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.
815634435 = 3*5*7*11*547*1291 is included as in base-2 (A007088) it is written as 110000100111011001100000000011_2, thus A000120(815634435) = 12, while its nonnegative deficiency (A033879) is 2*815634435 - sigma(815634435) = 6 < 12.
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; }; A011772(n) = { if(n==1, return(1)); my(f=factor(if(n%2, n, 2*n)), step=vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); forstep(m=step, 2*n, step, if(m*(m-1)/2%n==0, return(m-1)); if(m*(m+1)/2%n==0, return(m))); }; \\ From A011772 A344769(n) = (A005187(n) - A011772(n));
from itertools import combinations from math import prod from sympy import factorint, divisors from sympy.ntheory.modular import crt def A344769(n): c = 2*n-bin(n).count('1') plist = [p**q for p, q in factorint(2*n).items()] if len(plist) == 1: return int(c+1+(plist[0] % 2 - 2)*n) return int(c-min(min(crt([m,2*n//m],[0,-1])[0],crt([2*n//m,m],[0,-1])[0]) for m in (prod(d) for l in range(1,len(plist)//2+1) for d in combinations(plist,l)))) # Chai Wah Wu, Jun 03 2021
G.f.: A(x) = 1 + 1/2*x - 1/8*x^2 + 5/16*x^3 - 53/128*x^4 + 127/256*x^5 - 677/1024*x^6 + 2221/2048*x^7 + ... + a(n)/2^A005187(n)*x^n + ... where 2^A005187(n) is also the denominator of [x^n] 1/sqrt(1-x). GENERATING METHOD. We can illustrate the generating method for g.f. A(x) as follows. Given F(n) = F(n-1)^2 + (2*x)^(2^n-1) for n >= 1 with F(0) = 1, the first few polynomials generated by F(n) begin F(0) = 1, F(1) = F(0)^2 + x^(2^1-1) = 1 + x, F(2) = F(1)^2 + x^(2^2-1) = 1 + 2*x + x^2 + x^3, F(3) = F(2)^2 + x^(2^3-1) = 1 + 4*x + 6*x^2 + 6*x^3 + 5*x^4 + 2*x^5 + x^6 + x^7. ... The 2^n-th roots of F(n) tend to the limit of the g.f.: F(1)^(1/2^1) = 1 + 1/2*x - 1/8*x^2 + 1/16*x^3 - 5/128*x^4 + 7/256*x^5 - 21/1024*x^6 + 33/2048*x^7 - 429/32768*x^8 + ... F(2)^(1/2^2) = 1 + 1/2*x - 1/8*x^2 + 5/16*x^3 - 53/128*x^4 + 127/256*x^5 - 677/1024*x^6 + 1965/2048*x^7 - 46797/32768*x^8 + ... F(3)^(1/2^3) = 1 + 1/2*x - 1/8*x^2 + 5/16*x^3 - 53/128*x^4 + 127/256*x^5 - 677/1024*x^6 + 2221/2048*x^7 - 61133/32768*x^8 + ... ... The limit of this process equals the g.f. A(x) of this sequence. Note: the sum of the coefficients in F(n) equals A003095(n): 1, 2 = 1 + 1, 5 = 1 + 2 + 1 + 1, 26 = 1 + 4 + 6 + 6 + 5 + 2 + 1 + 1, ... The last n coefficients in F(n) read backwards are Catalan numbers (A000108). POWERS OF A(x). The coefficients of x^k in the 2^n powers of the g.f. A(x) begin: A^(2^0) = [1, 1/2, -1/8, 5/16, -53/128, 127/256, -677/1024, 2221/2048, ...], A^(2^1) = [1, 1, 0, 1/2, -1/2, 1/2, -5/8, 9/8, -2, 53/16, -89/16, 155/16, ...], A^(2^2) = [1, 2, 1, 1, 0, 0, 0, 1/2, -1, 3/2, -5/2, 9/2, -8, 14, -197/8, 44, ...], A^(2^3) = [1, 4, 6, 6, 5, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1/2, -2, 5, ...], A^(2^4) = [1, 8, 28, 60, 94, 116, 114, 94, 69, 44, 26, 14, 5, 2, 1, 1, 0, 0, ...].
{a(n) = my(F=1,A,L); if(n==0,A=1,L=ceil(log(n+1)/log(2)); for(k=1,L, F = F^2 + x^(2^k-1) +x*O(x^n)); A = polcoeff(F^(1/2^L),n)); numerator(A)}
for(n=0,32, print1(a(n),", "))
1, 2, 2, 8, 8, 8, 16, 16, 16, 16.
Flatten[Table[Denominator[Binomial[2n, n]/4^n], {n, 0, 19}, {n + 1}]] (* Alonso del Arte, Jan 07 2013 *)
(* Checking with the antidiagonals *) diff = Table[ Differences[ CoefficientList[ Series[1/Sqrt[1 - x], {x, 0, 9}], x], n], {n, 0, 9}]; Table[ diff[[n-k+1,k]] // Denominator,{n,0,10},{k,1,n}] // Flatten (* Jean-François Alcover, Jan 07 2013 *)
Flatten[Table[2^IntegerExponent[(2*n)!, 2], {n, 0, 19}, {n + 1}]]; (* Jean-François Alcover, Mar 27 2013, after A005187 *)
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