A329031 -id:A329031 - cp's OEIS frontend

A327860 Arithmetic derivative of the primorial base exp-function: a(n) = A003415(A276086(n)).

0, 1, 1, 5, 6, 21, 1, 7, 8, 31, 39, 123, 10, 45, 55, 185, 240, 705, 75, 275, 350, 1075, 1425, 3975, 500, 1625, 2125, 6125, 8250, 22125, 1, 9, 10, 41, 51, 165, 12, 59, 71, 247, 318, 951, 95, 365, 460, 1445, 1905, 5385, 650, 2175, 2825, 8275, 11100, 30075, 4125, 12625, 16750, 46625, 63375, 166125, 14, 77, 91, 329, 420

Offset: 0

Antti Karttunen, Sep 30 2019

Are there any other fixed points after 0, 1, 7, 8 and 2556? (A328110, see also A351087 and A351088).
Out of the 30030 initial terms, 19220 are multiples of 5. (See A327865).
Proof that a(n) is even if and only if n is a multiple of 4: Consider Charlie Neder's Feb 25 2019 comment in A235992. As A276086 is never a multiple of 4, and as it toggles the parity, we only need to know when A001222(A276086(n)) = A276150(n) is even. The condition for that is given in the latter sequence by David A. Corneth's Feb 27 2019 comment. From this it also follows that A166486 gives similarly the parity of terms of A342002, A351083 and A345000. See also comment in A327858. - Antti Karttunen, May 01 2022

			2556 has primorial base expansion [1,1,1,1,0,0] as 1*A002110(5) + 1*A002110(4) + 1*A002110(3) + 1*A002110(2) = 2310 + 210 + 30 + 6 = 2556. That in turn is converted by A276086 to 13^1 * 11^1 * 7^1 * 5^1 = 5005, whose arithmetic derivative is 5' * 1001 + 1001' * 5 = 1*1001 + 311*5 = 2556, thus 2556 is one of the rare fixed points (A328110) of this sequence.

Antti Karttunen, Table of n, a(n) for n = 0..2310
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..30030
Index entries for sequences related to primorial base

Cf. A002110 (positions of 1's), A003415, A048103, A276086, A327858, A327859, A327865, A328110 (fixed points), A328233 (positions of primes), A328242 (positions of squarefree terms), A328388, A328392, A328571, A328572, A329031, A329032, A329041, A342002.
Cf. A345000, A351074, A351075, A351076, A351077, A351080, A351083, A351084, A351087 (numbers k such that a(k) is a multiple of k), A351088.
Coincides with A329029 on positions given by A276156.
Cf. A166486 (a(n) mod 2), A353630 (a(n) mod 4).
Cf. A267263, A276150, A324650, A324653, A324655 for omega, bigomega, phi, sigma and tau applied to A276086(n).
Cf. also A351950 (analogous sequence).

Block[{b = MixedRadix[Reverse@ Prime@ Range@ 12]}, Array[Function[k, If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]] ] &@ Abs[Times @@ Power @@@ # &@ Transpose@{Prime@ Range@ Length@ k, Reverse@ k}]]@ IntegerDigits[#, b] &, 65, 0]] (* Michael De Vlieger, Mar 12 2021 *)

A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A327860(n) = A003415(A276086(n));

A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); }; \\ (Standalone version) - Antti Karttunen, Nov 07 2019

a(n) = A003415(A276086(n)).
a(A002110(n)) = 1 for all n >= 0.
From Antti Karttunen, Nov 03 2019: (Start)
Whenever A329041(x,y) = 1, a(x + y) = A003415(A276086(x)*A276086(y)) = a(x)*A276086(y) + a(y)*A276086(x). For example, we have:
a(n) = a(A328841(n)+A328842(n)) = A329031(n)*A328572(n) + A329032(n)*A328571(n).
A051903(a(n)) = A328391(n).
A328114(a(n)) = A328392(n).
(End)
From Antti Karttunen, May 01 2022: (Start)
a(n) = A328572(n) * A342002(n).
For all n >= 0, A000035(a(n)) = A166486(n). [See comments]
(End)

Verbal description added to the definition by Antti Karttunen, May 01 2022

A069359 a(n) = n * Sum_{p|n} 1/p where p are primes dividing n.

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, 31, 1, 16, 14, 19, 12, 30, 1, 21, 16, 28, 1, 41, 1, 26, 24, 25, 1, 40, 7, 35, 20, 30, 1, 45, 16, 36, 22, 31, 1, 62, 1, 33, 30, 32, 18, 61, 1, 38, 26, 59, 1, 60, 1, 39, 40, 42, 18, 71, 1, 56

Offset: 1

Benoit Cloitre, Apr 15 2002

nonn

Coincides with arithmetic derivative on squarefree numbers: a(A005117(n)) = A068328(n) = A003415(A005117(n)). - Reinhard Zumkeller, Jul 20 2003, Clarified by Antti Karttunen, Nov 15 2019
a(n) = n-1 iff n = 1 or n is a primary pseudoperfect number A054377. - Jonathan Sondow, Apr 16 2014
a(1) = 0 by the standard convention for empty sums.
“Seva” on the MathOverflow link asks if the iterates of this sequence are all eventually 0. - Charles R Greathouse IV, Feb 15 2019

			a(12) = 10 because the prime divisors of 12 are 2 and 3 so we have: 12/2 + 12/3 = 6 + 4 = 10. - _Geoffrey Critzer_, Mar 17 2015

Antti Karttunen, Table of n, a(n) for n = 1..16384 (first 10000 terms from Franklin T. Adams-Watters)
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
MathOverflow, A recursion with a number-theoretic function (2019)
Joshua Zelinsky, The sum of the reciprocals of the prime divisors of an odd perfect or odd primitive non-deficient number, arXiv:2402.14234 [math.NT], 2024. See also Integers (2025) Vol. 25, Art. No. A59, page 2.

Cf. A003415, A005117, A068328, A010051, A000027, A054377, A180253, A230593, A292786, A306369, A326690, A329029, A329350, A329352.
Cf. A322068 (partial sums), A323599 (Inverse Möbius transform).
Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k = 0..10: A001221 (k=0), this sequence (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), A351262 (k=10).

[0] cat [n*&+[1/p: p in PrimeDivisors(n)]:n in [2..80]]; // Marius A. Burtea, Jan 21 2020

A069359 := n -> add(n/d, d = select(isprime, numtheory[divisors](n))):
seq(A069359(i), i = 1..20); # Peter Luschny, Jan 31 2012
# second Maple program:
a:= n-> n*add(1/i[1], i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 23 2019

f[list_, i_] := list[[i]]; nn = 100; a = Table[n, {n, 1, nn}]; b =
Table[If[PrimeQ[n], 1, 0], {n, 1, nn}]; Table[DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}] (* Geoffrey Critzer, Mar 17 2015 *)

a(n) = n*sumdiv(n, d, isprime(d)/d); \\ Michel Marcus, Mar 18 2015

a(n) = my(ps=factor(n)[,1]~);sum(k=1,#ps,n\ps[k]) \\ Franklin T. Adams-Watters, Apr 09 2015

from sympy import primefactors
def A069359(n): return sum(n//p for p in primefactors(n)) # Chai Wah Wu, Feb 05 2022

def A069359(n) :
    D = filter(is_prime, divisors(n))
    return add(n/d for d in D)
print([A069359(i) for i in (1..20)]) # Peter Luschny, Jan 31 2012

G.f.: Sum(x^p(j)/(1-x^p(j))^2,j>=1), where p(j) is the j-th prime. - Vladeta Jovovic, Mar 29 2006
a(n) = A230593(n) - n. a(n) = A010051(n) (*) A000027(n), where operation (*) denotes Dirichlet convolution, that is, convolution of type: a(n) = Sum_(d|n) b(d) * c(n/d) = Sum_{d|n} A010051(d) * A000027(n/d). - Jaroslav Krizek, Nov 07 2013
a(A054377(n)) = A054377(n) - 1. - Jonathan Sondow, Apr 16 2014
Dirichlet g.f.: zeta(s - 1)*primezeta(s). - Geoffrey Critzer, Mar 17 2015
Sum_{k=1..n} a(k) ~ A085548 * n^2 / 2. - Vaclav Kotesovec, Feb 04 2019
From Antti Karttunen, Nov 15 2019: (Start)
a(n) = Sum_{d|n} A008683(n/d)*A323599(d).
a(n) = A003415(n) - A329039(n) = A230593(n) - n = A306369(n) - A000010(n).
a(n) = A276085(A329350(n)) = A048675(A329352(n)).
a(A276086(n)) = A329029(n), a(A328571(n)) = A329031(n).
(End)
a(n) = Sum_{d|n} A000010(d) * A001221(n/d). - Torlach Rush, Jan 21 2020
a(n) = Sum_{k=1..n} omega(gcd(n, k)). - Ilya Gutkovskiy, Feb 21 2020
a(p^k) = p^(k-1) for p prime and k>=1. - Wesley Ivan Hurt, Jul 15 2025

A329029 a(n) = A069359(A276086(n)), where A276086 is the primorial base exp-function and A069359(n) = n * Sum_{p|n} 1/p.

Original entry on oeis.org

0, 1, 1, 5, 3, 15, 1, 7, 8, 31, 24, 93, 5, 35, 40, 155, 120, 465, 25, 175, 200, 775, 600, 2325, 125, 875, 1000, 3875, 3000, 11625, 1, 9, 10, 41, 30, 123, 12, 59, 71, 247, 213, 741, 60, 295, 355, 1235, 1065, 3705, 300, 1475, 1775, 6175, 5325, 18525, 1500, 7375, 8875, 30875, 26625, 92625, 7, 63, 70, 287, 210, 861, 84, 413, 497, 1729

Offset: 0

Antti Karttunen, Nov 07 2019

nonn

A380535 gives the indices n where a(n) is a multiple of A053669(n). This can be seen from the formula a(n) = A003557(A276086(n)) * A069359(A328571(n)). The left hand side of the product is a multiple of A053669(n) if and only if A276088(n) > 1, while the right hand side is never a multiple of A053669(n), as it is equal to A329031(n) = A003415(A007947(A276086(n))). - Antti Karttunen, Feb 11 2025

Antti Karttunen, Table of n, a(n) for n = 0..2310
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..32768
Index entries for sequences related to primorial base

Cf. A003415, A003557, A053669, A069359, A276086, A276088, A328571, A328572, A329031, A380535.

Coincides with A327860 on the positions given by A276156.

A329029(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); if(e, m *= (p^e); s += (1/p)); n = n\p; p = nextprime(1+p)); (s*m); };

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A069359(n) = (n*sumdiv(n, d, isprime(d)/d));
A329029(n) = A069359(A276086(n));

a(n) = A069359(A276086(n)).

a(n) = A328572(n) * A329031(n) = A003557(A276086(n)) * A069359(A328571(n)). - Antti Karttunen, Feb 11 2025

A329032 a(n) = A327860(A328842(n)).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 8, 8, 10, 10, 10, 10, 55, 55, 75, 75, 75, 75, 350, 350, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 8, 8, 10, 10, 10, 10, 55, 55, 75, 75, 75, 75, 350, 350, 1, 1, 1, 1, 10, 10, 1, 1, 1, 1, 10, 10, 12, 12, 12, 12, 71, 71, 95, 95, 95, 95, 460, 460, 650, 650, 650, 650, 2825, 2825, 14, 14, 14, 14

Offset: 0

Antti Karttunen, Nov 07 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..2589
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..32768
Index entries for sequences related to primorial base

Cf. A003415, A276086, A327860, A328572, A328842, A329031.

Cf. A276156 (positions of zeros).

A329032(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); if(e, m *= (p^(e-1)); s += ((e-1)/p)); n = n\p; p = nextprime(1+p)); (s*m); };

a(n) = A327860(A328842(n)) = A003415(A276086(A328842(n))) = A003415(A328572(n)).

cp's OEIS Frontend

A327860 Arithmetic derivative of the primorial base exp-function: a(n) = A003415(A276086(n)).

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A069359 a(n) = n * Sum_{p|n} 1/p where p are primes dividing n.

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A329029 a(n) = A069359(A276086(n)), where A276086 is the primorial base exp-function and A069359(n) = n * Sum_{p|n} 1/p.

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A329032 a(n) = A327860(A328842(n)).

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