A174070 -id:A174070 - 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)

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

A299646 a(n) = Sum_{k = n..2*n+1} k^2.

Original entry on oeis.org

1, 14, 54, 135, 271, 476, 764, 1149, 1645, 2266, 3026, 3939, 5019, 6280, 7736, 9401, 11289, 13414, 15790, 18431, 21351, 24564, 28084, 31925, 36101, 40626, 45514, 50779, 56435, 62496, 68976, 75889, 83249, 91070, 99366, 108151, 117439, 127244, 137580, 148461, 159901

Offset: 0

Bruno Berselli, Feb 20 2018

Inverse binomial transform is 1, 13, 27, 14, 0, 0, 0, ... (0 continued).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Subsequence of A008854, A047388, A174070 (after 1).

Cf. A050409: Sum_{k = n..2*n} k^2; A050410: Sum_{k = n..2*n-1} k^2.

List([0..50], n -> (n+2)*(14*n^2+11*n+3)/6);

[(n+2)*(14*n^2+11*n+3)/6: n in [0..50]];

seq((n + 2)*(14*n^2 + 11*n + 3)/6, n=0..50); # Peter Luschny, Feb 21 2018

Table[(n + 2) (14 n^2 + 11 n + 3)/6, {n, 0, 50}]
(* Second program: *)
LinearRecurrence[{4, -6, 4, -1}, {1, 14, 54, 135}, 41] (* Jean-François Alcover, Feb 21 2018 *)

makelist((n+2)*(14*n^2+11*n+3)/6, n, 0, 50);

a(n)=(n+2)*(14*n^2+11*n+3)/6 \\ Charles R Greathouse IV, Feb 21 2018

Vec((1 + 10*x + 4*x^2 - x^3)/(1 - x)^4 + O(x^60)) \\ Colin Barker, Feb 22 2018

[(n+2)*(14*n^2+11*n+3)/6 for n in (0..50)]

O.g.f.: (1 + 10*x + 4*x^2 - x^3)/(1 - x)^4.
E.g.f.: (6 + 78*x + 81*x^2 + 14*x^3)*exp(x)/6.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = (n + 2)*(14*n^2 + 11*n + 3)/6. Therefore:
a(6*k + r) = 504*k^3 + 18*(14*r + 13)*k^2 + (42*r^2 + 78*r + 25)*k + a(r), with 0 <= r <= 5. Example: for r=5, a(6*k + 5) = (6*k + 7)*(84*k^2 + 151*k + 68).

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

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A299646 a(n) = Sum_{k = n..2*n+1} k^2.

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