A005069 -id:A005069 - cp's OEIS frontend

A005067 Sum of cubes of odd primes dividing n.

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

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

A005078 Sum of primes = 1 mod 4 dividing n.

Original entry on oeis.org

0, 0, 0, 0, 5, 0, 0, 0, 0, 5, 0, 0, 13, 0, 5, 0, 17, 0, 0, 5, 0, 0, 0, 0, 5, 13, 0, 0, 29, 5, 0, 0, 0, 17, 5, 0, 37, 0, 13, 5, 41, 0, 0, 0, 5, 0, 0, 0, 0, 5, 17, 13, 53, 0, 5, 0, 0, 29, 0, 5, 61, 0, 0, 0, 18, 0, 0, 17, 0, 5, 0, 0, 73, 37, 5, 0, 0, 13, 0, 5, 0, 41, 0, 0, 22, 0, 29, 0, 89, 5, 13, 0, 0, 0, 5, 0, 97, 0, 0, 5, 101

Offset: 1

N. J. A. Sloane

nonn

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

Cf. A005069, A005079, A005080, A005081, A005082, A008472, A059841.

Array[DivisorSum[#, # &, And[PrimeQ@ #, Mod[#, 4] == 1] &] &, 101] (* Michael De Vlieger, Jul 11 2017 *)
f[p_, e_] := If[Mod[p, 4] == 1, p, 0]; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 21 2022 *)

a(n)=my(f=factor(n)[,1]); sum(i=1,#f,if(f[i]%4==1,f[i])) \\ Charles R Greathouse IV, Mar 11 2014

(define (A005078 n) (if (= 1 n) 0 (+ (if (= 1 (modulo (A020639 n) 4)) (A020639 n) 0) (A005078 (A028234 n))))) ;; Antti Karttunen, Jul 11 2017

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

a(n) = A008472(n) - A005082(n) - 2*A059841(n). - Antti Karttunen, Jul 11 2017

More terms from Antti Karttunen, Jul 11 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

A279051 Sum of odd nonprime divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 16, 1, 1, 10, 1, 1, 22, 1, 1, 1, 26, 1, 37, 1, 1, 16, 1, 1, 34, 1, 36, 10, 1, 1, 40, 1, 1, 22, 1, 1, 70, 1, 1, 1, 50, 26, 52, 1, 1, 37, 56, 1, 58, 1, 1, 16, 1, 1, 94, 1, 66, 34, 1, 1, 70, 36, 1, 10, 1, 1, 116, 1, 78, 40, 1, 1, 118, 1, 1, 22, 86, 1, 88, 1, 1, 70, 92, 1, 94, 1, 96, 1, 1

Offset: 1

Ilya Gutkovskiy, Jan 17 2017

nonn

			a(9) = 10 because 9 has 3 divisors {1, 3, 9} among which 2 are odd nonprime {1, 9} therefore 1 + 9 = 10.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Odd Divisor Function
Index entries for sequences related to sums of divisors

Cf. A000593, A014076, A023890, A005069, A033272, A093641.

with(numtheory):
a:= n-> add(`if`(d::even or d::prime, 0, d), d=divisors(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 18 2017

Table[DivisorSum[n, #1 &, Mod[#1, 2] == 1 && ! PrimeQ[#1] &], {n, 97}]
nmax = 97; Rest[CoefficientList[Series[Sum[k x^k/(1 + x^k), {k, 1, nmax}] - Sum[Prime[k] x^Prime[k]/(1 - x^Prime[k]), {k, 2, nmax}], {x, 0, nmax}], x]]

a(n) = sumdiv(n, d, !isprime(d)*(d%2)*d); \\ Michel Marcus, Sep 18 2017

G.f.: A(x) = B(x) - C(x), where B(x) = Sum_{k>=1} k*x^k/(1 + x^k), C(x) = Sum_{k>=2} prime(k)*x^prime(k)/(1 - x^prime(k)).
a(n) = Sum_{d|n, d odd nonprime} d.
a(A093641(n)) = 1.

A279910 a(n) = Sum_{k=1..n} prime(k+1)*floor(n/prime(k+1)).

Original entry on oeis.org

0, 0, 3, 3, 8, 11, 18, 18, 21, 26, 37, 40, 53, 60, 68, 68, 85, 88, 107, 112, 122, 133, 156, 159, 164, 177, 180, 187, 216, 224, 255, 255, 269, 286, 298, 301, 338, 357, 373, 378, 419, 429, 472, 483, 491, 514, 561, 564, 571, 576, 596, 609, 662, 665, 681, 688, 710, 739, 798, 806, 867, 898, 908, 908, 926, 940, 1007, 1024, 1050, 1062

Offset: 1

Ilya Gutkovskiy, Dec 24 2016

Sum of all odd prime divisors of all positive integers <= n.

			For n = 7 the odd prime divisors of the first seven positive integers are {0}, {0}, {3}, {0}, {5}, {3}, {7} so a(7) = 0 + 0 + 3 + 0 + 5 + 3 + 7 = 18.

Index entries for sequences related to sums of divisors

Cf. A005069, A008472, A024924, A052928.

Table[Sum[Prime[k + 1] Floor[n/Prime[k + 1]], {k, 1, n}], {n, 70}]
Rest[nmax = 70; CoefficientList[Series[(1/(1 - x)) Sum[Prime[k] x^Prime[k]/(1 - x^Prime[k]), {k, 2, nmax}], {x, 0, nmax}], x]]

G.f.: (1/(1 - x))*Sum_{k>=2} prime(k)*x^prime(k)/(1 - x^prime(k)).
a(n) = -2*floor(n/2) + Sum_{k=1..n} prime(k)*floor(n/prime(k)) .
a(n) = A024924(n) - A052928(n).

A284233 Sum of odd prime power divisors of n (not including 1).

Original entry on oeis.org

0, 0, 3, 0, 5, 3, 7, 0, 12, 5, 11, 3, 13, 7, 8, 0, 17, 12, 19, 5, 10, 11, 23, 3, 30, 13, 39, 7, 29, 8, 31, 0, 14, 17, 12, 12, 37, 19, 16, 5, 41, 10, 43, 11, 17, 23, 47, 3, 56, 30, 20, 13, 53, 39, 16, 7, 22, 29, 59, 8, 61, 31, 19, 0, 18, 14, 67, 17, 26, 12, 71, 12, 73, 37, 33, 19, 18, 16, 79, 5

Offset: 1

Ilya Gutkovskiy, Mar 23 2017

			a(15) = 8 because 15 has 4 divisors {1, 3, 5, 15} among which 2 are odd prime powers {3, 5} therefore 3 + 5 = 8.

Eric Weisstein's World of Mathematics, Prime Power.
Index entries for sequences related to sums of divisors.

Cf. A000961, A005069, A023888, A023889, A038712, A061345, A065091 (fixed points), A087436 (number of odd prime power divisors of n), A206787, A246655, A284117.

nmax = 80; Rest[CoefficientList[Series[Sum[Boole[PrimePowerQ[k] && Mod[k, 2] == 1] k x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Total[Select[Divisors[n], PrimePowerQ[#] && Mod[#, 2] == 1 &]], {n, 80}]
f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; f[2, e_] := 0; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 24 2024 *)

G.f.: Sum_{k>=1} A061345(k)*x^A061345(k)/(1 - x^A061345(k)).
a(n) = Sum_{d|n, d = p^k, p prime, p > 2, k > 0} d.
a(p^k) = p*(p^k - 1)/(p - 1) for p is a prime > 2.
a(2^k*p) = p for p is a prime > 2.
a(2^k) = 0.
Additive with a(2^e) = 0, and a(p^e) = (p^(e+1)-1)/(p-1) - 1 for an odd prime p. - Amiram Eldar, Jul 24 2024

A281905 Expansion of Sum_{i>=2} prime(i)*x^prime(i)/(1 - x^prime(i)) / Product_{j>=1} (1 - x^j).

Original entry on oeis.org

0, 0, 3, 3, 11, 17, 35, 49, 84, 124, 199, 280, 426, 594, 858, 1172, 1654, 2224, 3061, 4066, 5472, 7196, 9543, 12391, 16196, 20857, 26921, 34351, 43924, 55574, 70419, 88455, 111142, 138697, 173025, 214527, 265895, 327831, 403825, 495234, 606755, 740371, 902507, 1096215, 1329912, 1608445, 1942926, 2340203

Offset: 1

Ilya Gutkovskiy, Feb 01 2017

nonn

Total sum of odd prime parts in all partitions of n.

Convolution of the sequences A000041 and A005069.

			a(5) = 11 because we have [5], [4, 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1], [1, 1, 1, 1, 1] and 5 + 3 + 3 = 11.

Index entries for related partition-counting sequences

Cf. A000041, A005069, A065091, A066186, A073118.

nmax = 48; Rest[CoefficientList[Series[Sum[Prime[i] x^Prime[i]/(1 - x^Prime[i]), {i, 2, nmax}]/Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]]

G.f.: Sum_{i>=2} prime(i)*x^prime(i)/(1 - x^prime(i)) / Product_{j>=1} (1 - x^j).

A338490 Sum of indices of distinct odd prime factors of n.

Original entry on oeis.org

0, 0, 2, 0, 3, 2, 4, 0, 2, 3, 5, 2, 6, 4, 5, 0, 7, 2, 8, 3, 6, 5, 9, 2, 3, 6, 2, 4, 10, 5, 11, 0, 7, 7, 7, 2, 12, 8, 8, 3, 13, 6, 14, 5, 5, 9, 15, 2, 4, 3, 9, 6, 16, 2, 8, 4, 10, 10, 17, 5, 18, 11, 6, 0, 9, 7, 19, 7, 11, 7, 20, 2, 21, 12, 5, 8, 9, 8, 22, 3, 2, 13, 23, 6, 10, 14, 12, 5, 24, 5

Offset: 1

Ilya Gutkovskiy, Nov 09 2020

nonn

			a(60) = a(2^2 * 3 * 5) = a(prime(1)^2 * prime(2) * prime(3)) = 2 + 3 = 5.

Index entries for sequences computed from indices in prime factorization

Cf. A000265, A000720, A005069, A005087, A056239, A066328.

nmax = 90; CoefficientList[Series[Sum[k x^Prime[k]/(1 - x^Prime[k]), {k, 2, nmax}], {x, 0, nmax}], x] // Rest

a(n) = vecsum(apply(primepi, (factor(n >> valuation(n, 2))[, 1]))); \\ Michel Marcus, Nov 10 2020

G.f.: Sum_{k>=2} k * x^prime(k) / (1 - x^prime(k)).
a(n) = Sum_{p|n, p odd prime} A000720(p).
a(n) = A066328(A000265(n)).

cp's OEIS Frontend

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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A005066 Sum of squares of odd primes dividing n.

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A005078 Sum of primes = 1 mod 4 dividing n.

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A005082 Sum of primes = 3 mod 4 dividing n.

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A279051 Sum of odd nonprime divisors of n.

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A279910 a(n) = Sum_{k=1..n} prime(k+1)*floor(n/prime(k+1)).

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