A053186 -id:A053186 - cp's OEIS frontend

A056892 a(n) = square excess of the n-th prime.

1, 2, 1, 3, 2, 4, 1, 3, 7, 4, 6, 1, 5, 7, 11, 4, 10, 12, 3, 7, 9, 15, 2, 8, 16, 1, 3, 7, 9, 13, 6, 10, 16, 18, 5, 7, 13, 19, 23, 4, 10, 12, 22, 24, 1, 3, 15, 27, 2, 4, 8, 14, 16, 26, 1, 7, 13, 15, 21, 25, 27, 4, 18, 22, 24, 28, 7, 13, 23, 25, 29, 35, 6, 12, 18, 22, 28, 36, 1, 9, 19, 21

Offset: 1

Henry Bottomley, Jul 05 2000

nonn

			a(5) = 2 since the 5th prime is 11 = 3^2 + 2.
From _M. F. Hasler_, Oct 19 2018: (Start)
Written as a table, starting a new row when a square is reached, the sequence reads:
  1, 2,    // = 2 - 1, 3 - 1 = {primes between 1^2 = 1 and 2^2 = 4} - 1
  1, 3,     // = 5 - 4, 7 - 4 = {primes between 2^2 = 4 and 3^2 = 9} - 4
  2, 4,      // = 11 - 9, 13 - 9 = {primes between 3^2 = 9 and 4^2 = 16} - 9
  1, 3, 7,    // = 17 - 16, 19 - 16, 23 - 16 = {primes between 16 and 25} - 16
  4, 6,        // = 29 - 25, 31 - 25 = {primes between 5^2 = 25 and 6^2 = 36} - 25
  1, 5, 7, 11,  // = {37, 41, 43, 47: primes between 6^2 = 36 and 7^2 = 49} - 36
  4, 10, 12,    // = {53, 59, 61: primes between 7^2 = 49 and 8^2 = 64} - 49
  3, 7, 9, 15,  // = {67, 71, 73, 79: primes between 8^2 = 64 and 9^2 = 81} - 64
  2, 8, 16,     // = {83, 89, 97: primes between 9^2 = 81 and 10^2 = 100} - 81
  etc. (End)

M. F. Hasler, Table of n, a(n) for n = 1..10000

Cf. A000040, A000196, A002496, A048760, A053186, A056893 .. A056898.

When written as a table, row lengths are A014085, and row sums are A108314 - A014085 * A000290 = A320688.

lst={};Do[p=Prime[n];s=p^(1/2);f=Floor[s];a=f^2;d=p-a;AppendTo[lst,d],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)
#-Floor[Sqrt[#]]^2&/@Prime[Range[90]] (* Harvey P. Dale, Jul 06 2014 *)

A056892(n)={my(p=prime(n));p-sqrtint(p)^2} \\ M. F. Hasler, Oct 04 2009

a(n) = A053186(A000040(n)).

a(n) = A000040(n) - A000006(n)^2. - M. F. Hasler, Oct 04 2009

A055400 Cube excess: difference between n and largest cube <= n.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16

Offset: 0

Henry Bottomley, May 16 2000

			a(12) = 4 because 2^3 <= 12 < 3^3 and 12 - 2^3 = 4.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A048762, A053186, A055401.

A055400 := proc(n)
        n-A048762(n) ;
end proc:
seq(A055400(n),n=0..80) ; # R. J. Mathar, Nov 06 2011

Array[#-Floor[Power[#, (3)^-1]]^3&,90,0] (* Harvey P. Dale, Jan 18 2012 *)

a(n)=n-sqrtnint(n,3)^3 \\ Charles R Greathouse IV, Sep 02 2015

a(n) = n - A048762(n) = n - floor(n^(1/3))^3.

a(n) < 3*n^(2/3) for n > 0. - Charles R Greathouse IV, Sep 02 2015

A064784 Difference between n-th triangular number t(n) and the largest square <= t(n).

Original entry on oeis.org

0, 2, 2, 1, 6, 5, 3, 0, 9, 6, 2, 14, 10, 5, 20, 15, 9, 2, 21, 14, 6, 28, 20, 11, 1, 27, 17, 6, 35, 24, 12, 44, 32, 19, 5, 41, 27, 12, 51, 36, 20, 3, 46, 29, 11, 57, 39, 20, 0, 50, 30, 9, 62, 41, 19, 75, 53, 30, 6, 66, 42, 17, 80, 55, 29, 2, 69, 42, 14, 84, 56, 27, 100, 71, 41, 10, 87, 56

Offset: 1

Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 20 2001

The second differences of a(n) - (a(n)-a(n-1))-(a(n-1)-a(n-2)) - give 2, -2, -1, 6, -6, -1, -1, 12, -12, -1, 16, -16, -1 ... 82k+2, 82k-2, -1, 82k+6, 82k-6, -1, -1, 82k+12, 82k-12, -1, 82k+16, -82k-16, -1, 82k+20, -82k-20, -1, -1, 82k+26, -82k-26, -1, 82k+30, -82k-30, -1, -1, 82k+36, -82k-36, -1, 82k+40, -82k-40, -1, 82k+44, -82k-44, -1, -1, 82k+50, -82k-50, -1, 82k+54, -82k-54, -1, -1, 82k+60, -82k-60, -1, 82k+64, -82k-64, -1, -1, 82k+70, -82k-70, -1, 82k+74, -82k-74, -1, 82k+78, -82k-78, -1, -1, ...

			n = 5: A000217(5) = 28, largest square below that is 25, so a(5) = 28 - 25 = 3.

Harry J. Smith, Table of n, a(n) for n = 1..1000
J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The EKG sequence, Exper. Math. 11 (2002), 437-446.
J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The EKG sequence, arXiv:math/0204011 [math.NT], 2002.
Index entries for sequences related to EKG sequence

Cf. A001108, A076816, A128549, A230038. Unique values are in A230044.

seq(n*(n+1)/2-floor(sqrt(n*(n+1)/2))^2,n=0..100);

f[n_]:=n*(n+1)/2-Floor[Sqrt[n*(n+1)/2]]^2; lst={}; Do[AppendTo[lst,f[n]],{n,0,6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 17 2010 *)
#-Floor[Sqrt[#]]^2&/@Accumulate[Range[100]] (* Harvey P. Dale, Oct 15 2014 *)

{ default(realprecision, 100); for (n=1, 1000, t=n*(n + 1)/2; a=t - floor(sqrt(t))^2; write("b064784.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 25 2009

from math import isqrt
def A064784(n): return (m:=n*(n+1)>>1)-isqrt(m)**2 # Chai Wah Wu, Jun 01 2024

a(n) = n*(n+1)/2 - floor(sqrt(n*(n+1)/2))^2.
a(n) = A053186(A000217(n)). - R. J. Mathar, Sep 10 2016
a(A001108(n)) = 0. - Hugo Pfoertner, Jun 01 2024

Definition corrected by Harry J. Smith, Sep 25 2009

Terms corrected by Harry J. Smith, Sep 25 2009

A135932 Primes whose integer square root remainder is also prime.

Original entry on oeis.org

3, 7, 11, 19, 23, 41, 43, 47, 67, 71, 83, 103, 107, 113, 149, 151, 157, 163, 167, 199, 227, 263, 269, 331, 337, 347, 353, 419, 431, 443, 487, 491, 503, 521, 587, 593, 599, 607, 613, 617, 619, 683, 719, 787, 797, 821, 827, 907, 911, 919, 929, 937, 941, 947

Offset: 1

Harry J. Smith, Dec 07 2007

nonn

The integer square root of an integer x >= 0 can be defined as floor(sqrt(x)) and the remainder of this as x - (floor(sqrt(x)))^2.

Robert Israel, Table of n, a(n) for n = 1..10000
Wikipedia, Integer square root

Cf. A053186.

filter:= proc(p) isprime(p) and isprime(p - floor(sqrt(p))^2) end proc:
select(filter, [seq(i,i=3..10000,2)]); # Robert Israel, Apr 30 2025

f[n_]:=n-(Floor[Sqrt[n]])^2;lst={};Do[p=Prime[n];If[PrimeQ[f[p]],AppendTo[lst,p]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 25 2010 *)

{ forprime(p=2, 2000, isr = sqrtint(p); Rem = p - isr*isr; if (isprime(Rem), print1(p, ",") ) ) }

A056893 Smallest prime with square excess of n.

Original entry on oeis.org

2, 3, 7, 13, 41, 31, 23, 89, 73, 59, 47, 61, 113, 239, 79, 97, 593, 139, 163, 461, 277, 191, 167, 193, 281, 251, 223, 317, 353, 991, 431, 761, 433, 563, 359, 397, 521, 479, 439, 569, 617, 571, 619, 773, 829, 887, 947, 673, 1493, 1571, 727, 1013, 953, 1279

Offset: 1

Henry Bottomley, Jul 05 2000

nonn

			a(4)=13 since 13=3^2+4, while 2, 3, 5, 7 and 11 have square excesses of 1, 2, 1, 3 and 3 respectively.

Cf. A053186, A000196, A002496, A048760, A056892-A056898.

A056893 := proc(n)
    local p ;
    p :=2 ;
    while A053186(p) <> n do
        p := nextprime(p) ;
    end do:
    return p ;
end proc: # R. J. Mathar, Jul 28 2013

A056893(n)={
    local(p=2) ;
    while( A053186(p)!=n,
        p=nextprime(p+1)
    ) ;
    return(p)
} /* R. J. Mathar, Jul 28 2013 */

a(n) =n+A056894(n).

a(n) = min{p in A000040: A053186(p) = n}. - R. J. Mathar, Jul 28 2013

A066857 Smallest number k such that n! - k is a square.

Original entry on oeis.org

0, 1, 2, 8, 20, 44, 140, 320, 476, 3584, 12311, 4604, 74879, 414119, 2071775, 5703551, 11551671, 45680444, 442548224, 1960632176, 2657058876, 24923993276, 130518272975, 1478154932316, 5446454455004, 38610655379975

Offset: 1

Reinhard Zumkeller, Jan 21 2002

nonn

Sequence is not monotonic: a(n) < a(n-1) for n = 12, 71, 90, 143, 145, 151, 172, 218, 257. - Zak Seidov, Jun 25 2013

			a(10) = 3628800 - 1904 * 1904 = 3628800 - 3625216 = 3584.

T. D. Noe, Table of n, a(n) for n = 1..100

Cf. A068869.

Table[n! - Floor[Sqrt[n! ]]^2, {n, 1, 27}]

a(n)=my(N=n!); N-sqrtint(N)^2 \\ Charles R Greathouse IV, Jun 25 2013

a(n) = A053186(n!) = n!-A048760(n!) = n!-floor(sqrt(n!))^2 = n!-A055226(n)^2.

More terms from Vladeta Jovovic, Mar 21 2002

Edited by Robert G. Wilson v and N. J. A. Sloane, Mar 22 2002

A094765 a(n) = n + 2 * square excess of n.

Original entry on oeis.org

0, 1, 4, 7, 4, 7, 10, 13, 16, 9, 12, 15, 18, 21, 24, 27, 16, 19, 22, 25, 28, 31, 34, 37, 40, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 82, 85, 88, 91, 64, 67, 70, 73, 76, 79, 82, 85, 88, 91

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. A002061, A053186, A094766.

seq(3*n - 2*floor(sqrt(n))^2, n=0..1000); # Robert Israel, Oct 23 2015

a(n) = 3*n - 2*sqrtint(n)^2; \\ Michel Marcus, Oct 23 2015

a(n) = n + 2*A053186(n).

G.f.: 3*x/(1-x)^2 - (2/(1-x)) * Sum_{k>=1} (2*k-1)*x^(k^2). The sum is related to Jacobi theta functions. - Robert Israel, Oct 23 2015

A104492 Cube excess of the n-th prime.

Original entry on oeis.org

1, 2, 4, 6, 3, 5, 9, 11, 15, 2, 4, 10, 14, 16, 20, 26, 32, 34, 3, 7, 9, 15, 19, 25, 33, 37, 39, 43, 45, 49, 2, 6, 12, 14, 24, 26, 32, 38, 42, 48, 54, 56, 66, 68, 72, 74, 86, 7, 11, 13, 17, 23, 25, 35, 41, 47, 53, 55, 61, 65, 67, 77, 91, 95, 97, 101, 115, 121, 4, 6, 10, 16, 24, 30

Offset: 1

Jonathan Vos Post, Mar 10 2005

			a(48) = 7 because the 48th prime is 223 and 223 - 6^3 = 7, while 223 - 7^3 = -120.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A053186, A055400, A056892, A102821.

lst={};Do[p=Prime[n];s=p^(1/3);f=Floor[s];a=f^3;d=p-a;AppendTo[lst,d],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)
#-Floor[Surd[#,3]]^3&/@Prime[Range[80]] (* Harvey P. Dale, Feb 15 2018 *)

a(n) = A055400(A000040(n)).

a(n) = prime(n) - floor(prime(n)^(1/3))^3. - Jon E. Schoenfield, Jan 17 2015

A384688 Runs of t in the range 0 <= t <= k and the same parity as k, for successive k >= 0.

Original entry on oeis.org

0, 1, 0, 2, 1, 3, 0, 2, 4, 1, 3, 5, 0, 2, 4, 6, 1, 3, 5, 7, 0, 2, 4, 6, 8, 1, 3, 5, 7, 9, 0, 2, 4, 6, 8, 10, 1, 3, 5, 7, 9, 11, 0, 2, 4, 6, 8, 10, 12, 1, 3, 5, 7, 9, 11, 13, 0, 2, 4, 6, 8, 10, 12, 14, 1, 3, 5, 7, 9, 11, 13, 15, 0, 2, 4, 6, 8, 10, 12, 14, 16

Offset: 0

Kevin Ryde, Jun 07 2025

The corresponding k is A055086(n), or k+1 = A000267(n).
A run is 0, 2, 4, ..., k when k even, or 1, 3, 5, ..., k when k odd, and has length floor(k/2) + 1.
Runs start at quarter squares n = A002620(k+1), with those beginning 0 at oblong numbers n = A002378(i) and those starting 1 at the squares n = (i+1)^2 (for i >= 0 in both cases).
Starts to differ from A025643 at n=109.

			Runs and their corresponding k = A055086(n) begin,
  n          = 0  1  2    4    6      9
  a(n)       = 0, 1, 0,2, 1,3, 0,2,4, 1,3,5, ...
  A055086(n) = 0, 1, 2,2, 3,3, 4,4,4, 5,5,5, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..5001

Cf. A000196, A000267, A053186, A055086, A055087, A079813, A216607.

Cf. A002620, A002378 (indices of 0's), A000290 (indices of 1's).

ClearAll[a] a[n_Integer]:=Module[{s,r},s=Floor[Sqrt[n]]; r=n-s^2; If[rVincenzo Librandi, Jul 06 2025 *)

PARI
```
a(n) = my(r,s=sqrtint(n,&r)); if(r
				
```

a(n) = 2*r+1 if r < s or a(n) = 2*(r-s) otherwise, where square root and remainder n = s^2 + r being  s=A000196(n), r=A053186(n).
a(n) = ceiling(A053186(4*n+1) / 2).
a(n) = A055086(n) - 2*A216607(n+1).
a(n) = 2*A055087(n) + A079813(n+1).

A056894 If the smallest prime with a square excess of n is p then a(n)=p-n.

Original entry on oeis.org

1, 1, 4, 9, 36, 25, 16, 81, 64, 49, 36, 49, 100, 225, 64, 81, 576, 121, 144, 441, 256, 169, 144, 169, 256, 225, 196, 289, 324, 961, 400, 729, 400, 529, 324, 361, 484, 441, 400, 529, 576, 529, 576, 729, 784, 841, 900, 625, 1444, 1521, 676, 961, 900, 1225, 784

Offset: 1

Henry Bottomley, Jul 05 2000

nonn

			a(4)=9 because the smallest prime with a square excess of 4 is 13 and 13-4=9

Cf. A000040, A000196, A002496, A048760, A053186, A056892-A056898.

a(n) = A056893(n) - n = A048760(A056893(n)) = A056895(n)^2

cp's OEIS Frontend

A056892 a(n) = square excess of the n-th prime.

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A055400 Cube excess: difference between n and largest cube <= n.

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A064784 Difference between n-th triangular number t(n) and the largest square <= t(n).

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A135932 Primes whose integer square root remainder is also prime.

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A056893 Smallest prime with square excess of n.

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A066857 Smallest number k such that n! - k is a square.

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A094765 a(n) = n + 2 * square excess of n.

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