A069359 -id:A069359 - cp's OEIS frontend

A351245 a(n) = n^5 * Sum_{p|n, p prime} 1/p^5.

0, 1, 1, 32, 1, 275, 1, 1024, 243, 3157, 1, 8800, 1, 16839, 3368, 32768, 1, 66825, 1, 101024, 17050, 161083, 1, 281600, 3125, 371325, 59049, 538848, 1, 867151, 1, 1048576, 161294, 1419889, 19932, 2138400, 1, 2476131, 371536, 3232768, 1, 4629701, 1, 5154656, 818424, 6436375, 1

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

nonn

Dirichlet convolution of A010051(n) and n^5. - Wesley Ivan Hurt, Jul 15 2025

			a(6) = 275; a(6) = 6^5 * Sum_{p|6, p prime} 1/p^5 = 7776 * (1/2^5 + 1/3^5) = 275.

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

Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k = 0..10: A001221 (k=0), A069359 (k=1), A322078 (k=2), A351242 (k=3), A351244 (k=4), this sequence (k=5), A351246 (k=6), A351247 (k=7), A351248 (k=8), A351249 (k=9), A351262 (k=10).

Cf. A000040, A001221, A010051, A059378, A085966.

Array[#^5*DivisorSum[#, 1/#^5 &, PrimeQ] &, 47] (* Stefano Spezia, Jul 15 2025 *)

a(n) = my(f = factor(n)); sum(i = 1, #f~, (n/f[i,1])^5) \\ David A. Corneth, Jul 15 2025

a(A000040(n)) = 1.
Dirichlet g.f.: zeta(s-5)*primezeta(s). This follows because Sum_{n>=1} a(n)/n^s = Sum_{n>=1} (n^5/n^s) Sum_{p|n} 1/p^5. Since n = p*j, rewrite the sum as Sum_{p} Sum_{j>=1} 1/(p^5*(p*j)^(s-5)) = Sum_{p} 1/p^s Sum_{j>=1} 1/j^(s-5) = zeta(s-5)*primezeta(s). The result generalizes to higher powers of p. - Michael Shamos, Mar 03 2023
Sum_{k=1..n} a(k) ~ A085966 * n^6/6. - Vaclav Kotesovec, Mar 03 2023
a(n) = Sum_{d|n} A059378(d)*A001221(n/d). - Ridouane Oudra, Jul 14 2025
From Wesley Ivan Hurt, Jul 15 2025: (Start)
a(n) = Sum_{d|n} c(d) * (n/d)^5, where c = A010051.
a(p^k) = p^(5*k-5) for p prime and k>=1. (End)

A351246 a(n) = n^6 * Sum_{p|n, p prime} 1/p^6.

Original entry on oeis.org

0, 1, 1, 64, 1, 793, 1, 4096, 729, 15689, 1, 50752, 1, 117713, 16354, 262144, 1, 578097, 1, 1004096, 118378, 1771625, 1, 3248128, 15625, 4826873, 531441, 7533632, 1, 12437281, 1, 16777216, 1772290, 24137633, 133274, 36998208, 1, 47045945, 4827538, 64262144, 1, 93342313

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

nonn

Dirichlet convolution of A010051(n) and n^6. - Wesley Ivan Hurt, Jul 15 2025

			a(6) = 793; a(6) = 6^6 * Sum_{p|6, p prime} 1/p^6 = 46656 * (1/2^6 + 1/3^6) = 793.

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

Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k = 0..10: A001221 (k=0), A069359 (k=1), A322078 (k=2), A351242 (k=3), A351244 (k=4), A351245 (k=5), this sequence (k=6), A351247 (k=7), A351248 (k=8), A351249 (k=9), A351262 (k=10).

Cf. A000040, A001221, A010051, A069091.

Array[#^6*DivisorSum[#, 1/#^6 &, PrimeQ] &, 50] (* Wesley Ivan Hurt, Jul 15 2025 *)

a(A000040(n)) = 1.
a(n) = Sum_{d|n} A069091(d)*A001221(n/d). - Ridouane Oudra, Jul 14 2025
From Wesley Ivan Hurt, Jul 15 2025: (Start)
a(n) = Sum_{d|n} c(d) * (n/d)^6, where c = A010051.
a(p^k) = p^(6*k-6) for p prime and k>=1. (End)

A351247 a(n) = n^7 * Sum_{p|n, p prime} 1/p^7.

Original entry on oeis.org

0, 1, 1, 128, 1, 2315, 1, 16384, 2187, 78253, 1, 296320, 1, 823671, 80312, 2097152, 1, 5062905, 1, 10016384, 825730, 19487299, 1, 37928960, 78125, 62748645, 4782969, 105429888, 1, 181139311, 1, 268435456, 19489358, 410338801, 901668, 648051840, 1, 893871867, 62750704

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

nonn

Dirichlet convolution of A010051(n) and n^7. - Wesley Ivan Hurt, Jul 15 2025

			a(6) = 2315; a(6) = 6^7 * Sum_{p|6, p prime} 1/p^7 = 279936 * (1/2^7 + 1/3^7) = 2315.

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

Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k = 0..10: A001221 (k=0), A069359 (k=1), A322078 (k=2), A351242 (k=3), A351244 (k=4), A351245 (k=5), A351246 (k=6), this sequence (k=7), A351248 (k=8), A351249 (k=9), A351262 (k=10).

Cf. A000040, A001221, A010051, A069092.

Array[#^7*DivisorSum[#, 1/#^7 &, PrimeQ] &, 50] (* Wesley Ivan Hurt, Jul 15 2025 *)

a(A000040(n)) = 1.
a(n) = Sum_{d|n} A069092(d)*A001221(n/d). - Ridouane Oudra, Jul 15 2025
From Wesley Ivan Hurt, Jul 15 2025: (Start)
a(n) = Sum_{d|n} c(d) * (n/d)^7, where c = A010051.
a(p^k) = p^(7*k-7) for p prime and k>=1. (End)

A351248 a(n) = n^8 * Sum_{p|n, p prime} 1/p^8.

Original entry on oeis.org

0, 1, 1, 256, 1, 6817, 1, 65536, 6561, 390881, 1, 1745152, 1, 5765057, 397186, 16777216, 1, 44726337, 1, 100065536, 5771362, 214359137, 1, 446758912, 390625, 815730977, 43046721, 1475854592, 1, 2664570241, 1, 4294967296, 214365442, 6975757697, 6155426, 11449942272

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

nonn

Dirichlet convolution of A010051(n) and n^8. - Wesley Ivan Hurt, Jul 15 2025

			a(6) = 6817; a(6) = 6^8 * Sum_{p|6, p prime} 1/p^8 = 1679616 * (1/2^8 + 1/3^8) = 6817.

Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k = 0..10: A001221 (k=0), A069359 (k=1), A322078 (k=2), A351242 (k=3), A351244 (k=4), A351245 (k=5), A351246 (k=6), A351247 (k=7), this sequence (k=8), A351249 (k=9), A351262 (k=10).

Cf. A000040, A001221, A010051, A069093.

Array[#^8*DivisorSum[#, 1/#^8 &, PrimeQ] &, 36] (* Stefano Spezia, Jul 15 2025 *)

from sympy import primefactors
def A351248(n): return sum((n//p)**8 for p in primefactors(n)) # Chai Wah Wu, Feb 05 2022

a(A000040(n)) = 1.
a(n) = Sum_{d|n} A069093(d)*A001221(n/d). - Ridouane Oudra, Jul 15 2025
From Wesley Ivan Hurt, Jul 15 2025: (Start)
a(n) = Sum_{d|n} c(d) * (n/d)^8, where c = A010051.
a(p^k) = p^(8*k-8) for p prime and k>=1. (End)

A351249 a(n) = n^9 * Sum_{p|n, p prime} 1/p^9.

Original entry on oeis.org

0, 1, 1, 512, 1, 20195, 1, 262144, 19683, 1953637, 1, 10339840, 1, 40354119, 1972808, 134217728, 1, 397498185, 1, 1000262144, 40373290, 2357948203, 1, 5293998080, 1953125, 10604499885, 387420489, 20661308928, 1, 39453437071, 1, 68719476736, 2357967374, 118587877009, 42306732

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

nonn

Dirichlet convolution of A010051(n) and n^9. - Wesley Ivan Hurt, Jul 15 2025

			a(6) = 20195; a(6) = 6^9 * Sum_{p|6, p prime} 1/p^9 = 10077696 * (1/2^9 + 1/3^9) = 20195.

Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k = 0..10: A001221 (k=0), A069359 (k=1), A322078 (k=2), A351242 (k=3), A351244 (k=4), A351245 (k=5), A351246 (k=6), A351247 (k=7), A351248 (k=8), this sequence (k=9), A351262 (k=10).

Cf. A000040, A001221, A010051, A069094.

Array[#^9*DivisorSum[#, 1/#^9 &, PrimeQ] &, 50] (* Wesley Ivan Hurt, Jul 15 2025 *)

from sympy import primefactors
def A351249(n): return sum((n//p)**9 for p in primefactors(n)) # Chai Wah Wu, Feb 05 2022

a(A000040(n)) = 1.
a(n) = Sum_{d|n} A069094(d)*A001221(n/d). - Ridouane Oudra, Jul 15 2025
From Wesley Ivan Hurt, Jul 15 2025: (Start)
a(n) = Sum_{d|n} c(d) * (n/d)^9, where c = A010051.
a(p^k) = p^(9*k-9) for p prime and k>=1. (End)

A351262 a(n) = n^10 * Sum_{p|n, p prime} 1/p^10.

Original entry on oeis.org

0, 1, 1, 1024, 1, 60073, 1, 1048576, 59049, 9766649, 1, 61514752, 1, 282476273, 9824674, 1073741824, 1, 3547250577, 1, 10001048576, 282534298, 25937425625, 1, 62991106048, 9765625, 137858492873, 3486784401, 289255703552, 1, 586710856801, 1, 1099511627776, 25937483650

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

nonn

Dirichlet convolution of A010051(n) and n^10. - Wesley Ivan Hurt, Jul 15 2025

			a(6) = 60073; a(6) = 6^10 * Sum_{p|6, p prime} 1/p^10 = 60466176 * (1/2^10 + 1/3^10) = 60073.

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

Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k = 0..10: A001221 (k=0), A069359 (k=1), A322078 (k=2), A351242 (k=3), A351244 (k=4), A351245 (k=5), A351246 (k=6), A351247 (k=7), A351248 (k=8), A351249 (k=9), this sequence (k=10).

Cf. A000040, A001221, A010051, A069095.

f:= proc(n) local p;
  n^10 * add(1/p^10, p = numtheory:-factorset(n))
end proc:
map(f, [$1..40]); # Robert Israel, Sep 10 2024

Join[{0},Table[n^10 Total[1/FactorInteger[n][[;;,1]]^10],{n,2,40}]] (* Harvey P. Dale, Aug 10 2024 *)

a(n) = my(f=factor(n)); n^10*sum(k=1, #f~, 1/f[k,1]^10); \\ Michel Marcus, Sep 10 2024

from sympy import primefactors
def A351262(n): return sum((n//p)**10 for p in primefactors(n)) # Chai Wah Wu, Feb 05 2022

a(A000040(n)) = 1.
a(n) = Sum_{d|n} A069095(d)*A001221(n/d). - Ridouane Oudra, Jul 15 2025
From Wesley Ivan Hurt, Jul 15 2025: (Start)
a(n) = Sum_{d|n} c(d) * (n/d)^10, where c = A010051.
a(p^k) = p^(10*k-10) for p prime and k>=1. (End)

A068328 Arithmetic derivatives of the squarefree numbers.

Original entry on oeis.org

0, 1, 1, 1, 5, 1, 7, 1, 1, 9, 8, 1, 1, 10, 13, 1, 15, 1, 31, 1, 14, 19, 12, 1, 21, 16, 1, 41, 1, 25, 1, 20, 1, 16, 22, 31, 1, 1, 33, 18, 61, 1, 26, 59, 1, 1, 39, 18, 71, 1, 43, 1, 22, 45, 32, 1, 20, 34, 49, 24, 1, 1, 91, 1, 71, 55, 1, 1, 87, 40, 1, 101, 28, 61, 24, 63, 44, 1, 46

Offset: 1

Reinhard Zumkeller, Feb 27 2002

nonn

a(n) and A005117(n) are coprime, cf. A085731. - Reinhard Zumkeller, May 10 2011

			a(65) = d(A005117(65)) = d(105) = d(3*35) = 3*d(35)+d(3)*35 = 3*d(5*7)+1*35 = 3*d(5*7)+1*35 = 3*(5*d(7)+d(5)*7)+35 = 3*(5*1+1*7)+35 = 3*12+35 = 71, where d(n) = A003415(n).
With d(1)=0, d(prime) = 1 and d(m*n) = d(m)*n + m*d(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6 (2003), Article 03.3.4.

Cf. A003415, A005117, A069359, A085731.

a068328 = a003415 . a005117 -- Reinhard Zumkeller, May 10 2011

ad[n_] := ad[n] = n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); ad[1] = 0; Table[ad[k], {k, Select[Range[150], SquareFreeQ]}] (* Amiram Eldar, Mar 04 2024 *)

a(n) = A003415(A005117(n)).
a(n) = A069359(A005117(n)).
a(n) = Sum_{prime p | A005117(n)} A005117(n)/p.

A329350 a(n) = Product_{d|n} A276086(d)^A010051(n/d).

Original entry on oeis.org

1, 2, 2, 3, 2, 18, 2, 9, 6, 54, 2, 45, 2, 30, 108, 15, 2, 150, 2, 405, 60, 270, 2, 375, 18, 150, 30, 675, 2, 33750, 2, 225, 540, 1350, 180, 3125, 2, 750, 300, 5625, 2, 281250, 2, 10125, 4500, 6750, 2, 140625, 10, 56250, 2700, 16875, 2, 468750, 1620, 84375, 1500, 33750, 2, 65625, 2, 42, 22500, 21, 900, 236250, 2, 567, 13500, 425250, 2, 21875

Offset: 1

Antti Karttunen, Nov 12 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A010051, A069359, A276085, A276086, A329351 (rgs-transform).

Cf. also A329352, A329380.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A329350(n) = { my(m=1); fordiv(n,d,if(isprime(n/d), m *= A276086(d))); (m); };

a(n) = Product_{d|n} A276086(d)^A010051(n/d).

A276085(a(n)) = A069359(n).

A230593 a(n) = n * Sum_{q|n} 1 / q, where q are noncomposite numbers (A008578) dividing n.

Original entry on oeis.org

1, 3, 4, 6, 6, 11, 8, 12, 12, 17, 12, 22, 14, 23, 23, 24, 18, 33, 20, 34, 31, 35, 24, 44, 30, 41, 36, 46, 30, 61, 32, 48, 47, 53, 47, 66, 38, 59, 55, 68, 42, 83, 44, 70, 69, 71, 48, 88, 56, 85, 71, 82, 54, 99, 71, 92, 79, 89, 60, 122, 62, 95, 93, 96, 83, 127

Offset: 1

Jaroslav Krizek, Oct 25 2013

nonn

			For n = 6: a(6) = 6 * (1/1 + 1/2 + 1/3) = 11.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A080339, A000027, A008578, A329039.

Coincides with A129283 on squarefree numbers, A005117.

a[n_] := n * DivisorSum[n, 1/# &, !CompositeQ[#] &]; Array[a, 100] (* Amiram Eldar, Nov 12 2021 *)

A230593(n) = sumdiv(n,d,((1==d)||isprime(d))*(n/d)); \\ Antti Karttunen, Nov 12 2021

For n > 1, a(n) = n + n * Sum_(p|n) 1 / p, where p are primes dividing n.
a(n) = A069359(n) + n.
a(n) = Sum_{d|n}  A080339(d) * A000027(n/d).
a(n) = A080339(n) * A000027(n), where operation * denotes Dirichlet convolution, i.e. convolution of type: a(n) = Sum_{d|n} b(d) * c(n/d).
For p, q = distinct primes, a(p) = p + 1, a(pq) = pq - 1.
From Antti Karttunen, Nov 12 2021: (Start)
a(n) = A129283(n) - A329039(n).
a(A005117(n)) = A129283(A005117(n)), for all n >= 1.
(End)
For p prime, k>=1, a(p^k) = p^(k-1) * (p+1). - Bernard Schott, Nov 12 2021

A318320 a(n) = (psi(n) - phi(n))/2.

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 4, 3, 7, 1, 10, 1, 9, 8, 8, 1, 15, 1, 14, 10, 13, 1, 20, 5, 15, 9, 18, 1, 32, 1, 16, 14, 19, 12, 30, 1, 21, 16, 28, 1, 42, 1, 26, 24, 25, 1, 40, 7, 35, 20, 30, 1, 45, 16, 36, 22, 31, 1, 64, 1, 33, 30, 32, 18, 62, 1, 38, 26, 60, 1, 60, 1, 39, 40, 42, 18, 72, 1, 56, 27, 43, 1, 84, 22, 45, 32, 52, 1, 96

Offset: 1

Antti Karttunen, Aug 26 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Marcin Mazur and Bogdan V. Petrenko, Generalizations of Arnold's version of Euler's theorem for matrices, Japanese Journal of Mathematics, 5:183-189, 2010.

Cf. A000010, A001615, A291784, A292786, A318325, A318326, A318442.

Differs from A069359 for the first time at n=30, where a(30) = 32, while A069359(30) = 31.

psi[n_] := n * Times @@ (1 + 1/FactorInteger[n][[;; , 1]]); psi[1] = 1; a[n_] := (psi[n] - EulerPhi[n])/2; Array[a, 100] (* Amiram Eldar, Dec 05 2023 *)

A318320(n) = sumdiv(n,d,(-1==moebius(n/d))*d);

A318320(n) = ((n*sumdivmult(n, d, issquarefree(d)/d))-eulerphi(n))/2;

a(n) = (A001615(n) - A000010(n))/2 = A292786(n)/2.
a(n) = A291784(n) - A000010(n).
a(n) = A318326(n) + A318442(n).
Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = 9/(4*Pi^2) = 0.227972... . - Amiram Eldar, Dec 05 2023

cp's OEIS Frontend

A351245 a(n) = n^5 * Sum_{p|n, p prime} 1/p^5.

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A351246 a(n) = n^6 * Sum_{p|n, p prime} 1/p^6.

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A351247 a(n) = n^7 * Sum_{p|n, p prime} 1/p^7.

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A351248 a(n) = n^8 * Sum_{p|n, p prime} 1/p^8.

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A351249 a(n) = n^9 * Sum_{p|n, p prime} 1/p^9.

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A351262 a(n) = n^10 * Sum_{p|n, p prime} 1/p^10.

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A068328 Arithmetic derivatives of the squarefree numbers.

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