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.
2 is a term because it is not a solution of Sum_{d|k} A069359(d), k >= 1. See proof in Comments.
nn = 60; b = MixedRadix[Reverse@ Prime@ Range@ PrimePi[nn + 1]]; Table[FromDigits[#, b] &@ Reverse@ If[n == 1, {0}, Function[k, ReplacePart[Table[0, {PrimePi[k[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, k]]@ FactorInteger@ n], {n, nn}] (* Version 10.2, or *)
f[w_List] := Total[Times @@@ Transpose@ {Map[Times @@ # &, Prime@ Range@ Range[0, Length@ w - 1]], Reverse@ w}]; Table[f@ Reverse@ If[n == 1, {0}, Function[k, ReplacePart[Table[0, {PrimePi[k[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, k]]@ FactorInteger@ n], {n, 60}] (* Michael De Vlieger, Aug 30 2016 *)
A276085(n) = { my(f = factor(n), pr=1, i=1, s=0); for(k=1, #f~, while(i <= primepi(f[k, 1])-1, pr *= prime(i); i++); s += f[k, 2]*pr); (s); }; \\ Antti Karttunen, Nov 11 2024
from sympy import primorial, primepi, factorint
def a002110(n):
return 1 if n<1 else primorial(n)
def a(n):
f=factorint(n)
return sum(f[i]*a002110(primepi(i) - 1) for i in f)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 22 2017
a(12) = 10 because the prime divisors of 12 are 2 and 3 so we have: 12/2 + 12/3 = 6 + 4 = 10. - _Geoffrey Critzer_, Mar 17 2015
[0] cat [n*&+[1/p: p in PrimeDivisors(n)]:n in [2..80]]; // Marius A. Burtea, Jan 21 2020
A069359 := n -> add(n/d, d = select(isprime, numtheory[divisors](n))): seq(A069359(i), i = 1..20); # Peter Luschny, Jan 31 2012 # second Maple program: a:= n-> n*add(1/i[1], i=ifactors(n)[2]): seq(a(n), n=1..100); # Alois P. Heinz, Oct 23 2019
f[list_, i_] := list[[i]]; nn = 100; a = Table[n, {n, 1, nn}]; b =
Table[If[PrimeQ[n], 1, 0], {n, 1, nn}]; Table[DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}] (* Geoffrey Critzer, Mar 17 2015 *)
a(n) = n*sumdiv(n, d, isprime(d)/d); \\ Michel Marcus, Mar 18 2015
a(n) = my(ps=factor(n)[,1]~);sum(k=1,#ps,n\ps[k]) \\ Franklin T. Adams-Watters, Apr 09 2015
from sympy import primefactors def A069359(n): return sum(n//p for p in primefactors(n)) # Chai Wah Wu, Feb 05 2022
def A069359(n) : D = filter(is_prime, divisors(n)) return add(n/d for d in D) print([A069359(i) for i in (1..20)]) # Peter Luschny, Jan 31 2012
[p^2+1: p in PrimesUpTo(300)]; // Vincenzo Librandi, Oct 31 2014
Table[Prime[n]^2 + 1, {n, 41}] (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)
A066872(n) = (prime(n)^2 + 1); \\ Harry J. Smith, Apr 02 2010
Table[DivisorSum[n, #*PrimeNu[n/#] &, PrimeQ[#] &], {n, 100}]
A369744(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i,1]*omega(n/f[i, 1]))); \\ Antti Karttunen, Jan 23 2025
Table[Sum[d*PrimeNu[d], {d, Divisors[n]}], {n, 1, 100}] (* Vaclav Kotesovec, Aug 18 2021 *)
A329354(n) = sumdiv(n,d,omega(d)*d);
A329347(n) = sumdiv(n,d,(n/d)*bigomega(d));
a(4) = (1 + 2) + (2 + 4) = 9. a(120) = a(3*5*2^3) = 4*6*(3*8 + 4*4 + 4*2 + 3) = 1224.
divisorSumPrime[n_] := DivisorSum[n, 1+1/# &, PrimeQ[#] &]; a[n_] := DivisorSum[n, #*divisorSumPrime[#]& ]; Array[a, 70] (* Amiram Eldar, Dec 17 2018 *)
a(n) = sumdiv(n, d, d*sumdiv(d, p, isprime(p)*(1+1/p))); \\ Michel Marcus, Dec 17 2018
a(n) = my(f = factor(n), res = 0); fordiv(n, d, for(i = 1, #f~, v = valuation(d, f[i, 1]); res+=(d * ((v > 0) + (v < f[i, 2]))))); res \\ David A. Corneth, Dec 17 2018
up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A329380(n) = { my(m=1); fordiv(n,d,m *= A276086(d)^omega(n/d)); (m); };
v329381 = rgs_transform(vector(up_to, n, A329380(n)));
A329381(n) = v329381[n];
Comments