = A000188(A260443(n));
A284264(n) = bigomega(A283983(n));

(define (A284264 n) (/ (- (A002487 n) (A277700 n)) 2))

a(n) = A001222(A283983(n)).
Other identities and observations. For all n >= 0:
a(2n) = a(n).
a(n) = (1/2) * (A002487(n) - A277700(n)).
2*a(n) <= A284272(n).

A365549 The number of exponentially odd divisors of the square root of the largest square dividing n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Sep 08 2023

First differs from A278908, A307848, A323308 and A358260 at n = 64.

The number of exponentially odd divisors of the largest square dividing n is the same as the number of squares dividing n, A046951(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000188, A005117, A013662, A046951, A322483.

Cf. A278908, A307848, A323308, A358260.

f[p_, e_] := 2 + Floor[(e-2)/4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> 2 + (x-2)\4, factor(n)[, 2]));

a(n) = A322483(A000188(n)).
a(n) >= 1 with equality if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = 2 + floor((e-2)/4).
Dirichlet g.f.: zeta(s) * zeta(4*s) * Product_{p prime} (1 + 1/p^(2*s) - 1/p^(4*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(4) * Product_{p prime} (1 + 1/p^2 - 1/p^4) = 1.54211628314015874165... .

A381522 Sequence where k is appended after every k^2 occurrences of 1, with multiple values following a 1 listed in order.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 2, 4, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 5, 1, 1, 3, 1, 2, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 2, 3, 6, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 4, 1, 7, 1, 5, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 4, 8, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 6

Offset: 1

Jwalin Bhatt, Feb 26 2025

nonn

The frequencies of the terms follow the zeta distribution with parameter value 2.
The geometric mean approaches exp(-zeta'(2)/zeta(2)) A381456 in the limit. In general, if the sequence was formed by every k^s occurrences, it would approach e^(-zeta'(s)/zeta(s)).
Considered as an irregular triangle, the n-th row lists the divisors of the square root of the largest square dividing n.

			After every 4 ones we see a 2, after every 9 ones we see a 3 and so on.

Jwalin Bhatt, Table of n, a(n) for n = 1..10000
Wikipedia, Zeta distribution

Cf. A000188, A084580, A381456, A381900, A382093.

lista(n)={my(L=List()); for(n=1, n, fordiv(sqrtint(n/core(n)), d, listput(L,d))); Vec(L[1..n])} \\ Andrew Howroyd, Feb 26 2025

from itertools import islice
def zeta_distribution_generator():
    num_ones, num_reached = 0, 1
    while num_ones := num_ones+1:
        yield 1
        for num in range(2, num_reached+2):
            if num_ones % (num*num) == 0:
                yield num
                num_reached += num == num_reached+1
A381522 = list(islice(zeta_distribution_generator(), 120))

A229297 Number of solutions to x^2 == n (mod 2n) for 0 <= x < 2n.

Original entry on oeis.org

1, 0, 1, 2, 1, 0, 1, 0, 3, 0, 1, 2, 1, 0, 1, 4, 1, 0, 1, 2, 1, 0, 1, 0, 5, 0, 3, 2, 1, 0, 1, 0, 1, 0, 1, 6, 1, 0, 1, 0, 1, 0, 1, 2, 3, 0, 1, 4, 7, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 3, 8, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 5, 2, 1, 0, 1, 4, 9, 0, 1, 2, 1, 0

Offset: 1

José María Grau Ribas, Sep 22 2013

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A000188, A001620, A229294, A229295, A229296.

A[n_] := Sum[If[Mod[a^2, 2*n] == n, 1, 0], {a, 0, 2*n - 1}]; Array[A, 100]
f[p_, e_] := If[OddQ[e], p^((e - 1)/2), p^(e/2)]; f[2, e_] := If[OddQ[e], 0, 2^(e/2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 01 2023 *)

a(n)={sum(i=0, 2*n-1, i^2%(2*n)==n)} \\ Andrew Howroyd, Aug 06 2018

a(n)={if(valuation(n,2)%2==1, 0, core(n, 1)[2])} \\ Andrew Howroyd, Aug 07 2018

From Andrew Howroyd, Aug 07 2018: (Start)
Multiplicative with a(2^e) = 0 for odd e and 2^floor(e/2) for even e, and a(p^e) = p^floor(e/2) for p>=3. [corrected by Georg Fischer, Aug 01 2022].
a(n) = A000188(n) for odd n, a(2^k) = 1 + (-1)^k for k > 0. (End)
From Amiram Eldar, Jan 01 2023: (Start)
Dirichlet g.f.: zeta(2*s-1)*zeta(s)/(zeta(2*s)*(1+1/2^s)).
Sum_{k=1..n} a(k) ~ (n*log(n) + (3*gamma + log(2)/3 - 2*zeta'(2)/zeta(2) - 1)*n)*2/Pi^2, where gamma is Euler's constant (A001620). (End)

A254732 a(n) is the least k > n such that n divides k^2.

Original entry on oeis.org

2, 4, 6, 6, 10, 12, 14, 12, 12, 20, 22, 18, 26, 28, 30, 20, 34, 24, 38, 30, 42, 44, 46, 36, 30, 52, 36, 42, 58, 60, 62, 40, 66, 68, 70, 42, 74, 76, 78, 60, 82, 84, 86, 66, 60, 92, 94, 60, 56, 60, 102, 78, 106, 72, 110, 84, 114, 116, 118, 90, 122, 124, 84, 72

Offset: 1

Peter Kagey, Feb 06 2015

A073353(n) <= a(n) <= 2*n. Any prime that divides n must also divide a(n), and because n divides (2*n)^2.

Are all terms even? -Harvey P. Dale, Aug 07 2025

			a(12) = 18 because 12 divides 18^2, but 12 does not divide 13^2, 14^2, 15^2, 16^2, or 17^2.

Peter Kagey, Table of n, a(n) for n = 1..5000

Cf. A254733 (similar, with k^3), A254734 (similar, with k^4), A073353 (similar, with limit m->infinity of k^m).

Cf. A253905.

a254732 n = head [k | k <- [n + 1 ..], mod (k ^ 2) n == 0]
-- Reinhard Zumkeller, Feb 07 2015

lk[n_]:=Module[{k=n+1},While[!Divisible[k^2,n],k++];k]; Array[lk,70] (* Harvey P. Dale, Nov 05 2017 *)
Table[Module[{k=n+1},While[PowerMod[k,2,n]!=0,k++];k],{n,70}] (* Harvey P. Dale, Aug 07 2025 *)

a(n)=for(k=n+1,2*n,if(k^2%n==0,return(k)))
vector(100,n,a(n)) \\ Derek Orr, Feb 06 2015

a(n)=my(t=factorback(factor(n)[,1])); forstep(k=n+t,2*n,t,if(k^2%n==0, return(k))) \\ Charles R Greathouse IV, Feb 07 2015

def A254732(n):
    k = n + 1
    while pow(k,2,n):
        k += 1
    return k # Chai Wah Wu, Feb 15 2015

def a(n)
  (n+1..2*n).find { |k| k**2 % n == 0 }
end

a(n) = sqrt(n*A072905(n)).
a(n) = A019554(n)*(A000188(n)+1).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 + zeta(3)/zeta(2) = 1 + A253905 = 1.73076296940143849872... . - Amiram Eldar, Feb 17 2024

cp's OEIS Frontend

A336321 a(n) = A122111(A225546(n)).

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A056042 a(n) = n!/(k!)^2, where k is the largest number such that (k!)^2 divides n!.

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A059591 Squarefree part of n^2+1.

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A160696 Largest k such that k^2 divides prime(n)+1.

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A283484 Odd bisection of A283983; square root of the largest square dividing A277324.

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Mathematica

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A284264 a(n) = A001222(A283983(n)).

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A365549 The number of exponentially odd divisors of the square root of the largest square dividing n.

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Mathematica

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A381522 Sequence where k is appended after every k^2 occurrences of 1, with multiple values following a 1 listed in order.

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A229297 Number of solutions to x^2 == n (mod 2n) for 0 <= x < 2n.