A005066 -id:A005066 - cp's OEIS frontend

A005063 Sum of squares of primes dividing n.

0, 4, 9, 4, 25, 13, 49, 4, 9, 29, 121, 13, 169, 53, 34, 4, 289, 13, 361, 29, 58, 125, 529, 13, 25, 173, 9, 53, 841, 38, 961, 4, 130, 293, 74, 13, 1369, 365, 178, 29, 1681, 62, 1849, 125, 34, 533, 2209, 13, 49, 29, 298, 173, 2809, 13, 146, 53, 370, 845, 3481, 38, 3721

Offset: 1

N. J. A. Sloane

nonn

The set of these terms apart from 0 is A048261. - Bernard Schott, Feb 10 2022

Inverse Möbius transform of n^2 * c(n), where c(n) is the prime characteristic (A010051). - Wesley Ivan Hurt, Jun 22 2024

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000290, A005066, A005071, A005075, A005079, A005083, A059841, A079978.
Cf. A067666, A081403, A048261. - Franklin T. Adams-Watters, May 03 2009
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).
Cf. A010051.

A005063 := proc(n)
        add(d^2, d= numtheory[factorset](n)) ;
end proc;
seq(A005063(n),n=1..40) ; # R. J. Mathar, Nov 08 2011

a[n_] := Total[FactorInteger[n][[All, 1]]^2]; a[1]=0; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Mar 20 2017 *)
Array[DivisorSum[#, #^2 &, PrimeQ] &, 61] (* Michael De Vlieger, Jul 11 2017 *)

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

a(n) = vecsum(apply(x->x^2, factor(n)[, 1])); \\ Michel Marcus, Sep 19 2020

from sympy import primefactors
def a(n): return sum(p**2 for p in primefactors(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 11 2017

(define (A005063 n) (if (= 1 n) 0 (+ (A000290 (A020639 n)) (A005063 (A028234 n))))) ;; Antti Karttunen, Jul 10 2017

Additive with a(p^e) = p^2.
G.f.: Sum_{k>=1} prime(k)^2*x^prime(k)/(1 - x^prime(k)). - Ilya Gutkovskiy, Dec 24 2016
From Antti Karttunen, Jul 11 2017: (Start)
a(n) = A005066(n) + 4*A059841(n).
a(n) = A005079(n) + A005083(n) + 4*A059841(n).
a(n) = A005071(n) + A005075(n) + 9*A079978(n).
(End)
Dirichlet g.f.: primezeta(s-2)*zeta(s). - Benedict W. J. Irwin, Jul 11 2018
a(n) = Sum_{p|n, p prime} p^2. - Wesley Ivan Hurt, Feb 04 2022
a(n) = Sum_{d|n} d^2 * c(d), where c = A010051. - Wesley Ivan Hurt, Jun 22 2024

More terms from Franklin T. Adams-Watters, May 03 2009

A005069 Sum of odd primes dividing n.

Original entry on oeis.org

0, 0, 3, 0, 5, 3, 7, 0, 3, 5, 11, 3, 13, 7, 8, 0, 17, 3, 19, 5, 10, 11, 23, 3, 5, 13, 3, 7, 29, 8, 31, 0, 14, 17, 12, 3, 37, 19, 16, 5, 41, 10, 43, 11, 8, 23, 47, 3, 7, 5, 20, 13, 53, 3, 16, 7, 22, 29, 59, 8, 61, 31, 10, 0, 18, 14, 67, 17, 26, 12, 71, 3, 73, 37, 8, 19, 18, 16, 79

Offset: 1

N. J. A. Sloane

Primes counted without multiplicity. - Harvey P. Dale, Aug 28 2019

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000265, A005066, A005067, A005068, A005078, A005082, A008472, A020639, A028234, A059841.

a = {0, 0}; For[n = 3, n < 80, n++, su = 0; b = FactorInteger[n]; For[i = 1, i < Length[b] + 1, i++, If[OddQ[b[[i, 1]]], su = su + b[[i, 1]]]]; AppendTo[a, su]]; a (* Stefan Steinerberger, Jun 02 2007 *)
Array[DivisorSum[#, # &, And[PrimeQ@ #, OddQ@ #] &] &, 79] (* Michael De Vlieger, Jul 11 2017 *)
Join[{0},Table[Total[FactorInteger[n][[All,1]]/.(2->0)],{n,2,100}]] (* Harvey P. Dale, Aug 28 2019 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, if (((p=f[k,1])%2) == 1, p)); \\ Michel Marcus, Jul 11 2017

(define (A005069 n) (cond ((= 1 n) 0) ((even? n) (A005069 (/ n 2))) (else (+ (A020639 n) (A005069 (A028234 n)))))) ;; Antti Karttunen, Jul 10 2017

Additive with a(p^e) = 0 if p = 2, p otherwise.
G.f.: Sum_{k>=2} prime(k)*x^prime(k)/(1 - x^prime(k)). - Ilya Gutkovskiy, Dec 24 2016
From Antti Karttunen, Jul 10 & 11 2017: (Start)
a(1) = 0; after which, for even n: a(n) = a(n/2), for odd n: a(n) = A020639(n) + a(A028234(n)).
a(n) = A008472(A000265(n)) = A008472(n) - 2*A059841(n).
a(n) = A005078(n) + A005082(n).
(End)

More terms from Stefan Steinerberger, Jun 02 2007

A005067 Sum of cubes of odd primes dividing n.

Original entry on oeis.org

0, 0, 27, 0, 125, 27, 343, 0, 27, 125, 1331, 27, 2197, 343, 152, 0, 4913, 27, 6859, 125, 370, 1331, 12167, 27, 125, 2197, 27, 343, 24389, 152, 29791, 0, 1358, 4913, 468, 27, 50653, 6859, 2224, 125, 68921, 370, 79507, 1331, 152, 12167, 103823, 27, 343, 125, 4940, 2197, 148877, 27, 1456, 343, 6886, 24389, 205379, 152

Offset: 1

N. J. A. Sloane

nonn

Harvey P. Dale (terms 1 .. 1000) & Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000265, A000578, A005064, A005066, A005068, A005069, A020639, A028234, A065091.

Join[{0},Table[Total[Select[Transpose[FactorInteger[n]][[1]],OddQ]^3],{n,2,50}]] (* Harvey P. Dale, Jun 09 2016 *)
Array[DivisorSum[#, #^3 &, And[PrimeQ@ #, OddQ@ #] &] &, 60] (* Michael De Vlieger, Jul 11 2017 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, if (((p=f[k,1])%2) == 1, p^3)); \\ Michel Marcus, Jul 11 2017

(define (A005067 n) (cond ((= 1 n) 0) ((even? n) (A005067 (/ n 2))) (else (+ (A000578 (A020639 n)) (A005067 (A028234 n)))))) ;; Antti Karttunen, Jul 10 2017

Additive with a(p^e) = 0 if p = 2, p^3 otherwise.
G.f.: Sum_{k>=2} prime(k)^3*x^prime(k)/(1 - x^prime(k)). - Ilya Gutkovskiy, Jan 06 2017
From Antti Karttunen, Jul 10 2017: (Start)
a(1) = 0; after which, for even n: a(n) = a(n/2), for odd n: a(n) = A020639(n)^3 + a(A028234(n)).
a(n) = A005064(A000265(n)).
(End)

More terms from Antti Karttunen, Jul 10 2017

A005068 Sum of 4th powers of odd primes dividing n.

Original entry on oeis.org

0, 0, 81, 0, 625, 81, 2401, 0, 81, 625, 14641, 81, 28561, 2401, 706, 0, 83521, 81, 130321, 625, 2482, 14641, 279841, 81, 625, 28561, 81, 2401, 707281, 706, 923521, 0, 14722, 83521, 3026, 81, 1874161, 130321, 28642, 625, 2825761, 2482, 3418801, 14641, 706, 279841, 4879681, 81, 2401, 625, 83602, 28561, 7890481, 81

Offset: 1

N. J. A. Sloane

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000265, A000583, A005065, A005066, A005067, A005069, A020639, A028234.

Array[DivisorSum[#, #^4 &, And[PrimeQ@ #, OddQ@ #] &] &, 54] (* Michael De Vlieger, Jul 11 2017 *)
f[2, e_] := 0; f[p_, e_] := p^4; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 50] (* Amiram Eldar, Jun 20 2022 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, if (((p=f[k,1])%2) == 1, p^4)); \\ Michel Marcus, Jul 11 2017

(define (A005068 n) (cond ((= 1 n) 0) ((even? n) (A005068 (/ n 2))) (else (+ (A000583 (A020639 n)) (A005068 (A028234 n)))))) ;; Antti Karttunen, Jul 10 2017

Additive with a(p^e) = 0 if p = 2, p^4 otherwise.
From Antti Karttunen, Jul 10 2017: (Start)
a(1) = 0; after which, for even n: a(n) = a(n/2), for odd n: a(n) = A020639(n)^4 + a(A028234(n)).
a(n) = A005065(A000265(n)).
(End)
G.f.: Sum_{k>=2} prime(k)^4 * x^prime(k) / (1 - x^prime(k)). - Ilya Gutkovskiy, Aug 19 2021

More terms from Antti Karttunen, Jul 10 2017

A347156 Sum of squares of distinct prime divisors of n that are < sqrt(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 0, 4, 0, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 38, 0, 4, 9, 4, 25, 13, 0, 4, 9, 29, 0, 13, 0, 4, 34, 4, 0, 13, 0, 29, 9, 4, 0, 13, 25, 53, 9, 4, 0, 38, 0, 4, 58, 4, 25, 13, 0, 4, 9, 78, 0, 13, 0, 4, 34, 4, 49, 13, 0, 29

Offset: 1

Ilya Gutkovskiy, Aug 20 2021

nonn

Cf. A001157, A005063, A005066, A098002, A333806, A333808, A339353, A347157, A347158.

Table[DivisorSum[n, #^2 &, # < Sqrt[n] && PrimeQ[#] &], {n, 1, 80}]
nmax = 80; CoefficientList[Series[Sum[Prime[k]^2 x^(Prime[k] (Prime[k] + 1))/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

G.f.: Sum_{k>=1} prime(k)^2 * x^(prime(k)*(prime(k) + 1)) / (1 - x^prime(k)).

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A005063 Sum of squares of primes dividing n.

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A005069 Sum of odd primes dividing n.

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A005067 Sum of cubes of odd primes dividing n.

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A005068 Sum of 4th powers of odd primes dividing n.

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A347156 Sum of squares of distinct prime divisors of n that are < sqrt(n).

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