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A005063 Sum of squares of primes dividing n.

%I A005063 #58 Jun 22 2024 19:05:41
%S A005063 0,4,9,4,25,13,49,4,9,29,121,13,169,53,34,4,289,13,361,29,58,125,529,
%T A005063 13,25,173,9,53,841,38,961,4,130,293,74,13,1369,365,178,29,1681,62,
%U A005063 1849,125,34,533,2209,13,49,29,298,173,2809,13,146,53,370,845,3481,38,3721
%N A005063 Sum of squares of primes dividing n.
%C A005063 The set of these terms apart from 0 is A048261. - _Bernard Schott_, Feb 10 2022
%C A005063 Inverse Möbius transform of n^2 * c(n), where c(n) is the prime characteristic (A010051). - _Wesley Ivan Hurt_, Jun 22 2024
%H A005063 Alois P. Heinz, <a href="/A005063/b005063.txt">Table of n, a(n) for n = 1..10000</a>
%F A005063 Additive with a(p^e) = p^2.
%F A005063 G.f.: Sum_{k>=1} prime(k)^2*x^prime(k)/(1 - x^prime(k)). - _Ilya Gutkovskiy_, Dec 24 2016
%F A005063 From _Antti Karttunen_, Jul 11 2017: (Start)
%F A005063 a(n) = A005066(n) + 4*A059841(n).
%F A005063 a(n) = A005079(n) + A005083(n) + 4*A059841(n).
%F A005063 a(n) = A005071(n) + A005075(n) + 9*A079978(n).
%F A005063 (End)
%F A005063 Dirichlet g.f.: primezeta(s-2)*zeta(s). - _Benedict W. J. Irwin_, Jul 11 2018
%F A005063 a(n) = Sum_{p|n, p prime} p^2. - _Wesley Ivan Hurt_, Feb 04 2022
%F A005063 a(n) = Sum_{d|n} d^2 * c(d), where c = A010051. - _Wesley Ivan Hurt_, Jun 22 2024
%p A005063 A005063 := proc(n)
%p A005063         add(d^2, d= numtheory[factorset](n)) ;
%p A005063 end proc;
%p A005063 seq(A005063(n),n=1..40) ; # _R. J. Mathar_, Nov 08 2011
%t A005063 a[n_] := Total[FactorInteger[n][[All, 1]]^2]; a[1]=0; Table[a[n], {n, 1, 60}] (* _Jean-François Alcover_, Mar 20 2017 *)
%t A005063 Array[DivisorSum[#, #^2 &, PrimeQ] &, 61] (* _Michael De Vlieger_, Jul 11 2017 *)
%o A005063 (PARI) a(n)=local(fm,t);fm=factor(n);t=0;for(k=1,matsize(fm)[1],t+=fm[k,1]^2);t \\ _Franklin T. Adams-Watters_, May 03 2009
%o A005063 (PARI) a(n) = vecsum(apply(x->x^2, factor(n)[, 1])); \\ _Michel Marcus_, Sep 19 2020
%o A005063 (Scheme) (define (A005063 n) (if (= 1 n) 0 (+ (A000290 (A020639 n)) (A005063 (A028234 n))))) ;; _Antti Karttunen_, Jul 10 2017
%o A005063 (Python)
%o A005063 from sympy import primefactors
%o A005063 def a(n): return sum(p**2 for p in primefactors(n))
%o A005063 print([a(n) for n in range(1, 101)]) # _Indranil Ghosh_, Jul 11 2017
%Y A005063 Cf. A000290, A005066, A005071, A005075, A005079, A005083, A059841, A079978.
%Y A005063 Cf. A067666, A081403, A048261. - _Franklin T. Adams-Watters_, May 03 2009
%Y A005063 Sum of the k-th powers of the primes dividing n for k=0..10 : A001221 (k=0), A008472 (k=1), this sequence (k=2), A005064 (k=3), A005065 (k=4), A351193 (k=5), A351194 (k=6), A351195 (k=7), this sequence (k=8), A351197 (k=9), A351198 (k=10).
%Y A005063 Cf. A010051.
%K A005063 nonn
%O A005063 1,2
%A A005063 _N. J. A. Sloane_
%E A005063 More terms from _Franklin T. Adams-Watters_, May 03 2009