A005083 -id:A005083 - 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

A005079 Sum of squares of primes = 1 mod 4 dividing n.

Original entry on oeis.org

0, 0, 0, 0, 25, 0, 0, 0, 0, 25, 0, 0, 169, 0, 25, 0, 289, 0, 0, 25, 0, 0, 0, 0, 25, 169, 0, 0, 841, 25, 0, 0, 0, 289, 25, 0, 1369, 0, 169, 25, 1681, 0, 0, 0, 25, 0, 0, 0, 0, 25, 289, 169, 2809, 0, 25, 0, 0, 841, 0, 25, 3721, 0, 0, 0, 194, 0, 0, 289, 0, 25, 0, 0, 5329, 1369, 25, 0, 0, 169, 0, 25, 0, 1681, 0, 0, 314

Offset: 1

N. J. A. Sloane

nonn

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

Cf. A000290, A005063, A005078, A005080, A005081, A005083, A059841.

Array[DivisorSum[#, #^2 &, And[PrimeQ@ #, Mod[#, 4] == 1] &] &, 85] (* Michael De Vlieger, Jul 11 2017 *)
f[p_, e_] := If[Mod[p, 4] == 1, p^2, 0]; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 21 2022 *)

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

(define (A005079 n) (if (= 1 n) 0 (+ (if (= 1 (modulo (A020639 n) 4)) (A000290 (A020639 n)) 0) (A005079 (A028234 n))))) ;; Antti Karttunen, Jul 11 2017

Additive with a(p^e) = p^2 if p = 1 (mod 4), 0 otherwise.

a(n) = A005063(n) - A005083(n) - 4*A059841(n). - Antti Karttunen, Jul 11 2017

More terms from Antti Karttunen, Jul 11 2017

A005085 Sum of 4th powers of primes = 3 mod 4 dividing n.

Original entry on oeis.org

0, 0, 81, 0, 0, 81, 2401, 0, 81, 0, 14641, 81, 0, 2401, 81, 0, 0, 81, 130321, 0, 2482, 14641, 279841, 81, 0, 0, 81, 2401, 0, 81, 923521, 0, 14722, 0, 2401, 81, 0, 130321, 81, 0, 0, 2482, 3418801, 14641, 81, 279841, 4879681, 81, 2401, 0, 81, 0, 0, 81, 14641, 2401, 130402, 0, 12117361, 81, 0, 923521, 2482, 0, 0, 14722, 20151121, 0, 279922

Offset: 1

N. J. A. Sloane

nonn

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

Cf. A000583, A005065, A005081, A005082, A005083, A005084, A059841.

Array[DivisorSum[#, #^4 &, And[PrimeQ@ #, Mod[#, 4] == 3] &] &, 69] (* Michael De Vlieger, Jul 11 2017 *)
f[p_, e_] := If[Mod[p, 4] == 3, p^4, 0]; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 21 2022 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, if (((p=f[k,1])%4) == 3, p^4)); \\ Michel Marcus, Jul 11 2017

(define (A005085 n) (if (= 1 n) 0 (+ (if (= 3 (modulo (A020639 n) 4)) (A000583 (A020639 n)) 0) (A005085 (A028234 n))))) ;; Antti Karttunen, Jul 11 2017

Additive with a(p^e) = p^4 if p = 3 (mod 4), 0 otherwise.

a(n) = A005065(n) - A005081(n) - 16*A059841(n). - Antti Karttunen, Jul 11 2017

More terms from Antti Karttunen, Jul 11 2017

A005066 Sum of squares of odd primes dividing n.

Original entry on oeis.org

0, 0, 9, 0, 25, 9, 49, 0, 9, 25, 121, 9, 169, 49, 34, 0, 289, 9, 361, 25, 58, 121, 529, 9, 25, 169, 9, 49, 841, 34, 961, 0, 130, 289, 74, 9, 1369, 361, 178, 25, 1681, 58, 1849, 121, 34, 529, 2209, 9, 49, 25, 298, 169, 2809, 9, 146, 49, 370, 841, 3481, 34, 3721, 961, 58, 0, 194, 130, 4489, 289, 538, 74, 5041, 9, 5329, 1369

Offset: 1

N. J. A. Sloane

nonn

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

Cf. A000265, A000290, A005063, A005067, A005068, A005069, A005079, A005083, A020639, A028234.

Table[Total[Select[Divisors[n],OddQ[#]&&PrimeQ[#]&]^2],{n,60}] (* Harvey P. Dale, May 02 2012 *)
Array[DivisorSum[#, #^2 &, And[PrimeQ@ #, OddQ@ #] &] &, 74] (* Michael De Vlieger, Jul 11 2017 *)
f[2, e_] := 0; f[p_, e_] := p^2; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 20 2022 *)

a(n) = sumdiv(n, d, ((d%2) && isprime(d))*d^2); \\ Michel Marcus, Jan 04 2017

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

(define (A005066 n) (cond ((= 1 n) 0) ((even? n) (A005066 (/ n 2))) (else (+ (A000290 (A020639 n)) (A005066 (A028234 n)))))) ;; Antti Karttunen, Jul 10 2017

Additive with a(p^e) = 0 if p = 2, p^2 otherwise.
G.f.: Sum_{k>=2} prime(k)^2*x^prime(k)/(1 - x^prime(k)). - Ilya Gutkovskiy, Jan 04 2017
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)^2 + a(A028234(n)).
a(n) = A005063(A000265(n)).
a(n) = A005079(n) + A005083(n).
(End)

More terms from Antti Karttunen, Jul 10 2017

A005082 Sum of primes = 3 mod 4 dividing n.

Original entry on oeis.org

0, 0, 3, 0, 0, 3, 7, 0, 3, 0, 11, 3, 0, 7, 3, 0, 0, 3, 19, 0, 10, 11, 23, 3, 0, 0, 3, 7, 0, 3, 31, 0, 14, 0, 7, 3, 0, 19, 3, 0, 0, 10, 43, 11, 3, 23, 47, 3, 7, 0, 3, 0, 0, 3, 11, 7, 22, 0, 59, 3, 0, 31, 10, 0, 0, 14, 67, 0, 26, 7, 71, 3, 0, 0, 3, 19, 18, 3, 79, 0, 3, 0, 83, 10, 0, 43, 3, 11, 0, 3, 7, 23, 34, 47, 19, 3, 0, 7, 14, 0, 0, 3, 103

Offset: 1

N. J. A. Sloane

nonn

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

Cf. A005069, A005078, A005083, A005084, A005085, A008472, A059841.

Table[Total[Select[Transpose[FactorInteger[n]][[1]],Mod[#,4] == 3&]],{n,80}] (* Harvey P. Dale, Jan 18 2015 *)
Array[DivisorSum[#, # &, And[PrimeQ@ #, Mod[#, 4] == 3] &] &, 103] (* Michael De Vlieger, Jul 11 2017 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, if (((p=f[k,1])%4) == 3, p)); \\ Michel Marcus, Jul 11 2017

(define (A005082 n) (if (= 1 n) 0 (+ (* (A079978 (modulo (A020639 n) 4)) (A020639 n)) (A005082 (A028234 n))))) ;; Antti Karttunen, Jul 11 2017

Additive with a(p^e) = p if p = 3 (mod 4), 0 otherwise.
From Antti Karttunen, Jul 11 2017: (Start)
a(1) = 0; for n > 1, a(n) = (A079978(A020639(n) mod 4)*A020639(n)) + a(A028234(n)).
a(n) = A008472(n) - A005078(n) - 2*A059841(n).
(End)

More terms from Antti Karttunen, Jul 11 2017

A005084 Sum of cubes of primes = 3 mod 4 dividing n.

Original entry on oeis.org

0, 0, 27, 0, 0, 27, 343, 0, 27, 0, 1331, 27, 0, 343, 27, 0, 0, 27, 6859, 0, 370, 1331, 12167, 27, 0, 0, 27, 343, 0, 27, 29791, 0, 1358, 0, 343, 27, 0, 6859, 27, 0, 0, 370, 79507, 1331, 27, 12167, 103823, 27, 343, 0, 27, 0, 0, 27, 1331, 343, 6886, 0, 205379, 27, 0, 29791, 370, 0, 0, 1358, 300763, 0, 12194, 343, 357911

Offset: 1

N. J. A. Sloane

nonn

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

Cf. A000578, A005064, A005080, A005082, A005083, A005085, A059841.

Array[DivisorSum[#, #^3 &, And[PrimeQ@ #, Mod[#, 4] == 3] &] &, 71] (* Michael De Vlieger, Jul 11 2017 *)
f[p_, e_] := If[Mod[p, 4] == 3, p^3, 0]; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 21 2022 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, if (((p=f[k,1])%4) == 3, p^3)); \\ Michel Marcus, Jul 11 2017

(define (A005084 n) (if (= 1 n) 0 (+ (if (= 3 (modulo (A020639 n) 4)) (A000578 (A020639 n)) 0) (A005084 (A028234 n))))) ;;  Antti Karttunen, Jul 11 2017

Additive with a(p^e) = p^3 if p = 3 (mod 4), 0 otherwise.

a(n) = A005064(n) - A005080(n) - 8*A059841(n). - Antti Karttunen, Jul 11 2017

More terms from Antti Karttunen, Jul 11 2017

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A005063 Sum of squares of primes dividing n.

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A005079 Sum of squares of primes = 1 mod 4 dividing n.

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A005085 Sum of 4th powers of primes = 3 mod 4 dividing n.

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A005066 Sum of squares of odd primes dividing n.

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A005082 Sum of primes = 3 mod 4 dividing n.

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A005084 Sum of cubes of primes = 3 mod 4 dividing n.

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Mathematica

PARI

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