A005063 -id:A005063 - cp's OEIS frontend

A351198 Sum of the 10th powers of the primes dividing n.

0, 1024, 59049, 1024, 9765625, 60073, 282475249, 1024, 59049, 9766649, 25937424601, 60073, 137858491849, 282476273, 9824674, 1024, 2015993900449, 60073, 6131066257801, 9766649, 282534298, 25937425625, 41426511213649, 60073, 9765625, 137858492873, 59049, 282476273

Offset: 1

Wesley Ivan Hurt, Feb 04 2022

nonn

Inverse Möbius transform of n^10 * c(n), where c(n) is the prime characteristic (A010051). - Wesley Ivan Hurt, Jun 22 2024

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sum of the k-th powers of the primes dividing n for k=0..10 : A001221 (k=0), A008472 (k=1), A005063 (k=2), A005064 (k=3), A005065 (k=4), A351193 (k=5), A351194 (k=6), A351195 (k=7), A351196 (k=8), A351197 (k=9), this sequence (k=10).

Cf. A010051, A030629 (p^10).

Array[DivisorSum[#, #^10 &, PrimeQ] &, 50]
f[p_, e_] := p^10; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 20 2022 *)

a(n) = vecsum(apply(x->x^10, factor(n)[, 1])); \\ Michel Marcus, Feb 05 2022

from sympy import primefactors
def A351198(n): return sum(p**10 for p in primefactors(n)) # Chai Wah Wu, Feb 04 2022

a(n) = Sum_{p|n, p prime} p^10.
G.f.: Sum_{k>=1} prime(k)^10 * x^prime(k) / (1 - x^prime(k)). - Ilya Gutkovskiy, Feb 16 2022
Additive with a(p^e) = p^10. - Amiram Eldar, Jun 20 2022
a(n) = Sum_{d|n} d^10 * c(d), where c = A010051. - Wesley Ivan Hurt, Jun 22 2024

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

A005083 Sum of squares of primes = 3 mod 4 dividing n.

Original entry on oeis.org

0, 0, 9, 0, 0, 9, 49, 0, 9, 0, 121, 9, 0, 49, 9, 0, 0, 9, 361, 0, 58, 121, 529, 9, 0, 0, 9, 49, 0, 9, 961, 0, 130, 0, 49, 9, 0, 361, 9, 0, 0, 58, 1849, 121, 9, 529, 2209, 9, 49, 0, 9, 0, 0, 9, 121, 49, 370, 0, 3481, 9, 0, 961, 58, 0, 0, 130, 4489, 0, 538, 49, 5041, 9, 0, 0, 9, 361, 170, 9, 6241, 0, 9, 0, 6889, 58

Offset: 1

N. J. A. Sloane

nonn

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

Cf. A000290, A005063, A005079, A005082, A005084, A005085, A059841.

Array[DivisorSum[#, #^2 &, And[PrimeQ@ #, Mod[#, 4] == 3] &] &, 84] (* Michael De Vlieger, Jul 11 2017 *)
f[p_, e_] := If[Mod[p, 4] == 3, 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) == 3, p^2)); \\ Michel Marcus, Jul 11 2017

(define (A005083 n) (if (= 1 n) 0 (+ (if (= 3 (modulo (A020639 n) 4)) (A000290 (A020639 n)) 0) (A005083 (A028234 n))))) ;; Antti Karttunen, Jul 11 2017

Additive with a(p^e) = p^2 if p = 3 (mod 4), 0 otherwise.

a(n) = A005063(n) - A005079(n) - 4*A059841(n). - Antti Karttunen, Jul 11 2017

More terms from Antti Karttunen, Jul 11 2017

A351196 Sum of the 8th powers of the primes dividing n.

Original entry on oeis.org

0, 256, 6561, 256, 390625, 6817, 5764801, 256, 6561, 390881, 214358881, 6817, 815730721, 5765057, 397186, 256, 6975757441, 6817, 16983563041, 390881, 5771362, 214359137, 78310985281, 6817, 390625, 815730977, 6561, 5765057, 500246412961, 397442, 852891037441, 256

Offset: 1

Wesley Ivan Hurt, Feb 04 2022

nonn

Inverse Möbius transform of n^8 * c(n), where c(n) is the prime characteristic (A010051). - Wesley Ivan Hurt, Jun 22 2024

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sum of the k-th powers of the primes dividing n for k=0..10 : A001221 (k=0), A008472 (k=1), A005063 (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.

Array[DivisorSum[#, #^8 &, PrimeQ] &, 50]
f[p_, e_] := p^8; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 20 2022 *)

from sympy import primefactors
def A351196(n): return sum(p**8 for p in primefactors(n)) # Chai Wah Wu, Feb 05 2022

a(n) = Sum_{p|n, p prime} p^8.
G.f.: Sum_{k>=1} prime(k)^8 * x^prime(k) / (1 - x^prime(k)). - Ilya Gutkovskiy, Feb 16 2022
Additive with a(p^e) = p^8. - Amiram Eldar, Jun 20 2022
a(n) = Sum_{d|n} d^8 * c(d), where c = A010051. - Wesley Ivan Hurt, Jun 22 2024

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

A005075 Sum of squares of primes = 2 mod 3 dividing n.

Original entry on oeis.org

0, 4, 0, 4, 25, 4, 0, 4, 0, 29, 121, 4, 0, 4, 25, 4, 289, 4, 0, 29, 0, 125, 529, 4, 25, 4, 0, 4, 841, 29, 0, 4, 121, 293, 25, 4, 0, 4, 0, 29, 1681, 4, 0, 125, 25, 533, 2209, 4, 0, 29, 289, 4, 2809, 4, 146, 4, 0, 845, 3481, 29, 0, 4, 0, 4, 25, 125, 0, 293, 529, 29, 5041, 4, 0, 4, 25, 4, 121, 4, 0, 29, 0, 1685, 6889, 4, 314

Offset: 1

N. J. A. Sloane

nonn

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

Cf. A000290, A005063, A005071, A005074, A005076, A005077, A008472, A079978.

Array[DivisorSum[#, #^2 &, And[PrimeQ@ #, Mod[#, 3] == 2] &] &, 85] (* Michael De Vlieger, Jul 11 2017 *)
f[p_, e_] := If[Mod[p, 3] == 2, 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])%3) == 2, p^2)); \\ Michel Marcus, Jul 11 2017

(define (A005075 n) (if (= 1 n) 0 (+ (A000290 (if (= 2 (modulo (A020639 n) 3)) (A020639 n) 0)) (A005075 (A028234 n))))) ;; Antti Karttunen, Jul 10 2017

Additive with a(p^e) = p^2 if p = 2 (mod 3), 0 otherwise.

a(n) = A005063(n) - A005071(n) - 9*A079978(n). - Antti Karttunen, Jul 10 2017

More terms from Antti Karttunen, Jul 10 2017

A200768 Sum of the n-th powers of the distinct prime divisors of n.

Original entry on oeis.org

0, 4, 27, 16, 3125, 793, 823543, 256, 19683, 9766649, 285311670611, 535537, 302875106592253, 678223089233, 30531927032, 65536, 827240261886336764177, 387682633, 1978419655660313589123979, 95367432689201, 558545874543637210, 81402749386839765307625

Offset: 1

Michel Lagneau, Nov 22 2011

nonn

			a(6) = 793 because the distinct prime divisors of 6 are 2 and 3, and 2^6 + 3^6 = 793.

Robert Israel, Table of n, a(n) for n = 1..388

Cf. A005063, A008472, A199583.

f:= proc(n) local p;
   add(p^n, p = numtheory:-factorset(n))
end proc:
map(f, [$1..30]); # Robert Israel, Feb 20 2024

Prepend[Array[Plus@@First[Transpose[FactorInteger[#]^#]]&,100,2],0]
Join[{0},Table[Total[FactorInteger[n][[All,1]]^n],{n,2,25}]] (* Harvey P. Dale, Jan 23 2021 *)

a(n) = Sum_{p|n} p^n. - Wesley Ivan Hurt, Jun 14 2021

A114989 Numbers whose sum of squares of distinct prime factors is prime.

Original entry on oeis.org

6, 10, 12, 14, 18, 20, 24, 26, 28, 34, 36, 40, 48, 50, 52, 54, 56, 68, 72, 74, 80, 94, 96, 98, 100, 104, 105, 108, 112, 134, 136, 144, 146, 148, 160, 162, 188, 192, 194, 196, 200, 206, 208, 216, 224, 231, 250, 268, 272, 273, 274, 288, 292, 296, 315, 320, 324, 326

Offset: 1

Jonathan Vos Post, Feb 22 2006

A005063 is "sum of squares of primes dividing n." Hence this is the sum of squares of prime factors analog of A114522 "numbers n such that sum of distinct prime divisors of n is prime." Note the distinction between A005063 and A067666 is "sum of squares of prime factors of n (counted with multiplicity)."

			a(1) = 6 because 6 = 2 * 3 and 2^2 + 3^2 = 13 is prime.
a(2) = 10 because 10 = 2 * 5 and 2^2 + 5^2 = 29 is prime.
a(3) = 12 because 12 = 2^2 * 3 and 2^2 + 3^2 = 13 is prime (note that we are not counting the prime factors with multiplicity).
a(4) = 14 because 14 = 2 * 7 and 2^2 + 7^2 = 53 is prime.
a(8) = 26 because 26 = 2 * 3 and 2^2 + 13^2 = 173 is prime.
a(10) = 34 because 34 = 2 * 17 and 2^2 + 17^2 = 293 is prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A005063, A067666, A014612, A014613, A069273, A069279, A069281, A114522.

with(numtheory): a:=proc(n) local DPF: DPF:=factorset(n): if isprime(sum(DPF[j]^2,j=1..nops(DPF)))=true then n else fi end: seq(a(n),n=1..400); # Emeric Deutsch, Mar 07 2006

Select[Range[400],PrimeQ[Total[Transpose[FactorInteger[#]][[1]]^2]]&] (* Harvey P. Dale, Jan 16 2016 *)

PARI
```
is(n)=isprime(norml2(factor(n)[,1]))
```

{k such that A005063(k) is prime}. {k such that A005063(k) is an element of A000040}. {k = (for distinct i, j, ... prime(i)^e_1 * prime(j)^e_2 * ...) such that (prime(i)^2 * prime(j)^2 * ...) is prime}.

More terms from Emeric Deutsch, Mar 07 2006

A144711 Numbers n such that [sum_i=1..r (p(i)^2)]/r = c^2. p(i) prime divisors of n, c integer.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 119, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 161, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211

Offset: 1

Ctibor O. Zizka, Sep 19 2008

A005063(n)/A001221(n) = c^2.
Also numbers n such that the root mean square (quadratic mean) of the prime divisors of n is an integer.
These numbers are power of primes (p^k with k>=1) (A000961) or in A255580. - Daniel Lignon, Feb 26 2015

Daniel Lignon, Table of n, a(n) for n = 1..1000

Cf. A005063, A001221, A140480.

A005063 := proc(n) add(p^2,p=numtheory[factorset](n)) ; end: A001221 := proc(n) nops(numtheory[factorset](n)) ; end: isA144711 := proc(n) local sofpr ; sofpr := A001221(n) ; if sofpr <> 0 then issqr(A005063(n)/sofpr) ; else false ; fi; end: for n from 1 to 500 do if isA144711(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, Sep 20 2008

Select[Range[2,1000],IntegerQ[RootMeanSquare[Select[Divisors[#],PrimeQ]]]&] (* Daniel Lignon, Feb 26 2015 *)

More terms from R. J. Mathar, Sep 20 2008

A189120 Sum of squares of nonprime divisors of n.

Original entry on oeis.org

1, 1, 1, 17, 1, 37, 1, 81, 82, 101, 1, 197, 1, 197, 226, 337, 1, 442, 1, 517, 442, 485, 1, 837, 626, 677, 811, 997, 1, 1262, 1, 1361, 1090, 1157, 1226, 1898, 1, 1445, 1522, 2181, 1, 2438, 1, 2437, 2332, 2117, 1, 3397, 2402, 3226, 2602, 3397, 1, 4087, 3026, 4197, 3250, 3365

Offset: 1

Jonathan Vos Post, Apr 17 2011

a(p) = 1 for p prime.

			a(12) = 197 because the divisors of 12 are {1, 2, 3, 4, 6, 12}, the subset of nonprime divisors are {1, 4, 6, 12}, and 1^2 + 4^2 + 6^2 + 12^2 = 197.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A023890 (sum of the nonprime divisors of n).

Cf. A001157, A005063.

A189120 := proc(n) local a,d; a := 0 ; for d in numtheory[divisors](n) do if not isprime(d) then a := a+d^2 ; end if; end do: a ; end proc: # R. J. Mathar, Apr 17 2011

Table[Total[Select[Divisors[n], ! PrimeQ[#] &]^2], {n, 50}]

a(n) = Sum_{k|n, k not prime} k^2.
G.f.: Sum_{k>=1} k^2*x^(k+1)/(1 - x^k) - prime(k)^2*x^(prime(k)+1)/(1 - x^prime(k)). - Ilya Gutkovskiy, Jan 01 2017
a(n) = A001157(n) - A005063(n). - Wesley Ivan Hurt, Sep 04 2022

cp's OEIS Frontend

A351198 Sum of the 10th powers of the primes dividing n.

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

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A005083 Sum of squares of primes = 3 mod 4 dividing n.

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A351196 Sum of the 8th powers of the primes dividing n.

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A005066 Sum of squares of odd primes dividing n.

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A005075 Sum of squares of primes = 2 mod 3 dividing n.

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A200768 Sum of the n-th powers of the distinct prime divisors of n.

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