A053186 -id:A053186 - cp's OEIS frontend

A056895 If the smallest prime with a square excess of n is p then a(n)^2 = p - n.

1, 1, 2, 3, 6, 5, 4, 9, 8, 7, 6, 7, 10, 15, 8, 9, 24, 11, 12, 21, 16, 13, 12, 13, 16, 15, 14, 17, 18, 31, 20, 27, 20, 23, 18, 19, 22, 21, 20, 23, 24, 23, 24, 27, 28, 29, 30, 25, 38, 39, 26, 31, 30, 35, 28, 45, 34, 31, 42, 31, 34, 33, 32, 33, 36, 35, 34, 75, 40, 37, 36, 41, 48, 45

Offset: 1

Henry Bottomley, Jul 05 2000

nonn

			a(4)=3 because the smallest prime with a square excess of 4 is 13 and 13 - 4 = 3^2.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000040, A000196, A002496, A048760, A053186.

Cf. A056892, A056893, A056894, A056896, A056897, A056898.

a = {}; Do[p = 2; While[n != p - (r = Floor@Sqrt[p])^2, p = NextPrime[p]]; AppendTo[a, r], {n, 74}]; a (* Ivan Neretin, May 02 2019 *)

a(n) = {my(p=2); while(n != p-sqrtint(p)^2, p = nextprime(p+1)); sqrtint(p - n);} \\ Michel Marcus, May 05 2019

a(n) = sqrt(A056893(n)-n) = A000196(A056893(n)) = sqrt(A056894(n)).

A094761 a(n) = n + (square excess of n).

Original entry on oeis.org

0, 1, 3, 5, 4, 6, 8, 10, 12, 9, 11, 13, 15, 17, 19, 21, 16, 18, 20, 22, 24, 26, 28, 30, 32, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 64, 66, 68, 70, 72, 74, 76, 78

Offset: 0

N. J. A. Sloane, Jun 10 2004

The trajectory of n under iteration of m -> a(m) is eventually constant iff n is a perfect square.
Conjecture (verified up to 727): the numbers not in this sequence are those of A008865. - R. J. Mathar, Jan 23 2009
From Maon Wenders, Jul 01 2012: (Start)
Proof of conjecture:
(1) (n+2)^2 - n^2 = n^2 + 4n + 4 - n^2 = 4n + 4
(2) (n+1)^2 - n^2 = n^2 + 2n + 1 - n^2 = 2n + 1
(3) (n+1) + square excess of (n+1) - (n + square excess of n) = 2, except when (n+1) is a square, where a(n) collapses back to (n+1)
(4) so, cause of (2) and (3), the sequence has blocks of even and odd numbers starting with an even or odd square, m^2 and of length 2m+1:
    0,
    1, 3, 5,
    4, 6, 8, 10, 12,
    9, 11, 13, 15, 17, 19, 21,
    16, 18, 20, 22, 24, 26, 28, 30, 32,
    ...
(5) such a block of 2m+1 numbers fills in all even or odd numbers between
    n^2 and (n+2)^2
(6) but, because a block starts n^2 + 0, n^2 + 2, n^2 + 4, ..., the last number in such a block is n^2 + 2*(2n+1-1) = n^2 + 4n
(7) so the numbers n^2 + 4n + 2 = (n+2)^2 - 2 are missing.
End of proof. (End)

S. H. Weintraub, An interesting recursion, Amer. Math. Monthly, 111 (No. 6, 2004), 528-530.

Cf. A053186, A094763, A094764, A094765.

f[n_] := 2 n - (Floor@ Sqrt@ n)^2; Table[f@ n, {n, 0, 71}] (* Robert G. Wilson v, Jan 23 2009 *)

a(n)=2*n-sqrtint(n)^2 \\ Charles R Greathouse IV, Jul 01 2012

a(n) = n + A053186(n).

A094763 Trajectory of 2 under repeated application of the map n -> n + square excess of n.

Original entry on oeis.org

2, 3, 5, 6, 8, 12, 15, 21, 26, 27, 29, 33, 41, 46, 56, 63, 77, 90, 99, 117, 134, 147, 150, 156, 168, 192, 215, 234, 243, 261, 266, 276, 296, 303, 317, 345, 366, 371, 381, 401, 402, 404, 408, 416, 432, 464, 487, 490, 496, 508, 532, 535, 541, 553, 577, 578, 580, 584, 592, 608

Offset: 0

N. J. A. Sloane, Jun 10 2004

nonn

Robert Israel, Table of n, a(n) for n = 0..10000
S. H. Weintraub, An interesting recursion, Amer. Math. Monthly, 111 (No. 6, 2004), 528-530.

Cf. A053186, A094761, A094764.

a[0]:= 2:
for n from 1 to 100 do a[n]:= f(a[n-1]) od:
seq(a[n],n=0..100); # Robert Israel, Jan 28 2018

lista(nn) = {print1(n=2, ", "); for (k=2, nn, m = 2*n - sqrtint(n)^2; print1(m, ", "); n = m;);} \\ Michel Marcus, Oct 23 2015

A176035 Difference between product of two distinct primes and previous perfect square.

Original entry on oeis.org

2, 1, 5, 6, 5, 6, 1, 8, 9, 10, 2, 3, 10, 2, 6, 8, 9, 13, 1, 5, 10, 13, 1, 4, 5, 6, 10, 12, 13, 14, 6, 11, 15, 18, 19, 1, 2, 8, 12, 13, 20, 21, 22, 1, 2, 11, 14, 15, 17, 22, 8, 9, 14, 16, 18, 25, 5, 6, 7, 9, 10, 13, 17, 18, 19, 21, 22, 23, 25, 1, 10, 12, 22, 24, 28, 29, 3, 6, 9, 11, 18, 22, 31

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 06 2010

6-4=2, 10-9=1, 14-9=5, 15-9=6, 21-16=5,..

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A006881, A056892, A106044, A176032, A176033, A176034

A006881:= [n: n in [1..1000] | EulerPhi(n) + DivisorSigma(1, n) eq 2*(n+1)];
[A006881[n] - Floor(Sqrt(A006881[n]))^2: n in [1..100]]; // G. C. Greubel, Oct 26 2022

A006881=Sort@Flatten@Table[Prime[m]*Prime[n], {n, 2, 150}, {m, n-1}];
Table[A006881[[n]]-Floor[Sqrt[A006881[[n]]]]^2, {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Jun 02 2011 *)

A006881=[n for n in (1..750) if euler_phi(n) + sigma(n,1) == 2*n+2]
[A006881[n] - isqrt(A006881[n])^2 for n in range(101)] # G. C. Greubel, Oct 26 2022

a(n) = A053186(A006881(n)). - R. J. Mathar, Aug 25 2025

A262678 a(n) = n - A262690(n), where A262690(n) = largest square k <= n such that A002828(n-k) = A002828(n)-1.

Original entry on oeis.org

0, 0, 1, 2, 0, 1, 2, 3, 4, 0, 1, 2, 8, 4, 5, 6, 0, 1, 9, 10, 4, 5, 13, 14, 8, 0, 1, 2, 3, 4, 5, 6, 16, 8, 9, 10, 0, 1, 2, 3, 4, 16, 17, 18, 8, 9, 10, 11, 32, 0, 1, 2, 16, 4, 5, 6, 20, 8, 9, 10, 11, 25, 13, 14, 0, 1, 2, 18, 4, 5, 34, 22, 36, 9, 25, 26, 40, 13, 29, 30, 16, 0, 1, 2, 20, 4, 5, 6, 52, 25, 9, 10, 11, 29, 13, 14, 32, 16, 49, 18, 0

Offset: 0

Antti Karttunen, Oct 03 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65536

Cf. A002828, A262689, A262690.

Cf. also A053186.

Scheme
```
(define (A262678 n) (- n (A262690 n)))
```

a(n) = n - A262690(n).

A288969 Triangular array read by rows: row n is the list of the 2n-1 successive values taken by the function z = n - floor(x) floor(y) along the hyperbola with equation y = n/x, for 1 <= x <= n.

Original entry on oeis.org

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

Offset: 1

Luc Rousseau, Jun 20 2017

See A288966's links for explanations about the algorithm used to go along an hyperbola of equation y = n/x, with 1 <= x <= n.
When represented as a triangular array, internal zeros "0" correspond to factorizations of n.
This array appears to resemble a version of the sieve of Eratosthenes with zeros aligned.
A053186 and A293497 appear to intertwine into this sequence. The following will be denoted "assumption (1)": with t indexing columns, t=0 being central: T(n, 2k) = A053186(n+k^2) and T(n, 2k+1) = A293497(n+k(k+1)). - Luc Rousseau, Oct 11 2017
It would be nice to have a larger b-file, or an a-file. - N. J. A. Sloane, Oct 13 2017

			Array begins:
                0
              0 1 0
            0 1 2 1 0
          0 1 2 0 2 1 0
        0 1 2 3 1 3 2 1 0
      0 1 2 3 0 2 0 3 2 1 0
    0 1 2 3 4 1 3 1 4 3 2 1 0
  0 1 2 3 4 0 2 4 2 0 4 3 2 1 0

Luc Rousseau, The first 25 lines of the triangle array, formatted

Cf. A053186, A288966, A293497, A293578.

package oeis;
public class B {
public static void main(String[] args) {
for (int n = 1; n <= 8; n ++) {
hyberbolaTiles(n);
}
}
private static void hyberbolaTiles(int n) {
int x = 0, y = 0, p = 0, q = n;
do {
if (p != 0) {
System.out.println(n - p * q);
}
if (y < 0) { x = y + q; q --; }
if (y > 0) { p ++; x = y - p; }
if (y == 0) {
p ++;
x = 0; System.out.println("0");
q --;
}
y = x + p - q;
} while (q > 0);
}
}

(* Under assumption (1) *)
A288969[n_, t_] := Module[{x},
  x = Floor[(-t + Sqrt[t^2 + 4 n])/2];
  n - x (t + x)
] (* Luc Rousseau, Oct 11 2017 *)
(* or *)
FEven[x_] := x^ 2
InvFEven[x_] := Sqrt[x]
GEven[n_] := n - FEven[Floor[InvFEven[n]]]
FOdd[x_] := x*(x + 1)
InvFOdd[x_] := (Sqrt[1 + 4 x] - 1)/2
GOdd[n_] := n - FOdd[Floor[InvFOdd[n]]]
A288969[n_, t_] := Module[
  {e, k, x},
  e = EvenQ[t];
  k = If[e, t/2, (t - 1)/2];
  x = n + If[e, FEven[k], FOdd[k]];
  If[e, GEven[x], GOdd[x]]
] (* Luc Rousseau, Oct 11 2017 *)

htrow(n) = {my(x = 0, y = 0, p = 0, q = n); while (q>0, if (p, print1(n-p*q, ", ")); if (y < 0, x = y + q; q --); if (y > 0, p ++; x = y - p); if (y == 0, p++; x = 0; print1(0, ", "); q --;); y = x + p - q;);}
tabf(nn) = for (n=1, nn, htrow(n); print()); \\ Michel Marcus, Jun 21 2017

From Luc Rousseau, Oct 11 2017: (Start)
(All formulas under assumption (1))
With t indexing columns, t=0 being central,
T(n, 2k) = A053186(n+k^2).
T(n, 2k+1) = A293497(n+k(k+1)).
T(n, t) = n - x*(x+t) where x = floor((-t+sqrt(t^2+4n))/2).
With A293578 viewed as a 2D array T',
T'(n,t)=T(n-1,t)-T(n,t)+1 (define T(0,0) as 0).
(End)

More terms from Michel Marcus, Jun 21 2017

A061887 n + largest square less than or equal to n; numbers in the range [2k^2,2k^2+2k] for some k.

Original entry on oeis.org

0, 2, 3, 4, 8, 9, 10, 11, 12, 18, 19, 20, 21, 22, 23, 24, 32, 33, 34, 35, 36, 37, 38, 39, 40, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 128, 129

Offset: 0

Henry Bottomley, May 12 2001

nonn

			a(15)=15+9=24; a(16)=16+16=32; a(17)=17+16=33.

Cf. A060984, A061885.

Table[n+Floor[Sqrt[n]]^2,{n,0,70}] (* Harvey P. Dale, Aug 23 2012 *)

a(n) = n+[sqrt(n)]^2 = n+A048760(n) = 2n-A053186(n).

A064672 a(0) = 0, a(1) = 1; for a(n), n >= 2, write n = x^2 + y with y >= 0 as small as possible, then a(n) = a(x) + a(y).

Original entry on oeis.org

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

Offset: 0

Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 09 2001

Because of the definition of a(n), a(n^2) = a(n) and more generally a(n^(2m)) = a(n), so the sequence recursively contains itself.

a(A064689(n)) = n and a(m) < n for m < A064689(n).

			a(7) = 5 because 7 = 2^2 + 3, a(2) = 2 and a(3) = 3, giving 5

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A064689.

Cf. A048760.

a064672 n = a064672_list !! n
a064672_list = 0 : 1 : f (drop 2 a000196_list) 1 1 (tail a064672_list)
   where f (r:rs) r' u (v:vs)
           | r == r' = (u + v) : f rs r u vs
           | r /= r' = u' : f rs r u' (tail a064672_list)
           where u' = a064672 $ fromInteger r
-- Reinhard Zumkeller, Apr 27 2012

a[0]=0; a[1]=1; a[n_] := a[n] = a[ Floor[ Sqrt[n] ] ] + a[ n - Floor[ Sqrt[n] ]^2 ]; Table[a[n], {n, 0, 98}] (* Jean-François Alcover, May 23 2012, after Reinhard Zumkeller *)

For n > 1: a(n) = a(A000196(n)) + a(A053186(n)), a(0) = 0, a(1) = 1. [Reinhard Zumkeller, Apr 27 2012]

A094764 Trajectory of 7 under repeated application of the map n --> n + square excess of n.

Original entry on oeis.org

7, 10, 11, 13, 17, 18, 20, 24, 32, 39, 42, 48, 60, 71, 78, 92, 103, 106, 112, 124, 127, 133, 145, 146, 148, 152, 160, 176, 183, 197, 198, 200, 204, 212, 228, 231, 237, 249, 273, 290, 291, 293, 297, 305, 321, 353, 382, 403, 406, 412, 424, 448, 455, 469, 497, 510, 536, 543

Offset: 0

N. J. A. Sloane, Jun 10 2004

nonn

H. Brocard, Note 2837, L'Intermédiaire des Mathématiciens, 11 (1904), p. 239.

S. H. Weintraub, An interesting recursion, Amer. Math. Monthly, 111 (No. 6, 2004), 528-530.

Cf. A053186, A094761, A094763.

lista(nn) = {print1(n=7, ", "); for (k=2, nn, m = 2*n - sqrtint(n)^2; print1(m, ", "); n = m;);} \\ Michel Marcus, Oct 24 2015

A094766 Trajectory of 11 under repeated application of the map n -> n + 2*square excess of n (see A094765).

Original entry on oeis.org

11, 15, 27, 31, 43, 57, 73, 91, 111, 133, 157, 183, 211, 241, 273, 307, 343, 381, 421, 463, 507, 553, 601, 651, 703, 757, 813, 871, 931, 993, 1057, 1123, 1191, 1261, 1333, 1407, 1483, 1561, 1641, 1723, 1807, 1893, 1981, 2071, 2163, 2257, 2353, 2451, 2551, 2653

Offset: 0

N. J. A. Sloane, Jun 10 2004

nonn

The trajectory of 3 gives A002061 and 5 gives essentially the same trajectory as 3.

Robert Israel, Table of n, a(n) for n = 0..10000
S. H. Weintraub, An interesting recursion, Amer. Math. Monthly, 111 (No. 6, 2004), 528-530.
Index entries for linear recurrences with constant coefficients, signature (3, -3, 1).

Cf. A002061, A053186, A094765.

f:= n -> 3*n - 2*floor(sqrt(n))^2:
g:= proc(n) option remember; f(procname(n-1)) end proc:
g(0):= 11:
seq(g(n),n=0..100); # Robert Israel, Oct 23 2015

NestList[3*#-2*Floor[Sqrt[#]]^2&,11,50] (* Harvey P. Dale, Feb 26 2022 *)

lista(nn) = {print1(n=11, ", "); for (k=2, nn, m = 3*n - 2*sqrtint(n)^2; print1(m, ", "); n = m;);} \\ Michel Marcus, Oct 23 2015

Vec(4+2*x+6*x^2+(7-8*x+3*x^2)/(1-x)^3 + O(x^100)) \\ Altug Alkan, Oct 23 2015

Numbers given satisfy a(n) = n^2 + 5n + 7, for n>2. - Ralf Stephan, Dec 04 2004
From Robert Israel, Oct 23 2015: (Start)
This is because for x = m^2 + 5*m + 7, (m+2)^2 < x < (m+3)^2 so A094765(x) = x + 2*(x-(m+2)^2) = m^2 + 7*m + 13 = (m+1)^2 + 5*(m+1) + 7.
Similarly, for any positive integer k, the trajectory of k^2 + k + 1 is n^2 + (2k+1) n + k^2 + k + 1 for n >= 0.
G.f.: 4 + 2*x + 6*x^2 + (7-8*x+3*x^2)/(1-x)^3. (End)

cp's OEIS Frontend

A056895 If the smallest prime with a square excess of n is p then a(n)^2 = p - n.

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A094761 a(n) = n + (square excess of n).

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A094763 Trajectory of 2 under repeated application of the map n -> n + square excess of n.

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Maple

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A176035 Difference between product of two distinct primes and previous perfect square.

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Magma

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SageMath

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A262678 a(n) = n - A262690(n), where A262690(n) = largest square k <= n such that A002828(n-k) = A002828(n)-1.

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A288969 Triangular array read by rows: row n is the list of the 2*n-1 successive values taken by the function z = n - floor(x) * floor(y) along the hyperbola with equation y = n/x, for 1 <= x <= n.

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A061887 n + largest square less than or equal to n; numbers in the range [2k^2,2k^2+2k] for some k.

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Mathematica

Formula

A064672 a(0) = 0, a(1) = 1; for a(n), n >= 2, write n = x^2 + y with y >= 0 as small as possible, then a(n) = a(x) + a(y).

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A288969 Triangular array read by rows: row n is the list of the 2n-1 successive values taken by the function z = n - floor(x) floor(y) along the hyperbola with equation y = n/x, for 1 <= x <= n.