A111774 -id:A111774 - cp's OEIS frontend

A338428 Multiplicities for A111774(n), for n >= 1.

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

Offset: 1

Wolfdieter Lang, Nov 23 2020

nonn

See the triangle and array of A337940 for details.

The first n and N(n) = A111774(n) with multiplicity m are [m, n, N] = [1, 1, 6], [2, 6, 15], [3, 16, 30], [4, 26, 45], [5, 60, 90], [6, 72, 105], [7, 157, 210], ...

			For a(6) = 2 for  A111774(6) = 15 see the example given in A337940.
Similarly for a(24) = 3 for A111774(24) = 42.

Cf. A111774, A337940.

a(n) gives the number of possibilities to write A111774(n) as a sum of at least three consecutive positive integers.

A065091 Odd primes.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277

Offset: 1

Labos Elemer, Nov 12 2001

Rayes et al. prove that the a(n)-th Chebyshev-T polynomial, divided by x, is irreducible over the integers.
Odd primes can be written as a sum of no more than two consecutive positive integers. Powers of 2 do not have a representation as a sum of k consecutive positive integers (other than the trivial n=n, for k=1). See A111774. - Jaap Spies, Jan 04 2007
Intersection of A005408 and A000040. - Reinhard Zumkeller, Oct 14 2008
Primes which are the sum of two consecutive numbers. - Juri-Stepan Gerasimov, Nov 07 2009
The arithmetic mean of divisors of p^3, (1+p)(1+p^2)/4, for odd primes p is an integer. - Ctibor O. Zizka, Oct 20 2009
Primes == -+ 1 (mod 4). - Juri-Stepan Gerasimov, Apr 27 2010
a(n) = A053670(A179675(n)) and a(n) <> A053670(m) for m < A179675(n). - Reinhard Zumkeller, Jul 23 2010
Triads of the form <2*a(n+1), a(n+1), 3*a(n+1)> like <6,3,9>, <10,5,15>, <14,7,21> appear in the EKG sequence A064413, see Theorem (3) there. - Paul Curtz, Feb 13 2011.
Complement of A065090; abs(A151763(a(n))) = 1. - Reinhard Zumkeller, Oct 06 2011
Right edge of the triangle in A065305. - Reinhard Zumkeller, Jan 30 2012
Numbers with two odd divisors. - Omar E. Pol, Mar 24 2012
Odd prime p divides some (2^k + 1) or (2^k - 1), (k>0, minimal, cf. A003558) depending on the parity of A179480((p+1)/2) = r. This is a consequence of the Quasi-order theorem and corollaries, [Hilton and Pederson, pp. 260-264]: 2^k == (-1)^r mod b, b odd; and b divides 2^k - (-1)^r, where p is a subset of b. - Gary W. Adamson, Aug 26 2012
Subset of the arithmetic numbers (A003601). - Wesley Ivan Hurt, Sep 27 2013
Odd primes p satisfy the identity: p = (product(2*cos((2*k+1)*Pi/(2*p)), k=0..(p-3)/2))^2.  This follows from C(2*p, 0) = (-1)^((p-1)/2)*p, n>=2, with the minimal polynomial C(k,x) of rho(k) := 2*cos(Pi/k). See A187360 for C and the W. Lang link on the field Q(rho(n)), eqs. (20) and (37). - Wolfdieter Lang, Oct 23 2013
Numbers m > 1 such that m^2 divides (2m-1)!! + m. - Thomas Ordowski, Nov 28 2014
Numbers m such that m divides 2*(m-3)! + 1. - Thomas Ordowski, Jun 20 2015
Numbers m such that (2m-3)!! == m (mod m^2). - Thomas Ordowski, Jul 24 2016
Odd numbers m such that ((m-3)!!)^2 == +-1 (mod m). - Thomas Ordowski, Jul 27 2016
Primes of the form x^2 - y^2. - Thomas Ordowski, Feb 27 2017
Conjecture: a(n) is the smallest odd number m > prime(n) such that Sum_{k=1..prime(n)-1} k^(m-1) == prime(n)-1 (mod m). This is an extension of the Agoh-Giuga conjecture. - Thomas Ordowski, Aug 01 2018
Numbers k > 1 such that either Phi(k,x) == 1 (mod k) or Phi(k,x) == k (mod k^2) holds, where Phi(k,x) is the k-th cyclotomic polynomial. - Jianing Song, Aug 02 2018

Paulo Ribenboim, The little book of big primes, Springer 1991, p. 106.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)
M. O. Rayes, V. Trevisan and P. S. Wang, Factorization of Chebyshev Polynomials, ICM Report, 1998.
M. O. Rayes, V. Trevisan and P. S. Wang, Factorization of Chebyshev Polynomials, Computers & Mathematics with Applications, Volume 50, Issues 8-9, October-November 2005, Pages 1231-1240.
Eric Weisstein's World of Mathematics, Prime Number.

Cf. A000040, A033270, union of A002144 and A002145.

Cf. A230953 (boustrophedon transform).

a065091 n = a065091_list !! (n-1)
a065091_list = tail a000040_list  -- Reinhard Zumkeller, Jan 30 2012

[NthPrime(n): n in [2..100]]; // Vincenzo Librandi, Jun 21 2015

A065091 := proc(n) RETURN(ithprime(n+1)) end:

Prime[Range[2, 33]] (* Vladimir Joseph Stephan Orlovsky, Aug 22 2008 *)

forprime(p=3, 200, print1(p, ", ")) \\ Felix Fröhlich, Jun 30 2014

from sympy import prime
def A065091(n): return prime(n+1) # Chai Wah Wu, Jul 13 2024

def A065091_list(limit):  # after Minác's formula
    f = 3; P = [f]
    for n in range(3, limit, 2):
        if (f+1)>n*(f//n)+1: P.append(n)
        f = f*n
    return P
A065091_list(100)  # Peter Luschny, Oct 17 2013

a(n) = A000040(n+1). - M. F. Hasler, Oct 26 2013

More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Jan 05 2002

Edited (moved contributions from A000040 to here) by M. F. Hasler, Oct 26 2013

A368110 Numbers of which it is possible to choose a different divisor of each prime index.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97

Offset: 1

Gus Wiseman, Dec 15 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

By Hall's marriage theorem, k is a term if and only if there is no sub-multiset S of the prime indices of k such that fewer than |S| numbers are divisors of a member of S. Equivalently, there is no divisor of k in A370348. - Robert Israel, Feb 15 2024

			The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   6: {1,2}
   7: {4}
   9: {2,2}
  10: {1,3}
  11: {5}
  13: {6}
  14: {1,4}
  15: {2,3}
  17: {7}
  19: {8}
  21: {2,4}
  22: {1,5}
  23: {9}
  25: {3,3}
  26: {1,6}
  29: {10}
  30: {1,2,3}

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

Partitions of this type are counted by A239312, complement A370320.
Positions of nonzero terms in A355739.
Complement of A355740.
For just prime divisors we have A368100, complement A355529 (odd A355535).
A000005 counts divisors.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741 chooses prime factors of prime indices, variations A355744, A355745.
Cf. A000720, A076610, A111774, A335433, A335448, A340852, A355733, A355734, A355737, A355749, A370348.

filter:= proc(n) uses numtheory, GraphTheory; local B,S,F,D,E,G,t,d;
  F:= ifactors(n)[2];
  F:= map(t -> [pi(t[1]),t[2]], F);
  D:= `union`(seq(divisors(t[1]), t = F));
  F:= map(proc(t) local i;seq([t[1],i],i=1..t[2]) end proc,F);
  if nops(D) < nops(F) then return false fi;
  E:= {seq(seq({t,d},d=divisors(t[1])),t = F)};
  S:= map(t -> convert(t,name), [op(F),op(D)]);
  E:= map(e -> map(convert,e,name),E);
  G:= Graph(S,E);
  B:= BipartiteMatching(G);
  B[1] = nops(F);
end proc:
select(filter, [$1..100]); # Robert Israel, Feb 15 2024

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Select[Tuples[Divisors/@prix[#]],UnsameQ@@#&]!={}&]

Heinz numbers of the partitions counted by A239312.

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

Original entry on oeis.org

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

A109814 a(n) is the largest k such that n can be written as sum of k consecutive positive integers.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 2, 1, 3, 4, 2, 3, 2, 4, 5, 1, 2, 4, 2, 5, 6, 4, 2, 3, 5, 4, 6, 7, 2, 5, 2, 1, 6, 4, 7, 8, 2, 4, 6, 5, 2, 7, 2, 8, 9, 4, 2, 3, 7, 5, 6, 8, 2, 9, 10, 7, 6, 4, 2, 8, 2, 4, 9, 1, 10, 11, 2, 8, 6, 7, 2, 9, 2, 4, 10, 8, 11, 12, 2, 5, 9, 4, 2, 8, 10, 4, 6, 11, 2, 12, 13, 8, 6, 4, 10, 3, 2, 7, 11, 8, 2, 12

Offset: 1

David W. Wilson

nonn

n is the sum of at most a(n) consecutive positive integers. As suggested by David W. Wilson, Aug 15 2005: Suppose n is to be written as sum of k consecutive integers starting with m, then 2n = k(2m + k - 1). Only one of the factors is odd. For each odd divisor d of n there is a unique corresponding k = min(d,2n/d). a(n) is the largest among those k. - Jaap Spies, Aug 16 2005
The numbers that can be written as a sum of k consecutive positive integers are those in column k of A141419 (as a triangle). - Peter Munn, Mar 01 2019
The numbers that cannot be written as a sum of two or more consecutive positive integers are the powers of 2. So a(n) = 1 iff n = 2^k for k >= 0. - Bernard Schott, Mar 03 2019

			Examples provided by _Rainer Rosenthal_, Apr 01 2008:
1 = 1     ---> a(1) = 1
2 = 2     ---> a(2) = 1
3 = 1+2   ---> a(3) = 2
4 = 4     ---> a(4) = 1
5 = 2+3   ---> a(5) = 2
6 = 1+2+3 ---> a(6) = 3
a(15) = 5: 15 = 15 (k=1), 15 = 7+8 (k=2), 15 = 4+5+6 (k=3) and 15 = 1+2+3+4+5 (k=5). - _Jaap Spies_, Aug 16 2005

Donovan Johnson, Table of n, a(n) for n = 1..10000
K. S. Brown's Mathpages, Partitions into Consecutive Integers
A. Heiligenbrunner, Sum of adjacent numbers (in German).
Jaap Spies, Problem C NAW 5/6 nr. 2 June 2005, July 2005 (solution to problem below).
Jaap Spies, Sage program for computing A109814
Universitaire Wiskunde Competitie, Problem C, Nieuw Archief voor Wiskunde, 5/6, no. 2, Problems/UWC, Jun 2005, pp. 181-182.

Cf. A001227, A111774, A111775, A141419, A138591.

Cf. A000079 (powers of 2), A000217 (triangular numbers).

A109814:= proc(n) local m, k, d; m := 0; for d from 1 by 2 to n do if n mod d = 0 then k := min(d, 2*n/d): fi; if k > m then m := k fi: od; return(m); end proc; seq(A109814(i),i=1..150); # Jaap Spies, Aug 16 2005

a[n_] := Reap[Do[If[OddQ[d], Sow[Min[d, 2n/d]]], {d, Divisors[n]}]][[2, 1]] // Max; Table[a[n], {n, 1, 102}]

from sympy import divisors
def a(n): return max(min(d, 2*n//d) for d in divisors(n) if d&1)
print([a(n) for n in range(1, 103)]) # Michael S. Branicky, Dec 23 2022

[sloane.A109814(n) for n in range(1,20)]
# Jaap Spies, Aug 16 2005

From Reinhard Zumkeller, Apr 18 2006: (Start)
a(n)*(a(n)+2*A118235(n)-1)/2 = n;
a(A000079(n)) = 1;
a(A000217(n)) = n. (End)

Edited by N. J. A. Sloane, Aug 23 2008 at the suggestion of R. J. Mathar

A174069 Numbers that can be written as a sum of at least 2 squares of consecutive positive integers.

Original entry on oeis.org

5, 13, 14, 25, 29, 30, 41, 50, 54, 55, 61, 77, 85, 86, 90, 91, 110, 113, 126, 135, 139, 140, 145, 149, 174, 181, 190, 194, 199, 203, 204, 221, 230, 245, 255, 265, 271, 280, 284, 285, 294, 302, 313, 330, 355, 365, 366, 371, 380, 384, 385, 415, 421, 434, 446, 451

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 06 2010

Numbers are listed without multiplicity: 365 is the first term that is the sum of two or more squares in more than one way. See A062681 for other numbers of that form. - M. F. Hasler, Dec 22 2013

A subsequence of A212016. This sequence focuses on the squares of consecutive positive integers. - Altug Alkan, Dec 24 2015

			5 = 1^2 + 2^2
13 = 2^2 + 3^2
14 = 1^2 + 2^2 + 3^2
25 = 3^2 + 4^2

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A111774, A138591, A151557 (subset of squares), A163251 (subset of primes).

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 *)

A111775 Number of ways n can be written as a sum of at least three consecutive integers.

Original entry on oeis.org

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

Offset: 1

Jaap Spies, Aug 16 2005

Powers of 2 and (odd) primes cannot be written as a sum of at least three consecutive integers. a(n) strongly depends on the number of odd divisors of n (A001227): Suppose n is to be written as sum of k consecutive integers starting with m, then 2n = k(2m + k - 1). Only one of the factors is odd. For each odd divisor of n there is a unique corresponding k, k=1 and k=2 must be excluded.
When the initial 0 term is a(1), a(n) is the number of times n occurs after the second column in the square array of A049777. - Bob Selcoe, Feb 14 2014
For nonnegative integers x,y where x-y>=3: a(n) equals the number of ways n can be expressed as a function of (x*(x+1)-y*(y+1))/2 when the initial 0 term is a(1). - Bob Selcoe, Feb 14 2014

			a(15) = 2 because 15 = 4+5+6 and 15 = 1+2+3+4+5. The number of odd divisors of 15 is 4.
G.f. = x^6 + x^9 + x^10 + x^12 + x^14 + 2*x^15 + 2*x^18 + x^20 + 2*x^21 + x^22 + ...
a(30) = 3 because there are 3 ways to satisfy (x*(x+1)-y*(y+1))/2 = 30 when x-y>=3: x=8, y=3; x=9, y=5; and x=11, y=8. - _Bob Selcoe_, Feb 14 2014

Nieuw Archief voor Wiskunde 5/6 nr. 2 Problems/UWC Problem C part 4, Jun 2005, p. 181-182

Alois P. Heinz, Table of n, a(n) for n = 1..10000
K. S. Brown's Mathpages, Partitions into Consecutive Integers
A. Heiligenbrunner, Sum of adjacent numbers (in German)
Nieuw Archief voor Wiskunde 5/6 nr. 2 Problems/UWC, Problem C: solution of this Problem
J. Spies, Sage program for computing A111775

Cf. A111774, A001227 (number of odd divisors), A069283.

A001227:= proc(n) local d, s; s := 0: for d from 1 by 2 to n do if n mod d = 0 then s:=s+1 fi: end do: return(s); end proc; A111775:= proc(n) local k; if n=1 then return(0) fi: k := A001227(n): if type(n,even) then k:=k-1 else k:=k-2 fi: return k; end proc; seq(A111775(i), i=1..150);

a[n_] := If[n == 1, 0, Total[Mod[Divisors[n], 2]] - Mod[n, 2] - 1];
a /@ Range[1, 100] (* Jean-François Alcover, Oct 14 2019 *)

{a(n) = local(m); if( n<1, 0, sum( i=0, n, m=0; if( issquare( 1 + 8*(n + i * (i + 1)/2), &m), m\2 > i+2)))}; /* Michael Somos, Aug 27 2012 */

If n is even then a(n) = A001227(n)-1 = A069283(n) otherwise a(n) = A001227(n)-2, for n > 1.

G.f.: Sum_{n >= 2} x^(3*n)/(1 - x^(2*n)). - Peter Bala, Jan 12 2021

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

A219839 a(n) is the number of odd integers in 2..(n-1) that have a common factor (other than 1) with n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 2, 0, 1, 3, 0, 0, 3, 0, 2, 4, 1, 0, 4, 2, 1, 4, 2, 0, 7, 0, 0, 6, 1, 5, 6, 0, 1, 7, 4, 0, 9, 0, 2, 10, 1, 0, 8, 3, 5, 9, 2, 0, 9, 7, 4, 10, 1, 0, 14, 0, 1, 13, 0, 8, 13, 0, 2, 12, 11, 0, 12, 0, 1, 17, 2, 8, 15, 0, 8, 13, 1, 0, 18

Offset: 1

Lei Zhou, Nov 29 2012

a(n) is also the number of linearly dependent diagonal/side length ratios R(n,k), in the regular n-gon. The following will explain this. In the regular n-gon inscribed in a circle the number of distinct diagonals including the side is floor(n/2). Not all of the corresponding length ratios R(n,k) = d(n,k)/d(n,1), k = 1..floor(n/2), with d(n,1) = s(n) (the length of the side), d(n,2) the length of the smallest diagonal, etc., are linearly independent because C(n,R(n,2)) = 0, where C is the minimal polynomial of R(n,2) = 2*cos(Pi/n) (see A187360) with degree delta(n) = A055034(n). Thus every ratio R(n,j), with j = delta(n)+1, ..., floor(n/2) can be expressed as a linear combination of the independent R(n,k), k=1, ..., delta(n). See the comment from Sep 21 2013 on A053121 for powers of R(n,2) (called there rho(N)). Therefore, a(n) = floor(n/2) - delta(n) is, for n>=2, the number of linearly dependent ratios R(n,k) in the regular n-gon. - Wolfdieter Lang, Sep 23 2013
From Wolfdieter Lang, Nov 23 2020: (Start)
This sequence gives the difference between the number of odd numbers in the smallest nonnegative residue system modulo n (called here RS(n)) and the smallest nonnegative restricted residue system (called here RRS(n), but RRS(1) = {1}, not {0}).
This sequence can be used to find sequence A111774 by recording the positions of the entries >= 1. See a W. Lang comment there, and also A337940, for the proof. Hence the complement of A111774, given in A174090, is given by the numbers m with a(m) = 0. (End)

			n=1: there is no odd number greater than 2 but smaller than 1-1=0, so a(1)=0.
Same for n=2,3.
n=4: 3 is the only odd number in 2..(4-1), and GCD(3,4)=1, so a(4)=0.
For any prime numbers and numbers in the form of 2^n, no odd number in 2..(n-1) has common factor with n, so a(p)=0 and a(2^n)=0, n>0.
n=6: 3,5 are odd numbers in 2..(6-1), and GCD(3,6)=3>1 and GCD(5,6)=1, so a(6)=1.
n=15: candidates are 3,5,7,9,11,13.  3, 5, and 9 have greater than 1 common factors with 15, so a(15)=3
From _Wolfdieter Lang_, Sep 23 2013: (Start)
Example n = 15 for a(n) = floor(n/2) - delta(n): 1, 3, 5, 7, 9, 11, 13 take out 1, 7, 9, 11, leaving 3, 5, 13. Therefore, a(15) = 7 - 4 = 3. See the formula above for delta.
In the regular 15-gon the 3 (= a(15)) diagonal/side ratios R(15, 5), R(15, 6) and R(15,7) can be expressed as linear combinations of the R(15,j), j=1..4.  See the n-gon comment above. (End)
From _Wolfdieter Lang_, Nov 23 2020: (Start)
n = 1: RS(1) = {0}, RRS(1) = {1}, hence a(1) = 0 - 1 = 0. Here RRS(1) is not {0}(standard) because delta(1) := 1 (the degree of minimal polynomial for 2*cos(Pi//1) = -2 which is x+2, see A187360).
n = 6: RS(6) = {0, 1, 2, 3, 4, 5} and RRS(6) = {1,5}, hence a(6) = 3 - 2 = 1, and A111774(1) = 6 = A337940(1, 1).
a(15) = 7 - 4 = 3, and A111774(6) = 15 = A337940(3, 3) = A337940(4, 1) (multiplicity 2 = A338428(6)). (End)

Lei Zhou, Table of n, a(n) for n = 1..10000

Cf. A000010, A016035 (see 1st comment there), A004526, A055034, A111774, A174090, A190357, A337940, A338428.

Table[Count[Range[3, n - 1, 2], ?(GCD[n, #] > 1 &)], {n, 100}] (* _T. D. Noe, Nov 30 2012 *)
a[1] = 0; a[n_] := Floor[n/2] - EulerPhi[2*n]/2; Array[a, 80] (* Amiram Eldar, Nov 28 2020 *)

a(n) = sum(i=2, n-1, (i%2) && (gcd(i,n)!=1)); \\ Michel Marcus, Aug 07 2018

a(n) = floor(n/2) - delta(n), with floor(n/2) = A004526 and delta(n) = A055034(n) = phi(2*n)/2, for n >= 2, with Euler's phi A000010. See the Aug 17 2011 comment on A055034. For n = 1 this would be -1, not 0, because delta(1) = 1. - Wolfdieter Lang, Sep 23 2013

Sum_{k=1..n} a(k) ~ c*n^2, where c = 1/4 - 2/Pi^2 = 0.04735763... (A190357). - Amiram Eldar, Feb 23 2025

cp's OEIS Frontend

A338428 Multiplicities for A111774(n), for n >= 1.

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A065091 Odd primes.

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A368110 Numbers of which it is possible to choose a different divisor of each prime index.

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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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A109814 a(n) is the largest k such that n can be written as sum of k consecutive positive integers.

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A174069 Numbers that can be written as a sum of at least 2 squares of consecutive positive integers.

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