A363735 -id:A363735 - cp's OEIS frontend

A363421 a(n) = Sum_{k=0..n}(n^[not(k | n)] - n^[k | n]), where '[ ]' denotes the Iverson bracket.

1, 0, -1, 0, -3, 8, -5, 24, 7, 32, 27, 80, 11, 120, 91, 112, 105, 224, 119, 288, 171, 280, 315, 440, 207, 480, 475, 520, 459, 728, 435, 840, 651, 832, 891, 952, 665, 1224, 1147, 1216, 975, 1520, 1107, 1680, 1419, 1496, 1755, 2024, 1363, 2112, 1911, 2200, 2091

Offset: 0

Peter Luschny, Jun 27 2023

sign

Peter Luschny, Table of n, a(n) for n = 0..10000

Cf. A000005, A113704, A363734, A363735.

A363421[n_]:=If[n==0,1,n^2-2(n-1)DivisorSigma[0,n]-1];Array[A363421,100,0] (* Paolo Xausa, Aug 06 2023 *)

from sympy import divisor_count
def A363421(n): return n**2-2*(n-1)*divisor_count(n)-1 if n else 1 # Chai Wah Wu, Jun 28 2023

print([sum(n^(not k.divides(n)) - n^k.divides(n) for k in srange(n+1)) for n in srange(53)])

a(n) = n^2 - 2*(n - 1)*tau(n) - 1 for n >= 1, where tau = A000005.

A363734 a(n) = Sum_{k=0..n} n^divides(k, n), where divides(k, n) = 1 if k divides n, otherwise 0.

Original entry on oeis.org

0, 2, 5, 8, 14, 14, 27, 20, 37, 34, 47, 32, 79, 38, 67, 72, 92, 50, 121, 56, 135, 102, 107, 68, 209, 98, 127, 132, 191, 86, 263, 92, 219, 162, 167, 172, 352, 110, 187, 192, 353, 122, 371, 128, 303, 310, 227, 140, 519, 194, 345, 252, 359, 158, 479, 272, 497

Offset: 0

Peter Luschny, Jun 27 2023

nonn

Peter Luschny, Table of n, a(n) for n = 0..10000

Cf. A113704, A000005, A363735, A363421.

A363734[n_]:=If[n==0,0,n+1+(n-1)DivisorSigma[0,n]];Array[A363734,100,0] (* Paolo Xausa, Aug 06 2023 *)

from sympy import divisor_count
def A363734(n): return (n-1)*divisor_count(n)+n+1 if n else 0 # Chai Wah Wu, Jun 28 2023

print([sum(n^k.divides(n) for k in srange(n+1)) for n in srange(57)])

a(n) = (n - 1) * tau(n) + n + 1 for n >= 1, where tau = A000005.
a(n) + A363735(n) = (n + 1)^2.
A363735(n) - a(n) = A363421(n).

A363733 Array read by upwards antidiagonals. The family of polynomials generated by the divisibility matrix (A113704) evaluated over the nonnegative integers.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 6, 3, 1, 0, 3, 10, 12, 4, 1, 0, 2, 22, 30, 20, 5, 1, 0, 4, 34, 93, 68, 30, 6, 1, 0, 2, 78, 246, 276, 130, 42, 7, 1, 0, 4, 130, 768, 1028, 655, 222, 56, 8, 1, 0, 3, 278, 2190, 4180, 3130, 1338, 350, 72, 9, 1

Offset: 0

Peter Luschny, Jun 27 2023

The name expresses the 'row view' of the array. The 'column view' regards the array as the collection of the inverse Möbius transforms of the power sequences k^n = 0^n, 1^n, 2^n, .... (n >= 0). Viewed this way, the array is a generalization of the number of divisors sequence tau (A000005), to which it reduces in the case k = 1.

The array has offset (0, 0). It uses the usual definition of 'k divides n' as described in Apostol, rather than the shortened version, which restricts to values k > 0 as some programs do (but not SageMath). Such a restriction makes sense in the context of rational numbers but not in the case of natural numbers.

			Array A(n, k) starts:
  [0] 1, 1,   1,    1,     1,      1,       1,       1,        1, ... A000012
  [1] 0, 1,   2,    3,     4,      5,       6,       7,        8, ... A001477
  [2] 0, 2,   6,   12,    20,     30,      42,      56,       72, ... A002378
  [3] 0, 2,  10,   30,    68,    130,     222,     350,      520, ... A034262
  [4] 0, 3,  22,   93,   276,    655,    1338,    2457,     4168, ...
  [5] 0, 2,  34,  246,  1028,   3130,    7782,   16814,    32776, ... A131471
  [6] 0, 4,  78,  768,  4180,  15780,   46914,  118048,   262728, ...
  [7] 0, 2, 130, 2190, 16388,  78130,  279942,  823550,  2097160, ... A190578
  [8] 0, 4, 278, 6654, 65812, 391280, 1680954, 5767258, 16781384, ...
   A000005,A055895,A363913, ...                             A066108 (diagonal)
.
Triangle T(n, k) starts:
  [0] 1;
  [1] 0, 1;
  [2] 0, 1,   1;
  [3] 0, 2,   2,   1;
  [4] 0, 2,   6,   3,    1;
  [5] 0, 3,  10,  12,    4,   1;
  [6] 0, 2,  22,  30,   20,   5,   1;
  [7] 0, 4,  34,  93,   68,  30,   6,  1;
  [8] 0, 2,  78, 246,  276, 130,  42,  7, 1;
  [9] 0, 4, 130, 768, 1028, 655, 222, 56, 8, 1;

Tom M. Apostol, Introduction to Analytic Number Theory, Springer 1976, p. 14.

Cf. A113704 (in compact form A113705), A000005 (column 1), A055895 (column 2), A363913 (column 3), A001477 (row 1), A002378 (row 2), A034262 (row 3), A131471 (row 5), A190578 (row 7), A363912 (row sums), A066108 (main diagonal of array).

Cf. A363734, A363735, A363421.

divides := (k, n) -> ifelse(k = n or (k > 0 and irem(n, k) = 0), 1, 0):
A := (n, k) -> local j; add(divides(j, n) * k^j, j = 0 ..n):
for n from 0 to 8 do seq(A(n, k), k = 0..8) od;
# If we introduce the 'inverse Möbius transform' InvMoebius acting on s ...
InvMoebius := (s, n) -> local j; add(divides(j, n) * s(j), j = 0 ..n):
# ... the transposed array is given by applying InvMoebius to the powers r^m:
seq(lprint(seq(InvMoebius(m -> r^m, n), n = 0..8)), r = 0..8);
# For instance we see that the number of divisors is the inverse
# Moebius transform of the constant sequence s = 1.

def A(n, k): return sum(j.divides(n) * k^j for j in (0..n))
for n in srange(9): print([A(n, k) for k in (0..8)])

A(n, k) = Sum_{j=0..n} divides(j, n) * k^j, where divides(k, n) <-> [k = n or (k > 0 and n mod k = 0)], and '[ ]' denotes the Iverson bracket.

The columns are the inverse Möbius transforms of the powers x^n, x >= 0.

A363913 a(n) = Sum_{k=0..n} divides(k, n) * 3^k, where divides(k, n) = 1 if k divides n, otherwise 0.

Original entry on oeis.org

1, 3, 12, 30, 93, 246, 768, 2190, 6654, 19713, 59304, 177150, 532290, 1594326, 4785168, 14349180, 43053375, 129140166, 387440940, 1162261470, 3486843786, 10460355420, 31381236768, 94143178830, 282430075332, 847288609689, 2541867422664, 7625597504700, 22876797240210

Offset: 0

Peter Luschny, Jun 28 2023

nonn

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

Cf. A000007 (m = 0), A000005 (m = 1), A055895 (m = 2), this sequence (m = 3).

Cf. A113704, A363733, A363734, A363735, A363421.

A363913:= func< n | n eq 0 select 1 else 3*(&+[3^(d-1): d in Divisors(n)]) >;
[A363913(n): n in [0..40]]; // G. C. Greubel, Jun 26 2024

divides := (k, n) -> ifelse(k = n or (k > 0 and irem(n, k) = 0), 1, 0):
a := n -> local j; add(divides(j, n) * 3^j, j = 0 ..n): seq(a(n), n = 0..28);

A363913[n_]:= If[n==0, 1, 3*DivisorSum[n, 3^(#-1) &]];
Table[A363913[n], {n,0,40}] (* G. C. Greubel, Jun 26 2024 *)

from sympy import divisors
def A363913(n): return sum(3**k for k in divisors(n,generator=True)) if n else 1 # Chai Wah Wu, Jun 28 2023

def a(n): return sum(3^k * k.divides(n) for k in srange(n+1))
print([a(n) for n in range(29)])

a(n) = Sum_{j=0..n} A113704(j, n) * m^j for m = 3; for other cases see the crossreferences.

a(n) = 3*A034730(n), n>=1. - R. J. Mathar, Jul 04 2023

cp's OEIS Frontend

A363421 a(n) = Sum_{k=0..n}(n^[not(k | n)] - n^[k | n]), where '[ ]' denotes the Iverson bracket.

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A363734 a(n) = Sum_{k=0..n} n^divides(k, n), where divides(k, n) = 1 if k divides n, otherwise 0.

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A363733 Array read by upwards antidiagonals. The family of polynomials generated by the divisibility matrix (A113704) evaluated over the nonnegative integers.

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A363913 a(n) = Sum_{k=0..n} divides(k, n) * 3^k, where divides(k, n) = 1 if k divides n, otherwise 0.

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