A174069 -id:A174069 - cp's OEIS frontend

A174090 Powers of 2 and odd primes; alternatively, numbers that cannot be written as a sum of at least three consecutive positive integers.

1, 2, 3, 4, 5, 7, 8, 11, 13, 16, 17, 19, 23, 29, 31, 32, 37, 41, 43, 47, 53, 59, 61, 64, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 256

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 07 2010, and Omar E. Pol, Feb 24 2014

From Omar E. Pol, Feb 24 2014: (Start)
Also the odd noncomposite numbers (A006005) and the powers of 2 with positive exponent, in increasing order.
If a(n) is composite and a(n) - a(n-1) = 1 then a(n-1) is a Mersenne prime (A000668), hence a(n-1)*a(n)/2 is a perfect number (A000396) and a(n-1)*a(n) equals the sum of divisors of a(n-1)*a(n)/2.
If a(n) is even and a(n+1) - a(n) = 1 then a(n+1) is a Fermat prime (A019434). (End)

Robert Israel, Table of n, a(n) for n = 1..10000
Jaap Spies, A Bit of Math, The Art of Problem Solving, Jaap Spies Publishers (2019).
Nieuw Archief voor Wiskunde, Problems/UWC, Problem C, Vol. 5/6, No. 2.

Numbers not in A111774.
Equals A000079 UNION A065091.
Equals A067133 \ {6}.
Cf. A000040, A000203, A000396, A000668, A006005, A019434, A092506.
Cf. also A138591, A174069, A174070, A174071.

N:= 300: # to get all terms <= N
S:= {seq(2^i,i=0..ilog2(N))} union select(isprime,{ 2*i+1 $ i=1..floor((N-1)/2) }):
sort(convert(S,list)); # Robert Israel, Jun 18 2015

a[n_] := Product[GCD[2 i - 1, n], {i, 1, (n - 1)/2}] - 1;
Select[Range[242], a[#] == 0 &] (* Gerry Martens, Jun 15 2015 *)

list(lim)=Set(concat(concat(1,primes(lim)), vector(logint(lim\2,2),i,2^(i+1)))) \\ Charles R Greathouse IV, Sep 19 2024

select( {is_A174090(n)=isprime(n)||n==1<M. F. Hasler, Oct 24 2024

from sympy import primepi
def A174090(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x+(0 if x<=1 else 1-primepi(x))-x.bit_length())
    return bisection(f,n,n) # Chai Wah Wu, Sep 19 2024

a(n) ~ n log n. - Charles R Greathouse IV, Sep 19 2024

This entry is the result of merging an old incorrect entry and a more recent correct version. N. J. A. Sloane, Dec 07 2015

A027578 Sums of five consecutive squares: a(n) = n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2 + (n+4)^2.

Original entry on oeis.org

30, 55, 90, 135, 190, 255, 330, 415, 510, 615, 730, 855, 990, 1135, 1290, 1455, 1630, 1815, 2010, 2215, 2430, 2655, 2890, 3135, 3390, 3655, 3930, 4215, 4510, 4815, 5130, 5455, 5790, 6135, 6490, 6855, 7230, 7615, 8010, 8415, 8830, 9255, 9690, 10135, 10590, 11055

Offset: 0

Patrick De Geest

a(n) is defined for n < 0 and a(-n) = a(n-4) for any n; a(-3) = a(-1) = 15, a(-2) = 10. - Jean-Christophe Hervé, Nov 11 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Patrick De Geest, Palindromic Sums of Squares of Consecutive Integers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Index entries for two-way infinite sequences.

Cf. A000290, A001844, A027575, A027865, A120328, A260637, A276026.
Cf. A027580.
Subsequence of A174069, A174070, A174071.

[n^2+(n+1)^2+(n+2)^2+(n+3)^2+(n+4)^2: n in [0..50] ]; // Vincenzo Librandi, Jun 17 2011

A027578:=n->5*(n+2)^2+10: seq(A027578(n), n=0..50); # Wesley Ivan Hurt, Nov 12 2015

Table[5 (n + 2)^2 + 10, {n, 0, 50}] (* Bruno Berselli, Jul 29 2015 *)
Total/@Partition[Range[0,50]^2,5,1] (* or *) LinearRecurrence[{3,-3,1},{30,55,90},50] (* Harvey P. Dale, Mar 06 2018 *)

vector(100, n, n--; n^2+(n+1)^2+(n+2)^2+(n+3)^2+(n+4)^2) \\ Altug Alkan, Nov 11 2015

[i^2+(i+1)^2+(i+2)^2+(i+3)^2+(i+4)^2 for i in range(0,50)] # Zerinvary Lajos, Jul 03 2008

a(n) = 5*A059100(n+2).
From Colin Barker, Mar 29 2012: (Start)
G.f.: 5*(6-7*x+3*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. (End)
a(n) = 5*(n + 2)^2 + 10. a(n) is never square. - Bruno Berselli, Jul 29 2015
E.g.f.: 5*(6 + 5*x + x^2)*exp(x). - G. C. Greubel, Aug 24 2022
From Amiram Eldar, Sep 15 2022: (Start)
Sum_{n>=0} 1/a(n) = coth(sqrt(2)*Pi)*Pi/(10*sqrt(2)) - 7/60.
Sum_{n>=0} (-1)^n/a(n) = cosech(sqrt(2)*Pi)*Pi/(10*sqrt(2)) + 1/60. (End)

A151557 Squares which are the sum of two or more consecutive squares.

Original entry on oeis.org

25, 841, 4900, 5929, 8464, 11236, 19044, 20449, 24964, 28561, 33124, 38025, 60025, 64009, 75076, 127449, 148225, 170569, 184900, 193600, 245025, 281961, 422500, 425104, 429025, 461041, 524176, 620944, 632025, 970225, 1024144, 1044484, 1113025, 1283689

Offset: 1

N. J. A. Sloane, May 21 2009

nonn

			25 = 5^2 = 3^2 + 4^2
841 = 29^2 = 20^2 + 21^2
4900 is the sum of the first 24 squares.

Donovan Johnson, Table of n, a(n) for n = 1..5077 (terms < 10^15)
Thomas Andrews, Sum of consecutive squares equal to a square
K. S. Brown, Sum of Consecutive Nth Powers Equals an Nth Power
Index entries for sequences related to sums of squares

Cf. A001032. Superset is A174069.

(* First run the program for A097812 *) Union[A097812]^2 (* Alonso del Arte, Nov 19 2012 *)

a(n) = A097812(n)^2.

A062681 Numbers that are sums of 2 or more consecutive squares in more than 1 way.

Original entry on oeis.org

365, 1405, 1730, 2030, 3281, 3655, 3740, 4510, 4705, 4760, 5244, 5434, 5915, 7230, 7574, 8415, 9385, 11055, 11900, 12325, 12524, 14905, 16745, 17484, 18879, 19005, 19855, 20449, 20510, 21790, 22806, 23681, 25580, 25585, 27230, 27420, 28985

Offset: 1

Erich Friedman, Jul 04 2001

A subsequence of A174069; A111044 is the subsequence of numbers allowing at least 3 representations of the given form. - M. F. Hasler, Dec 23 2013

			365 = 10^2 + 11^2 + 12^2 = 13^2 + 14^2.

David W. Wilson, Table of n, a(n) for n = 1..10000
G. Campbell et al., Sums of consecutive squares in two or more ways, discussion in Number Theory group on LinkedIn, Dec 2013

Cf. A174069, A111044.

{N=200;A174069=[];A062681=[];for(i=2,N,for(k=1,i-1,setsearch(A174069,t=sum(j=i-k,i,j^2))&&(A062681=setunion(A062681,Set(t)))&&next;t>N^2*2&&break;A174069=setunion(A174069,Set(t))));A062681} \\ M. F. Hasler, Dec 22 2013

A111044 Integers which can be written as a sum of at least 2 consecutive squares in at least 3 different ways.

Original entry on oeis.org

147441, 910805, 1026745, 2403800, 2513434, 3198550, 11739805, 15053585, 18646301, 33313175, 93812510, 102939515, 134910295, 136448235, 151443110, 163998695, 195435485, 197780465, 213872920, 267043455, 461498779, 482204660, 554503705, 559990541, 601704095

Offset: 1

Sébastien Dumortier, Oct 06 2005

nonn

The smallest number which can be expressed in 4 such ways is 554503705, which is equal to the sum of squares of the integers in the closed intervals (480,1210), (3570,3612), (3613,3654) and (7442,7451). - Giovanni Resta, Jul 25 2007

			147441 = 85^2 + 86^2 + ... + 101^2 = 29^2 + 30^2 + ... + 77^2 = 18^2 + 19^2 + ... + 76^2;
910805 = 550^2 + 551^2 + 552^2 = 144^2 + 145^2 + ... + 178^2 = 35^2 + 36^2 + ... + 140^2;
1026745 = 716^2 + 717^2 = 51^2 + 52^2 + ... + 147^2 = 1^2 + 2^2 + ... + 145^2;
2403800 = 583^2 + 584^2 + ... + 589^2 = 368^2 + 369^2 + ... + 384^2 = 298^2 + 299^2 + ... + 322^2;
2513434 = 473^2 + 474^2 + ... + 483^2 = 286^2 + 287^2 + ... + 313^2 = 66^2 + 67^2 + ... + 198^2;
3198550 = 225^2 + 226^2 + ... + 275^2 = 127^2 + 128^2 + ... + 226^2 = 1^2 + 2^2 + ... + 212^2.

David W. Wilson, Table of n, a(n) for n = 1..754
Sebastien DUMORTIER, Perfect Numbers [Broken link?]

See also: A001653, A097812, A062681, A174069.

More terms from Giovanni Resta, Jul 25 2007

A163251 Primes that are sum of (at least two) consecutive squares.

Original entry on oeis.org

5, 13, 29, 41, 61, 113, 139, 149, 181, 199, 271, 313, 421, 509, 613, 677, 761, 811, 1013, 1201, 1279, 1301, 1459, 1741, 1861, 1877, 2113, 2381, 2521, 2539, 2791, 3121, 3331, 3613, 3677, 3919, 4231, 4513, 5101, 7159, 7321, 8011, 8429, 8581, 9661, 9749, 9859

Offset: 1

Gaurav Kumar, Jul 23 2009

nonn

Let S(n,k) = (n+1)^2 + (n+2)^2 +... + (n+k)^2, n>=0, k>=2. S(n,k) is always composite for k=4 (2 | S), k=5 (5 | S), and k >= 7 (see A256503). So a(n) is the sum of 2, 3, or 6 consecutive squares. The smallest a(n) that cannot be written as a sum of fewer than 6 consecutive squares is a(7)=139. - Vladimir Shevelev, Apr 08 2015

			5 = 1^2 + 2^2.
13 = 2^2 + 3^2.
29 = 2^2 + 3^2 + 4^2.

Donovan Johnson, Table of n, a(n) for n = 1..1000

A027862 is a subsequence.

Subsequence of A174069.

lst = {}; Do[p = m^2; Do[p += n^2; If[PrimeQ[p] && p <= 101701, AppendTo[lst, p]], {n, m + 1, 6!, 1}], {m, 6!}]; Take[Union@lst, 5!] (* Vladimir Joseph Stephan Orlovsky, Sep 15 2009 *)
Select[Union[Flatten[Table[Total/@Partition[Range[100]^2,n,1],{n,2,10}]]],PrimeQ] (* Harvey P. Dale, Mar 12 2015 *)

Offset corrected by Donovan Johnson, Nov 05 2012

A174070 Numbers that can be written as a sum of at least 3 consecutive squares.

Original entry on oeis.org

14, 29, 30, 50, 54, 55, 77, 86, 90, 91, 110, 126, 135, 139, 140, 149, 174, 190, 194, 199, 203, 204, 230, 245, 255, 271, 280, 284, 285, 294, 302, 330, 355, 365, 366, 371, 380, 384, 385, 415, 434, 446, 451, 476, 492, 501, 505, 506, 509, 510, 534, 559, 590, 595

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 06 2010

nonn

Numbers of the form (a(a+1)(2a+1)-b(b+1)(2b+1))/6 where a >= b+3 and b >= 0. - Robert Israel, Jul 18 2017

			14 = 1^2 + 2^2 + 3^2, 29 = 2^2 + 3^2 + 4^2.
30 = 1^2 + 2^2 + 3^2 + 4^2, 50 = 3^2 + 4^2 + 5^2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A111774, A138591, A174069.

N:= 1000: # to get all terms <= N
R:= [seq(b*(b+1)*(2*b+1)/6, b=0..ceil(sqrt(N/3)))]:
sort(convert(select(`<=`, {seq(seq(R[i]-R[j],j=1..i-3),i=1..nops(R))},N),list)); # Robert Israel, Jul 18 2017

max=50^2;lst={};Do[z=n^2+(n+1)^2;Do[z+=(n+x)^2;If[z>max,Break[]];AppendTo[lst,z],{x,2,max/2}],{n,max/2}];Union[lst]
(* Second program: *)
Function[s, Function[t, Union@ Flatten@ Map[TakeWhile[#, # < t[[1, -1]] &] &, t]]@ Map[Total /@ Partition[s, #, 1] &, Range[3, Length@ s]]][Range[16]^2] (* Michael De Vlieger, Jul 18 2017 *)
Module[{nn=30,sq},sq=Range[nn]^2;Take[Union[Flatten[Table[Total/@ Partition[ sq,n,1],{n,3,nn-2}]]],2nn]] (* Harvey P. Dale, Nov 16 2017 *)

A212016 Sums of the squares of two or more consecutive integers.

Original entry on oeis.org

1, 2, 5, 6, 10, 13, 14, 15, 19, 25, 28, 29, 30, 31, 35, 41, 44, 50, 54, 55, 56, 60, 61, 69, 77, 85, 86, 90, 91, 92, 96, 105, 110, 113, 121, 126, 135, 139, 140, 141, 145, 146, 149, 154, 170, 174, 181, 182, 190, 194, 195, 199, 203, 204, 205, 209, 218

Offset: 1

Max Alekseyev, Apr 26 2012

Subsequence of A062861.
Contains A212015 as a subsequence.
A174069 is a subsequence. - Altug Alkan, Dec 24 2015

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A062861, A174069, A212015.

N:= 1000: # to get all terms <=N
g:= x -> x*(x+1)*(2*x+1)/6:
S:= select(`<=`,{seq(seq(g(b)-g(a), a= -b-1 .. b-2), b = 1..floor((sqrt(2*N-1)+1)/2))},N):
sort(convert(S,list)); # Robert Israel, Jan 05 2016

{ isA212016(t) = fordiv(6*t,k, if(k==1,next); z=(k^2-1)/3; if(issquare(4*t/k-z), return(k)); if(z>4*t/k,break); ); 0 }

A174071 Numbers that can be written as a sum of at least 4 consecutive positive squares.

Original entry on oeis.org

30, 54, 55, 86, 90, 91, 126, 135, 139, 140, 174, 190, 199, 203, 204, 230, 255, 271, 280, 284, 285, 294, 330, 355, 366, 371, 380, 384, 385, 415, 446, 451, 476, 492, 501, 505, 506, 510, 534, 559, 595, 615, 620, 630, 636, 645, 649, 650, 679, 728, 730, 734, 764

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 06 2010

nonn

Numbers of the form m*(6*k^2 + 6*k*m + 2*m^2 - 6*k - 3*m + 1)/6 for some m>=4 and k>=1. - Robert Israel, May 06 2019

			30=1^2+2^2+3^2+4^2, 54=2^2+3^2+4^2+5^2, 55=1^2+2^2+3^2+4^2+5^2, ...

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A111774, A138591, A174069, A174070, A307937.

N:= 1000: # to get all terms <= N
Res:= NULL:
for m from 4 while m*(m+1)*(2*m+1)/6 <= N do
   for k from 1 do
       v:= m*(6*k^2 + 6*k*m + 2*m^2 - 6*k - 3*m + 1)/6;
       if v > N then break fi;
       Res:= Res, v;
od od:
sort(convert({Res},list));  Robert Israel, May 06 2019

max=60^2;lst={};Do[z=n^2+(n+1)^2+(n+2)^2;Do[z+=(n+x)^2;If[z>max,Break[]];AppendTo[lst,z],{x,3,Sqrt[max]/2}],{n,Sqrt[max]/2}];Union[lst]

Edited by Robert Israel, May 06 2019

A234311 Least number which can be written as a sum of at least 2 consecutive squares in at least n different ways.

Original entry on oeis.org

5, 365, 147441, 554503705

Offset: 1

M. F. Hasler, Dec 23 2013

No more terms < 10^18. [Lars Blomberg, Jun 02 2014]

			a(1) = 5 = 1^2 + 2^2.
a(2) = 365 = 10^2 + 11^2 + 12^2 = 13^2 + 14^2.
a(3) = 147441 = 85^2+...+101^2 = 29^2+...+77^2 = 18^2+...+76^2.
a(4) = 554503705 = 480^2+...+1210^2 = 3570^2+...+3612^2 = 3613^2+...+3654^2 = 7442^2+...+7451^2.

Cf. A174069, A062681, A111044.

cp's OEIS Frontend

A174090 Powers of 2 and odd primes; alternatively, numbers that cannot be written as a sum of at least three consecutive positive integers.

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A027578 Sums of five consecutive squares: a(n) = n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2 + (n+4)^2.

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A151557 Squares which are the sum of two or more consecutive squares.

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A062681 Numbers that are sums of 2 or more consecutive squares in more than 1 way.

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A111044 Integers which can be written as a sum of at least 2 consecutive squares in at least 3 different ways.

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A163251 Primes that are sum of (at least two) consecutive squares.

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Mathematica

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A174070 Numbers that can be written as a sum of at least 3 consecutive squares.

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