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.
a(12) = a(2^2) * a(3) = (2#)^2 * (3#) = 2^2 * 6 = 24 a(45) = (3#)^2 * (5#) = (2*3)^2 * (2*3*5) = 1080 (as 45 = 3^2 * 5).
a[n_] := a[n] = Module[{f = FactorInteger[n], p, e}, If[Length[f]>1, Times @@ a /@ Power @@@ f, {{p, e}} = f; Times @@ (Prime[Range[PrimePi[p]]]^e)]]; a[1] = 1; Table[a[n], {n, 1, 42}] (* Jean-François Alcover, Feb 24 2015 *)
Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}], {n, 42}] (* Michael De Vlieger, Mar 18 2017 *)
primorial(n)=prod(i=1,primepi(n),prime(i)) a(n)=my(f=factor(n)); prod(i=1,#f~, primorial(f[i,1])^f[i,2]) \\ Charles R Greathouse IV, Jun 28 2015
from sympy import primerange, factorint
from operator import mul
def P(n): return reduce(mul, [i for i in primerange(2, n + 1)])
def a(n):
f = factorint(n)
return 1 if n==1 else reduce(mul, [P(i)**f[i] for i in f])
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, May 14 2017
def sharp_primorial(n): return sloane.A002110(prime_pi(n)) def p(f): return sharp_primorial(f[0])^f[1] [prod(p(f) for f in factor(n)) for n in range (1,51)] # Giuseppe Coppoletta, Feb 07 2015
The top left corner of the array: k= 1 2 3 4 5 6 7 8 9 10 11 12 2k = 2 4 6 8 10 12 14 16 18 20 22 24 ----+-------------------------------------------------------------------------- n=1 | 3, 7, 2, 15, 9, 7, 12, 31, 13, 21, 18, 5, 2 | 4, 13, 8, 40, 32, 26, 16, 121, 124, 104, 56, 16, 3 | 6, 31, 48, 156, 72, 248, 84, 781, 342, 372, 108, 1248, 4 | 8, 57, 96, 400, 16, 684, 144, 2801, 152, 114, 160, 4800, 5 | 12, 133, 168, 1464, 216, 1862, 240, 16105, 2196, 2394, 288, 20496, 6 | 14, 183, 252, 2380, 280, 3294, 336, 30941, 4298, 3660, 420, 2520, 7 | 18, 307, 360, 5220, 432, 6140, 540, 88741, 6858, 7368, 576, 104400, 8 | 20, 381, 480, 7240, 600, 9144, 640, 137561, 11060, 11430, 40, 173760, 9 | 24, 553, 720, 12720, 768, 16590, 912, 292561, 20904, 17696, 1008, 381600, etc.
up_to = 105; A246278sq(row,col) = if(1==row,2*col, my(f = factor(2*col)); for(i=1, #f~, f[i,1] = prime(primepi(f[i,1])+(row-1))); factorback(f)); A017665(n) = numerator(sigma(n)/n); A341605sq(row,col) = A017665(A246278sq(row,col)); A341605list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A341605sq(col,(a-(col-1))))); (v); }; v341605 = A341605list(up_to); A341605(n) = v341605[n];
For n = 120, sigma(120) = 360 >= 2*120, thus 120 is not deficient, and we get the next number by applying the prime shift, A003961(120) = 945, and sigma(945) = 1920 >= 945*2, so neither 945 is deficient, so we prime shift once again, and A003961(945) = 9625, which is deficient, as sigma(9625) = 14976 < 2*9625. Thus after two iteration steps we encounter a deficient number, and therefore a(120) = 2.
Array[-1 + Length@ NestWhileList[If[# == 1, 1, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}]] &, #, DivisorSigma[1, #] >= 2 # &] &, 120] (* Michael De Vlieger, Aug 27 2020 *)
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961 A336835(n) = { my(i=0); while(sigma(n) >= (n+n), i++; n = A003961(n)); (i); };
The top left corner of the array begins as: n/k | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 ----|---------------------------------------------------------------------- 0 | 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 3, 3, 3, 1, 3, 2, 3, 3, 3, 3, 3, 1 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, etc. A337474 gives the distance in each column to the first 1 in that column, being 0 for columns 1, 2, 4, 8, 16, ..., where 1 is already in the top row.
up_to = 105858-1; \\ Or 105-1. A337473sq(n, k) = if(1==k,k, my(f=factor(k), h = #f~, prevpid=primepi(f[h,1]), e=f[h,2], p, s=1); forstep(i=h-1,0,-1, if(!i,pid=0,pid=primepi(f[i,1])); forstep(j=prevpid,(1+pid),-1, p=prime(j+n); s *= ((p^(1+e)-1)/((p-1)*(p^e)))); if(!pid,return(floor(s))); prevpid = pid; e += f[i,2]); floor(s)); A337473list(up_to) = { my(v = vector(1+up_to), i=0); for(a=0, oo, for(b=1, a, i++; if(i > #v, return(v)); v[i] = A337473sq(b-1, (a-(b-1))))); (v); }; v337473 = A337473list(up_to); A337473(n) = v337473[1+n];
Comments