A285730 -id:A285730 - cp's OEIS frontend

A285731 Transpose of square array A285730.

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

Offset: 1

Antti Karttunen, Apr 28 2017

See A285730.

			The top left 25 x 4 corner of the array:
  1 2 3 2 5 3 7 2 3  5 11  3 13  7  5  2 17  3 19  5  7 11 23  3  5
  1 2 3 4 5 6 7 4 9 10 11  6 13 14 15  4 17  9 19 10 21 22 23  6 25
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  8 17 18 19 20 21 22 23 12 25
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Antti Karttunen, Table of n, a(n) for n = 1..7260; the first 120 antidiagonals of array

Transpose of A285730.

Cf. A006530 (the topmost row).

With[{nn = 14}, Function[s, Table[s[[n, #]] &[k - n + 1], {k, nn}, {n, k, 1, -1}]]@ MapIndexed[PadRight[#1, nn, First@ #2] &, Table[FoldList[Times, Reverse@ Flatten[FactorInteger[n] /. {p_, e_} /; e > 0 :> ConstantArray[p, e]]], {n, nn}]]] // Flatten (* Michael De Vlieger, Apr 28 2017 *)

from sympy import primefactors
def a006530(n): return 1 if n==1 else max(primefactors(n))
def A(n, k): return a006530(n) if k==1 else a006530(n)*A(n//a006530(n), k - 1)
for n in range(1, 21): print([A(n - k + 1, k) for k in range(1, n + 1)]) # Indranil Ghosh, Apr 28 2017

(define (A285731 n) (A285730bi (A004736 n) (A002260 n))) ;; For A285730bi see A285730.

A285388 a(n) = numerator of ((1/n) * Sum_{k=0..n^2-1} binomial(2k,k)/4^k).

Original entry on oeis.org

1, 35, 36465, 300540195, 79006629023595, 331884405207627584403, 22292910726608249789889125025, 11975573020964041433067793888190275875, 411646257111422564507234009694940786177843149765, 56592821660064550728377610673427602421565368547133335525825

Offset: 1

Ralf Steiner, Apr 18 2017

Editorial comment: This sequence arose from Ralf Steiner's attempt to prove Legendre's conjecture that there is a prime between N^2 and (N+1)^2 for all N. - N. J. A. Sloane, May 01 2017

Indranil Ghosh, Table of n, a(n) for n = 1..40

Cf. A000079, A000265, A056220, A060757, A201555, A285389 (denominators), A285406, A280655 (similar), A190732 (2/sqrt(Pi)), A285738 (greatest prime factor), A285717, A285730, A285786, A286264, A000290 (n^2), A056220 (2*n^2 -1), A286127 (sum a(n-1)/a(n)).

[Numerator( n*(n^2+1)*Catalan(n^2)/2^(2*n^2-1) ): n in [1..21]]; // G. C. Greubel, Dec 11 2021

Table[Numerator[Sum[Binomial[2k,k]/4^k,{k,0,n^2-1}]/n],{n,1,10}]
Numerator[Table[2^(1-2 n^2) n Binomial[2 n^2,n^2],{n,1,10}]] (* Ralf Steiner, Apr 22 2017 *)

A285388(n) = numerator((2^(1 - 2*(n^2)))*n*binomial(2*(n^2), n^2)); \\ Antti Karttunen, Apr 27 2017

a(n) = m=n*binomial(2*n^2, n^2);m>>valuation(m,2) \\ David A. Corneth, Apr 27 2017

from sympy import binomial, Integer
def a(n): return (Integer(2)**(1 - 2*n**2)*n*binomial(2*n**2, n**2)).numerator # Indranil Ghosh, Apr 27 2017

[numerator( n*(n^2+1)*catalan_number(n^2)/2^(2*n^2-1) ) for n in (1..20)] # G. C. Greubel, Dec 11 2021

a(n) is numerator of n*binomial(2 n^2, n^2)/2^(2*n^2 - 1). - Ralf Steiner, Apr 26 2017
a(n) = numerator(n*A201555(n) / (A060757(n)/2)) = n*A201555(n) / 2^(A285717(n)) = A000265(n*A201555(n)). [Using Ralf Steiner's formula and A285717(n) <= A056220(n), cf. A285406.] - Antti Karttunen, Apr 27 2017
Limit_{i->oo} a(i)*A285389(i+1)/(a(i+1)*A285389(i)) = 1. - Ralf Steiner, May 03 2017

Edited (including the removal of the author's claim that this leads to a proof of the Legendre conjecture) by N. J. A. Sloane, May 01 2017
Formula section edited by M. F. Hasler, May 02 2017
Edited by N. J. A. Sloane, May 10 2017

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A285731 Transpose of square array A285730.

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A285388 a(n) = numerator of ((1/n) * Sum_{k=0..n^2-1} binomial(2k,k)/4^k).

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