cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A286248 Triangle A286249 reversed.

Original entry on oeis.org

1, 2, 3, 4, 0, 3, 7, 0, 5, 10, 11, 0, 0, 0, 3, 16, 0, 0, 8, 5, 21, 22, 0, 0, 0, 0, 0, 3, 29, 0, 0, 0, 12, 0, 14, 36, 37, 0, 0, 0, 0, 0, 8, 0, 10, 46, 0, 0, 0, 0, 17, 0, 0, 5, 21, 56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 67, 0, 0, 0, 0, 0, 23, 0, 12, 19, 27, 78, 79, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 92, 0, 0, 0, 0, 0, 0, 30, 0, 0, 0, 0, 5, 21, 106, 0, 0, 0, 0, 0, 0, 0, 0, 0, 17, 0, 8, 0, 21
Offset: 1

Views

Author

Antti Karttunen, May 06 2017

Keywords

Comments

See A286249.

Examples

			Triangle begins:
   1,
   2, 3,
   4, 0, 3,
   7, 0, 5, 10,
  11, 0, 0,  0, 3,
  ...

Links

Crossrefs

Transpose: A286249 (triangle reversed).

Programs

  • Python
    from sympy import factorint
import math
def T(n, m): return ((n + m)**2 - n - 3*m + 2)//2
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def t(n, k): return 0 if n%k!=0 else T(a046523(n//k), k)
for n in range(1, 21): print([t(n, k) for k in range(1, n + 1)][::-1]) # Indranil Ghosh, May 08 2017
  • Scheme
    (define (A286248 n) (A286249tr (A002024 n) (A004736 n))) ;; For A286249tr, see A286249.

Formula

T(n,k) = A286249(k,n).

A286247 Triangular table: T(n,k) = 0 if k does not divide n, otherwise T(n,k) = P(A046523(k), n/k), where P is sequence A000027 used as a pairing function N x N -> N. Table is read by rows as T(1,1), T(2,1), T(2,2), etc.

Original entry on oeis.org

1, 2, 3, 4, 0, 3, 7, 5, 0, 10, 11, 0, 0, 0, 3, 16, 8, 5, 0, 0, 21, 22, 0, 0, 0, 0, 0, 3, 29, 12, 0, 14, 0, 0, 0, 36, 37, 0, 8, 0, 0, 0, 0, 0, 10, 46, 17, 0, 0, 5, 0, 0, 0, 0, 21, 56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 67, 23, 12, 19, 0, 27, 0, 0, 0, 0, 0, 78, 79, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 92, 30, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 21, 106, 0, 17, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21
Offset: 1

Views

Author

Antti Karttunen, May 06 2017

Keywords

Comments

Equally: square array A(n,k) = P(A046523(n), (n+k-1)/n) if n divides (n+k-1), 0 otherwise, read by descending antidiagonals as A(1,1), A(1,2), A(2,1), etc. Here P is a two-argument form of sequence A000027 used as a pairing function N x N -> N.
When viewed as a triangular table, this sequence packs the values of A046523(k) [which essentially stores the prime signature of k] and quotient n/k (when it is integral) to a single value with the pairing function A000027. The two "components" can be accessed with functions A002260 & A004736, which allows us to generate from this sequence (among other things) various sums related to the enumeration of aperiodic necklaces, because Moebius mu (A008683) obtains the same value on any representative of the same prime signature.
For example, we have:
Sum_{i=A000217(n-1) .. A000217(n)} [a(i) > 0] * mu(A002260(a(i))) * 2^(A004736(a(i))) = A027375(n).
and
Sum_{i=A000217(n-1) .. A000217(n)} [a(i) > 0] * mu(A002260(a(i))) * 3^(A004736(a(i))) = A054718(n).
Triangle A286249 has the same property.

Examples

			The first fifteen rows of triangle:
    1,
    2,  3,
    4,  0,  3,
    7,  5,  0, 10,
   11,  0,  0,  0,  3,
   16,  8,  5,  0,  0, 21,
   22,  0,  0,  0,  0,  0,  3,
   29, 12,  0, 14,  0,  0,  0, 36,
   37,  0,  8,  0,  0,  0,  0,  0, 10,
   46, 17,  0,  0,  5,  0,  0,  0,  0, 21,
   56,  0,  0,  0,  0,  0,  0,  0,  0,  0, 3,
   67, 23, 12, 19,  0, 27,  0,  0,  0,  0,  0, 78,
   79,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  3,
   92, 30,  0,  0,  0,  0,  5,  0,  0,  0,  0,  0,  0, 21,
  106,  0, 17,  0,  8,  0,  0,  0,  0,  0,  0,  0,  0,  0, 21
  (Note how triangle A286249 contains on each row the same numbers
  in the same "divisibility-allotted" positions, but in reverse order).
  ---------------------------------------------------------------
In the following examples: a = this sequence interpreted as a one-dimensional sequence, T = interpreted as a triangular table, A = interpreted as a square array, P = A000027 interpreted as a two-argument pairing function N x N -> N.
---
a(7) = T(4,1) = A(1,4) = P(A046523(1),4/1) = P(1,4) = 1+(((1+4)^2 - 1 - (3*4))/2) = 7.
a(30) = T(8,2) = A(2,7) = P(A046523(2),8/2) = P(2,4) = (1/2)*(2 + ((2+4)^2) - 2 - 3*4) = 12.

Links

Crossrefs

Cf. A000027, A002024, A002260, A004736, A008683, A027375, A046523, A051731, A054718, A286245, A286249, A286237.
Cf. A000124 (left edge of the triangle), A000217 (every number at the right edge is a triangular number).

Programs

  • Python
    from sympy import factorint
def T(n, m): return ((n + m)**2 - n - 3*m + 2)//2
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def t(n, k): return 0 if n%k!=0 else T(a046523(k), n//k)
for n in range(1, 21): print([t(n, k) for k in range(1, n + 1)]) # Indranil Ghosh, May 08 2017
  • Scheme
    (define (A286247 n) (A286247bi (A002260 n) (A004736 n)))
(define (A286247bi row col) (if (not (zero? (modulo (+ row col -1) row))) 0 (let ((a (A046523 row)) (b (quotient (+ row col -1) row))) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2)))))
;; Alternatively, with triangular indexing:
(define (A286247 n) (A286247tr (A002024 n) (A002260 n)))
(define (A286247tr n k) (if (not (zero? (modulo n k))) 0 (let ((a (A046523 k)) (b (/ n k))) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2)))))

Formula

As a triangle (with n >= 1, 1 <= k <= n):
T(n,k) = 0 if k does not divide n, otherwise T(n,k) = (1/2)*(2 + ((A046523(k)+(n/k))^2) - A046523(k) - 3*(n/k)).
T(n,k) = A051731(n,k) * A286245(n,k).

A286239 Triangular table: T(n,k) = 0 if k does not divide n, otherwise T(n,k) = P(A000010(n/k), k), where P is sequence A000027 used as a pairing function N x N -> N. Table is read by rows as T(1,1), T(2,1), T(2,2), etc.

Original entry on oeis.org

1, 1, 2, 3, 0, 4, 3, 2, 0, 7, 10, 0, 0, 0, 11, 3, 5, 4, 0, 0, 16, 21, 0, 0, 0, 0, 0, 22, 10, 5, 0, 7, 0, 0, 0, 29, 21, 0, 8, 0, 0, 0, 0, 0, 37, 10, 14, 0, 0, 11, 0, 0, 0, 0, 46, 55, 0, 0, 0, 0, 0, 0, 0, 0, 0, 56, 10, 5, 8, 12, 0, 16, 0, 0, 0, 0, 0, 67, 78, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 79, 21, 27, 0, 0, 0, 0, 22, 0, 0, 0, 0, 0, 0, 92, 36, 0, 19, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0, 0, 106
Offset: 1

Views

Author

Antti Karttunen, May 06 2017

Keywords

Comments

This sequence packs the values of phi(n/k) and k (whenever k divides n) to a single value, with the pairing function A000027. The two "components" can be accessed with functions A002260 & A004736, which allows us generate from this sequence various sums related to necklace enumeration (among other things).
For example, we have:
Sum_{i=A000217(n-1) .. A000217(n)} [a(i) > 0] * A002260(a(i)) * 2^(A004736(a(i))) = A053635(n).
and
Sum_{i=A000217(n-1) .. A000217(n)} [a(i) > 0] * A002260(a(i)) * 3^(A004736(a(i))) = A054610(n)
Triangle A286237 has the same property.

Examples

			The first fifteen rows of triangle:
   1,
   1,  2,
   3,  0,  4,
   3,  2,  0,  7,
  10,  0,  0,  0, 11,
   3,  5,  4,  0,  0, 16,
  21,  0,  0,  0,  0,  0, 22,
  10,  5,  0,  7,  0,  0,  0, 29,
  21,  0,  8,  0,  0,  0,  0,  0, 37,
  10, 14,  0,  0, 11,  0,  0,  0,  0, 46,
  55,  0,  0,  0,  0,  0,  0,  0,  0,  0, 56,
  10,  5,  8, 12,  0, 16,  0,  0,  0,  0,  0, 67,
  78,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, 79,
  21, 27,  0,  0,  0,  0, 22,  0,  0,  0,  0,  0,  0, 92,
  36,  0, 19,  0, 17,  0,  0,  0,  0,  0,  0,  0,  0,  0, 106
   -------------------------------------------------------------
Note how triangle A286237 contains on each row the same numbers in the same "divisibility-allotted" positions, but in reverse order.

Links

Crossrefs

Transpose: A286238.
Cf. A000124 (the right edge of the triangle).
Cf. A000010, A000027, A053635, A054610, A054523, A286156, A286237, A286249.

Programs

  • Python
    from sympy import totient
def T(n, m): return ((n + m)**2 - n - 3*m + 2)//2
def t(n, k): return 0 if n%k!=0 else T(totient(n//k), k)
for n in range(1, 21): print([t(n, k) for k in range(1, n + 1)]) # Indranil Ghosh, May 09 2017
  • Scheme
    (define (A286239 n) (A286239tr (A002024 n) (A002260 n)))
(define (A286239tr n k) (if (not (zero? (modulo n k))) 0 (let ((a (A000010 (/ n k))) (b k)) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2)))))

Formula

As a triangle (with n >= 1, 1 <= k <= n):
T(n,k) = 0 if k does not divide n, otherwise T(n,k) = (1/2)*(2 + ((A000010(n/k)+k)^2) - A000010(n/k) - 3*k).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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