A014601 -id:A014601 - cp's OEIS frontend

A350395 Numbers m such that a term with the largest coefficient in Product_{k=1..m} (1 + x^k) is unique.

0, 3, 8, 11, 12, 15, 16, 19, 20, 23, 24, 27, 28, 31, 32, 35, 36, 39, 40, 43, 44, 47, 48, 51, 52, 55, 56, 59, 60, 63, 64, 67, 68, 71, 72, 75, 76, 79, 80, 83, 84, 87, 88, 91, 92, 95, 96, 99, 100, 103, 104, 107, 108, 111, 112, 115, 116, 119, 120, 123, 124, 127, 128, 131, 132, 135, 136, 139, 140, 143, 144, 147, 148, 151, 152, 155, 156, 159, 160, 163, 164, 167, 168, 171, 172, 175, 176, 179, 180, 183, 184, 187, 188, 191, 192, 195, 196

Offset: 1

Max Alekseyev, Dec 28 2021

nonn

Numbers m such that A350393(m) = A350394(m).

Apparently, a(n) = A014601(n+1) for n >= 3. - Hugo Pfoertner, Dec 30 2021

Complement of A350396.
Cf. A025591 (largest coefficient), A350393, A350394.
Cf. A014601.

A053438 Expansion of (1+x+2x^3)/((1-x)(1-x^2)).

Original entry on oeis.org

1, 2, 3, 6, 7, 10, 11, 14, 15, 18, 19, 22, 23, 26, 27, 30, 31, 34, 35, 38, 39, 42, 43, 46, 47, 50, 51, 54, 55, 58, 59, 62, 63, 66, 67, 70, 71, 74, 75, 78, 79, 82, 83, 86, 87, 90, 91, 94, 95, 98, 99, 102, 103, 106, 107, 110, 111, 114, 115, 118, 119, 122

Offset: 0

N. J. A. Sloane, Jan 12 2000

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A010684 (first differences), A263511 (partial sums).

I:=[2,3,6]; [1] cat [n le 3 select I[n] else Self(n-1) +Self(n-2) -Self(n-3): n in [1..30]]; // G. C. Greubel, May 26 2018

A053438 := proc(n)
    if n > 0 then
        2*n -(1+(-1)^n)/2 ;
    else
        1 ;
    end if
end proc:
seq(A053438(n),n=0..30) ; # R. J. Mathar, Oct 27 2020

CoefficientList[Series[(1+x+2*x^3)/((1-x)*(1-x^2)), {x, 0, 50}], x] (* or *) Join[{1}, LinearRecurrence[{1,1,-1}, {2,3,6}, 50]] (* G. C. Greubel, May 26 2018 *)

a(n)=abs(n\2*4+n%2*3-1) \\ Charles R Greathouse IV, Dec 08 2011

a(n) = 2*n -(1+(-1)^n)/2 if n>=1. - Frank Ellermann, Feb 12 2002
a(n) = A042964(n), n>0. - R. J. Mathar, Oct 13 2008
a(n) = A014601(n) - 1 for n>0. - Hugo Pfoertner, Oct 26 2020

A104569 Triangle read by rows: T(i,j) is the (i,j)-entry (1 <= j <= i) of the product Q*R of the infinite lower triangular matrices Q = [1; 1,3; 1,3,1; 1 3,1,3; ...] and R = [1; 1,1; 1,1,1; 1,1,1,1; ...].

Original entry on oeis.org

1, 4, 3, 5, 4, 1, 8, 7, 4, 3, 9, 8, 5, 4, 1, 12, 11, 8, 7, 4, 3, 13, 12, 9, 8, 5, 4, 1, 16, 15, 12, 11, 8, 7, 4, 3, 17, 16, 13, 12, 9, 8, 5, 4, 1, 20, 19, 16, 15, 12, 11, 8, 7, 4, 3, 21, 20, 17, 16, 13, 12, 9, 8, 5, 4, 1, 24, 23, 20, 19, 16, 15, 12, 11, 8, 7, 4, 3, 25, 24, 21, 20, 17, 16, 13, 12, 9, 8, 5, 4, 1

Offset: 1

Gary W. Adamson, Mar 16 2005

			The first few rows of the triangle are:
  1;
  4, 3;
  5, 4, 1;
  8, 7, 4, 3;
  9, 8, 5, 4, 1;
  ...

Cf. A035608, A074377, A104570.

Row sums yield A074377. Columns 1, 3, 5, ... (starting at the diagonal entry) yield A042948. Columns 2, 4, 6, ... (starting at the diagonal entry) yield A014601. The product R*Q yields A104570.

T:=proc(i,j) if j>i then 0 elif i+j mod 2 = 1 then 2*(i-j)+2 elif i mod 2 = 1 and j mod 2 = 1 then 2*(i-j)+1 elif i mod 2 = 0 and j mod 2 = 0 then 2*(i-j)+3 else fi end: for i from 1 to 13 do seq(T(i,j),j=1..i) od; # yields sequence in triangular form # Emeric Deutsch, Mar 23 2005

Q[i_, j_] := If[j <= i, 2 + (-1)^j, 0];
R[i_, j_] := If[j <= i, 1, 0];
T[i_, j_] := Sum[Q[i, k]*R[k, j], {k, 1, 13}];
Table[T[i, j], {i, 1, 13}, {j, 1, i}] // Flatten (* Jean-François Alcover, Jul 24 2024 *)

For 1<=j<=i: T(i, j)=2(i-j+1) if i and j are of opposite parity; T(i, j)=2(i-j)+1 if both i and j are odd; T(i, j)=2(i-j)+3 if both i and j are even. - Emeric Deutsch, Mar 23 2005

More terms from Emeric Deutsch, Mar 23 2005

A274918 Numbers n such that the sum of numbers less than n that do not divide n is odd.

Original entry on oeis.org

4, 5, 6, 8, 10, 13, 14, 16, 17, 21, 22, 26, 29, 30, 32, 33, 34, 36, 37, 38, 41, 42, 45, 46, 53, 54, 57, 58, 61, 62, 64, 65, 66, 69, 70, 72, 73, 74, 77, 78, 82, 85, 86, 89, 90, 93, 94, 97, 100, 101, 102, 105, 106, 109, 110, 113, 114, 117, 118, 122, 125, 126, 128, 129, 130, 133, 134, 137, 138, 141, 142, 144, 145, 146, 149, 150

Offset: 1

Ilya Gutkovskiy, Dec 10 2016

Numbers n such that A000035(A024816(n)) = 1 or A000035(A000217(n)-A000203(n)) = 1.

There are 2 cases when n belongs to this sequence: 1) if n congruent to 0 or 3 mod 4 (A014601) and n is square or twice square (A028982); 2) if n congruent to 1 or 2 mod 4 (A042963) and n is not square and is not twice square (A028983).

			6 is in the sequence because 6 has 4 divisors {1,2,3,6} therefore 2 non-divisors {4,5}, 4 + 5 = 9 and 9 is odd.

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

Cf. A000035, A000203, A000217, A014601, A024816, A028982, A028983, A042963, A053868, A053869, A279064.

filter:= n -> evalb(n+1 mod 4 <= 1) = (issqr(n) or issqr(n/2)):
select(filter, [$1..200]); # Robert Israel, Dec 11 2016

Select[Range[150], Mod[#1 ((#1 + 1)/2) - DivisorSigma[1, #1], 2] == 1 & ]
Select[Range[150],OddQ[Total[Complement[Range[#],Divisors[#]]]]&] (* Harvey P. Dale, Jul 29 2024 *)

A279064 Numbers n such that the sum of numbers less than n that do not divide n is even.

Original entry on oeis.org

1, 2, 3, 7, 9, 11, 12, 15, 18, 19, 20, 23, 24, 25, 27, 28, 31, 35, 39, 40, 43, 44, 47, 48, 49, 50, 51, 52, 55, 56, 59, 60, 63, 67, 68, 71, 75, 76, 79, 80, 81, 83, 84, 87, 88, 91, 92, 95, 96, 98, 99, 103, 104, 107, 108, 111, 112, 115, 116, 119, 120, 121, 123, 124, 127, 131

Offset: 1

Ilya Gutkovskiy, Dec 10 2016

Numbers n such that A000035(A024816(n)) = 0 or A000035(A000217(n)-A000203(n)) = 0.

There are 2 cases when n belongs to this sequence: 1) if n congruent to 0 or 3 mod 4 (A014601) and n is not square and is not twice square (A028983); 2) if n congruent to 1 or 2 mod 4 (A042963) and n is square or twice square (A028982).

			12 is in the sequence because 12 has 6 divisors {1,2,3,4,6,12} therefore 6 non-divisors {5,7,8,9,10,11}, 5 + 7 + 8 + 9 + 10 + 11 = 50 and 50 is even.

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Cf. A000035, A000203, A000217, A014601, A024816, A028982, A028983, A042963, A053868, A053869, A274918.

Select[Range[150], Mod[#1 ((#1 + 1)/2) - DivisorSigma[1, #1], 2] == 0 & ]
Select[Range[150],EvenQ[(#(#+1))/2-DivisorSigma[1,#]]&] (* Harvey P. Dale, Oct 21 2018 *)

isok(n) = (sum(k=1, n-1, k*((n % k) != 0)) % 2) == 0; \\ Michel Marcus, Dec 11 2016

A298364 Permutation of the natural numbers partitioned into quadruples [4k-2, 4k-1, 4k-3, 4k] for k > 0.

Original entry on oeis.org

2, 3, 1, 4, 6, 7, 5, 8, 10, 11, 9, 12, 14, 15, 13, 16, 18, 19, 17, 20, 22, 23, 21, 24, 26, 27, 25, 28, 30, 31, 29, 32, 34, 35, 33, 36, 38, 39, 37, 40, 42, 43, 41, 44, 46, 47, 45, 48, 50, 51, 49, 52, 54, 55, 53, 56, 58, 59, 57, 60, 62, 63, 61, 64, 66, 67, 65

Offset: 1

Guenther Schrack, Jan 18 2018

Partition the natural number sequence into quadruples starting with (1,2,3,4); swap the first and second elements, then swap the second and third elements; repeat for all quadruples.

Guenther Schrack, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Inverse: A292576.
Sequence of fixed points: A008586(n) for n > 0.
First differences: (-1)^floor(n^2/4)*A068073(n-1) for n > 0.
Subsequences:
  elements with odd index: A042963(A103889(n)) for n > 0.
  elements with even index A014601(n) for n > 0.
  odd elements: A166519(n-1) for n > 0.
  indices of odd elements: A042964(n) for n > 0.
  even elements: A005843(n) for n > 0.
  indices of even elements: A042948(n) for n > 0.
Other similar permutations: A116966, A284307, A292576.

a = [2 3 1 4];
max = 10000;    % Generation of b-file.
for n := 5:max
   a(n) = a(n-4) + 4;
end;

Nest[Append[#, #[[-4]] + 4] &, {2, 3, 1, 4}, 63] (* or *)
Array[# + ((-1)^# + ((-1)^(# (# - 1)/2)) (1 - 2 (-1)^#))/2 &, 67] (* Michael De Vlieger, Jan 23 2018 *)
LinearRecurrence[{1,0,0,1,-1},{2,3,1,4,6},70] (* Harvey P. Dale, Dec 12 2018 *)

for(n=1, 100, print1(n + ((-1)^n + ((-1)^(n*(n-1)/2))*(1 - 2*(-1)^n))/2, ", "))

O.g.f.: (3*x^3 - 2*x^2 + x + 2)/(x^5 - x^4 - x - 1).
a(1) = 2, a(2) = 3, a(3) = 1, a(4) = 4, a(n) = a(n-4) + 4 for n > 4.
a(n) = n + ((-1)^n + ((-1)^(n*(n-1)/2))*(1 - 2*(-1)^n))/2.
a(n) = n + (cos(n*Pi) - cos(n*Pi/2) + 3*sin(n*Pi/2))/2.
a(n) = 2*floor((n+1)/2) - 4*floor((n+1)/4) + floor(n/2) + 2*floor(n/4).
a(n) = n + (-1)^floor((n-1)^2/4)*A140081(n) for n > 0.
a(n) = A056699(n+1) - 1, n > 0.
a(n+2) = A168269(n+1) - a(n), n > 0.
a(n+2) = a(n) + (-1)^floor((n+1)^2/4)*A132400(n+2) for n > 0.
Linear recurrence: a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
First differences: periodic, (1, -2, 3, 2) repeat.
Compositions:
  a(n) = A080412(A116966(n-1)) for n > 0.
  a(n) = A284307(A256008(n)) for n > 0.
  a(A067060(n)) = A133256(n) for n > 0.
  A116966(a(n+1)-1) = A092486(n) for n >= 0.
  A056699(a(n)) = A256008(n) for n > 0.

A355278 Lower left of the Cayley table for the primes when made into a group using the bijection (2, 3, 5, 7, ...) -> (0, +1, -1, +2, ...) into (Z, +); read by rows.

Original entry on oeis.org

2, 3, 7, 5, 2, 11, 7, 13, 3, 19, 11, 5, 17, 2, 23, 13, 19, 7, 29, 3, 37, 17, 11, 23, 5, 31, 2, 41, 19, 29, 13, 37, 7, 43, 3, 53, 23, 17, 31, 11, 41, 5, 47, 2, 59, 29, 37, 19, 43, 13, 53, 7, 61, 3, 71, 31, 23, 41, 17, 47, 11, 59, 5, 67, 2, 73, 37, 43, 29

Offset: 1

M. F. Hasler, Sep 08 2022

The primes can be given the structure of a group via a bijection with the group (Z, +). The simplest such bijection is to assign the integers (0, 1, -1, 2, -2, ...) to the primes, in increasing order, i.e., the composition of the prime counting function A000720, decreased by 1, with the canonical enumeration A001057.
Since this is an abelian group, the table (an infinite square matrix) is symmetric, T(m,n) = T(n,m), and it is sufficient to list only the lower left part, m >= n >= 1.
Each row and each column (of the complete square matrix) is a permutation of the primes, as always for Cayley tables. The inverse of a prime p, for the group operation *, is found at the top of the column in which 2 appears in the row starting with p, cf. 3rd formula.
Other simple bijections from the primes into (Z, +) would be to associate the negative integers to the primes congruent to 1 (mod 4) or to those congruent to 1 (mod 3), and the nonnegative integers to the others, in increasing order.

			The correspondence of primes p with integers m is as follows:
   p |  2 |  3 |  5 |  7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | ...
  ---+----+----+----+----+----+----+----+----+----+----+----+----
   m |  0 |  1 | -1 |  2 | -2 |  3 | -3 |  4 | -4 |  5 | -5 | ...
The table of the abelian group (P, *) based on this correspondence with (Z, +) starts as follows:
  [ 2  3  5  7 11 13 17 ...]
  [ 3  7  2 13  5 19 11 ...]
  [ 5  2 11  3 17  7 23 ...]
  [ 7 13  3 19  2 29  5 ...]
  [11  5 17  2 23  3 31 ...]
  [13 19  7 29  3 37  2 ...]
  [17 11 23  5 31  2 41 ...]
  [...    ...    ...    ...]
This means that 3 * 3 = 7, 3 * 5 = 2, 3 * 7 = 13, etc.: for example, primes 3 and 7 are associated to integers 1 and 2, so 3 * 7 is the prime associated to the integer 1 + 2 = 3, which yields 13.
Since 2, associated to 0 in Z, is the neutral element in the group (P, *), the first row and column is identical to the list of the primes themselves. Therefore we do not need to show row and columns headings in addition to the first row & column of the body of the table.
Since the table is symmetric,  T(m,n) = T(n,m)  <=>  p * q = q * p, the sequence lists only the lower left part: 2; 3, 7; 5, 2, 11; 7, 13, 3, 19; ...
The list of inverses of the primes 2, 3, 5, 7, 11, ... with respect to this group operation is 2, 5, 3, 11, 7, 17, 13, ... = A000040(A065190(n)), i.e., after the initial 2, list the primes with the two members of each subsequent pair swapped: swap 3 and 5, 7 and 11, 13 and 17, etc.

Antonio (u/hilberts12th), An example of a binary operation * such that (Set of prime numbers, *) is a group?, in r/abstractalgebra on reddit.com, May 18 2022

Cf. A000040, A000720, A001057.

A355278(m,n) = inv(f(prime(m))+f(prime(n)))
inv(x, p)=while(!mapisdefined(INV,x,&p), mapput(INV, f(p=prime(#INV+1)), p)); p
INV=Map(); f(p)=(p=primepi(p))\2*(-1)^(p%2)

T(m, n) = T(n, m), so only m >= n >= 1 is listed: row m has length m.
T(n, n) = A000040(A042948(n)) = A000040(A014601(n-1)+1)
T(m, 1) * T(k, 1) = 2 (neutral element of the group operation *) <=> T(k, 1) is the inverse of T(m, 1) <=> T(m, k) = 2.

A374080 Expansion of Product_{k>=1} (1 - x^(4k-1)) (1 - x^(4*k)).

Original entry on oeis.org

1, 0, 0, -1, -1, 0, 0, 0, -1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, -1, -1, 0, 0, 0, -1, 0, 1, 0, -1, -1, 0, 0, -1, -1, 1, 1, 0, -1, 0, 1, 0, -1, 0, 1, 1, -1, -1, 1, 1, 0, 0, 1, 2, -1, -1, 0, 1, 0, -2, 0, 2, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -2, 0, 1, -1, -2, 1, 2, 0, -2, -1, 2, 0, -2

Offset: 0

Ilya Gutkovskiy, Jun 27 2024

sign

Ilya Gutkovskiy, Scatterplot of a(n) up to n=10000

Cf. A010815, A014601, A035364, A081362, A082303, A106459, A137569, A274719, A284316, A301505, A374058, A374060, A374081.

nmax = 85; CoefficientList[Series[Product[(1 - x^(4 k - 1)) (1 - x^(4 k)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -(1/n) Sum[DivisorSum[k, # &, Or[Mod[#, 4] == 0, Mod[#, 4] == 3] &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 85}]

A030443 Nonzero coefficients in theta series of {E_7}* lattice.

Original entry on oeis.org

1, 56, 126, 576, 756, 1512, 2072, 4032, 4158, 5544, 7560, 12096, 11592, 13664, 16704, 24192, 24948, 27216, 31878, 44352, 39816, 41832, 55944, 72576, 66584, 67536, 76104, 100800, 99792, 101304, 116928, 145728, 133182, 126504, 160272, 205632, 177660, 176456, 205128, 249984, 249480, 234360

Offset: 0

N. J. A. Sloane

nonn

In the Eichler and Zagier reference this is e_4(A014601(n)), n >= 0, (p. 141), where e_4 is obtained from e_{4,1}(n,r), eq. (7), p. 22, depending only on 4*n-r^2 >= 0 (for integers n and r), i.e. on A014601(n), n >= 0 (with a new notation for n). - Wolfdieter Lang, Jan 08 2016

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 125.
M. Eichler and D. Zagier, The Theory of Jacobi Forms, Birkhäuser, 1985, p. 141.

N. Elkies and B. H. Gross, Embeddings into the integral octonions, Olga Taussky-Todd: in memoriam, Pacific J. Math. 1997, Special Issue, 147-158.

Cf. A003781.

f(n) = local(A); if( n<0, 0, A = sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); polcoeff( A^3 * (A^4 + 7 * subst(A, x, -x)^4) / 8, n)); \\ A003781
lista(nn) = select(x->(x>0), vector(nn, k, f(k-1))); \\ Michel Marcus, Nov 11 2023

More terms from Michel Marcus, Nov 11 2023

A105173 Numbers k such that either k(k+1)/4 + 1 or k(k+1)/4 - 1 or both are primes.

Original entry on oeis.org

3, 7, 8, 15, 16, 23, 24, 32, 39, 40, 47, 48, 55, 56, 63, 64, 71, 79, 80, 87, 95, 103, 104, 111, 112, 119, 120, 128, 135, 136, 143, 151, 152, 159, 167, 175, 176, 183, 199, 208, 216, 223, 224, 231, 232, 239, 240, 247, 248, 255, 256, 263, 264, 271, 279, 287, 288

Offset: 1

Pierre CAMI, Apr 11 2005

nonn

The primes are (sum of numbers from 1 to k)/2 + or - 1.

			3*4/4 = 3, 3-1 = 2 is prime so a(1) = 3.
7*8/4 = 14, 14-1 = 13 is prime so a(2) = 7.
8*9/4 = 18, 18-1 = 17 is prime, 18+1 = 19 is prime, so a(3) = 8.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A014601.

Select[Range[300], Or @@ PrimeQ[#*(#+1)/4 + {-1, 1}] &] (* Amiram Eldar, Jul 18 2021 *)

More terms from Amiram Eldar, Jul 18 2021

cp's OEIS Frontend

A350395 Numbers m such that a term with the largest coefficient in Product_{k=1..m} (1 + x^k) is unique.

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A053438 Expansion of (1+x+2*x^3)/((1-x)*(1-x^2)).

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A104569 Triangle read by rows: T(i,j) is the (i,j)-entry (1 <= j <= i) of the product Q*R of the infinite lower triangular matrices Q = [1; 1,3; 1,3,1; 1 3,1,3; ...] and R = [1; 1,1; 1,1,1; 1,1,1,1; ...].

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A274918 Numbers n such that the sum of numbers less than n that do not divide n is odd.

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A279064 Numbers n such that the sum of numbers less than n that do not divide n is even.

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A298364 Permutation of the natural numbers partitioned into quadruples [4k-2, 4k-1, 4k-3, 4k] for k > 0.

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A355278 Lower left of the Cayley table for the primes when made into a group using the bijection (2, 3, 5, 7, ...) -> (0, +1, -1, +2, ...) into (Z, +); read by rows.

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A374080 Expansion of Product_{k>=1} (1 - x^(4*k-1)) * (1 - x^(4*k)).

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A053438 Expansion of (1+x+2x^3)/((1-x)(1-x^2)).

A374080 Expansion of Product_{k>=1} (1 - x^(4k-1)) (1 - x^(4*k)).

A105173 Numbers k such that either k(k+1)/4 + 1 or k(k+1)/4 - 1 or both are primes.