A348507 -id:A348507 - cp's OEIS frontend

A349140 a(n) = Sum_{d|n} d * A348507(n/d), where A348507(n) = A003959(n) - n, where A003959 is fully multiplicative with a(p) = (p+1).

0, 1, 1, 7, 1, 11, 1, 33, 10, 15, 1, 61, 1, 19, 17, 131, 1, 77, 1, 89, 21, 27, 1, 263, 16, 31, 67, 117, 1, 145, 1, 473, 29, 39, 25, 379, 1, 43, 33, 395, 1, 189, 1, 173, 137, 51, 1, 997, 22, 155, 41, 201, 1, 443, 33, 527, 45, 63, 1, 743, 1, 67, 177, 1611, 37, 277, 1, 257, 53, 265, 1, 1541, 1, 79, 187, 285, 37, 321

Offset: 1

Antti Karttunen, Nov 08 2021

nonn

Dirichlet convolution of A348507 with the identity function, A000027.

Dirichlet convolution of sigma with A348971.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000027, A000203, A003959, A038040, A348507, A348971, A349141 (Möbius transform), A349142, A349143, A349170.

Cf. also A347130, A348980.

f[p_, e_] := (p + 1)^e; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; a[n_] := DivisorSum[n, #*s[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348507(n) = (A003959(n) - n);
A349140(n) = sumdiv(n,d,d*A348507(n/d));

a(n) = Sum_{d|n} d * A348507(n/d).
a(n) = Sum_{d|n} A000203(d) * A348971(n/d).
a(n) = Sum_{d|n} A349141(d).
For all n >= 1, a(n) >= A347130(n) >= A348980(n).
a(n) = A349170(n) - A038040(n). - Antti Karttunen, Nov 15 2021

A349141 a(n) = Sum_{d|n} phi(n/d) * A348507(d), where A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).

Original entry on oeis.org

0, 1, 1, 6, 1, 9, 1, 26, 9, 13, 1, 44, 1, 17, 15, 98, 1, 57, 1, 68, 19, 25, 1, 176, 15, 29, 57, 92, 1, 105, 1, 342, 27, 37, 23, 252, 1, 41, 31, 280, 1, 141, 1, 140, 111, 49, 1, 636, 21, 125, 39, 164, 1, 309, 31, 384, 43, 61, 1, 480, 1, 65, 147, 1138, 35, 213, 1, 212, 51, 209, 1, 960, 1, 77, 155, 236, 35, 249, 1, 1028

Offset: 1

Antti Karttunen, Nov 08 2021

nonn

Dirichlet convolution of Euler phi (A000010) with A348507.

Möbius transform of A349140.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A003959, A008683, A018804, A348507, A349140 (inverse Möbius transform), A349142, A349143, A349171.

Cf. also A347131, A348981.

f[p_, e_] := (p + 1)^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, (s[#] - #) * EulerPhi[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348507(n) = (A003959(n) - n);
A349141(n) = sumdiv(n,d,eulerphi(d)*A348507(n/d));

a(n) = Sum_{d|n} A000010(n/d) * A348507(d).
a(n) = Sum_{d|n} A008683(n/d) * A349140(d).
a(n) = Sum_{k=1..n} A348507(gcd(n,k)).
For all n >= 1, a(n) >= A347131(n) >= A348981(n).
a(n) = A349171(n) - A018804(n). - Antti Karttunen, Nov 14 2021

A349142 a(n) = Sum_{d|n} psi(n/d) * A348507(d), where psi is Dedekind psi (A001615), A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).

Original entry on oeis.org

0, 1, 1, 8, 1, 13, 1, 40, 11, 17, 1, 80, 1, 21, 19, 164, 1, 99, 1, 112, 23, 29, 1, 364, 17, 33, 77, 144, 1, 191, 1, 604, 31, 41, 27, 528, 1, 45, 35, 524, 1, 243, 1, 208, 165, 53, 1, 1424, 23, 187, 43, 240, 1, 597, 35, 684, 47, 65, 1, 1072, 1, 69, 209, 2084, 39, 347, 1, 304, 55, 327, 1, 2244, 1, 81, 221, 336, 39, 399

Offset: 1

Antti Karttunen, Nov 08 2021

nonn

Dirichlet convolution of A001615 with A348507.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A001615, A003959, A327251, A348507, A349140, A349141, A349143, A349172.

Cf. also A347132, A348982.

f1[p_, e_] := (p + 1)*p^(e - 1); psi[1] = 1; psi[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := (p + 1)^e; s[1] = 1; s[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := DivisorSum[n, (s[#] - #)*psi[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348507(n) = (A003959(n) - n);
A349142(n) = sumdiv(n,d,A001615(d)*A348507(n/d));

a(n) = Sum_{d|n} A001615(n/d) * A348507(d).
For all n >= 1, a(n) >= A347132(n) >= A348982(n).
a(n) = A349172(n) - A327251(n). - Antti Karttunen, Nov 14 2021

A349143 a(n) = Sum_{d|n} A038040(d) * A348507(n/d), where A038040(n) = n*tau(n), A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).

Original entry on oeis.org

0, 1, 1, 9, 1, 16, 1, 51, 13, 22, 1, 114, 1, 28, 25, 233, 1, 145, 1, 168, 31, 40, 1, 590, 21, 46, 106, 222, 1, 310, 1, 939, 43, 58, 37, 915, 1, 64, 49, 896, 1, 406, 1, 330, 262, 76, 1, 2570, 29, 297, 61, 384, 1, 1012, 49, 1202, 67, 94, 1, 2040, 1, 100, 340, 3489, 55, 598, 1, 492, 79, 574, 1, 4457, 1, 118, 360, 546, 55

Offset: 1

Antti Karttunen, Nov 08 2021

nonn

Dirichlet convolution of A348507 with A038040, which is the Dirichlet convolution of the identity function (A000027) with itself.
Dirichlet convolution of the identity function (A000027) with A349140.
Dirichlet convolution of sigma (A000203) with A349141.
Dirichlet convolution of A060640 with A348971.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000005, A000027, A000203, A003959, A038040, A060640, A348507, A348971, A349140, A349141, A349142.

Cf. also A349123, A348983.

f[p_, e_] := (p + 1)^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, #*DivisorSigma[0, #]*(s[n/#] - n/#) &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)

A038040(n) = (n*numdiv(n));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348507(n) = (A003959(n) - n);
A349143(n) = sumdiv(n,d,A038040(d)*A348507(n/d));

a(n) = Sum_{d|n} A038040(n/d) * A348507(d).
a(n) = Sum_{d|n} d * A349140(n/d).
a(n) = Sum_{d|n} A000203(d) * A349141(n/d).
a(n) = Sum_{d|n} A060640(d) * A348971(n/d).
For all n >= 1, a(n) >= A349123(n) >= A348983(n).

A348950 a(n) = A348507(A276086(n)), where A348507(n) = A003959(n) - n, A003959 is multiplicative with a(p^e) = (p+1)^e, and A276086 gives the prime product form of primorial base expansion of n.

Original entry on oeis.org

0, 1, 1, 6, 7, 30, 1, 8, 9, 42, 51, 198, 11, 58, 69, 282, 351, 1278, 91, 398, 489, 1842, 2331, 8118, 671, 2638, 3309, 11802, 15111, 50958, 1, 10, 11, 54, 65, 258, 13, 74, 87, 366, 453, 1674, 113, 514, 627, 2406, 3033, 10674, 853, 3434, 4287, 15486, 19773, 67194, 5993, 22354, 28347, 98166, 126513, 418914, 15, 94, 109

Offset: 0

Antti Karttunen, Nov 06 2021

Antti Karttunen, Table of n, a(n) for n = 0..11550
Index entries for sequences related to primorial base

Cf. A003959, A276086, A348507, A348949, A348999.

Cf. also A327860.

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A348507(n) = (A003959(n) - n);
A348950(n) = A348507(A276086(n));

A348950(n) = { my(m1=1, m2=1, p=2); while(n, m1 *= (p^(n%p)); m2 *= ((1+p)^(n%p)); n = n\p; p = nextprime(1+p)); (m2-m1); };

a(n) = A348949(n) - A276086(n) = A348507(A276086(n)).

A349139 a(n) = Sum_{d|n} A322582(d) * A348507(n/d), where A322582(n) = n - A003958(n) and A348507(n) = A003959(n) - n.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 8, 1, 2, 0, 18, 0, 2, 2, 41, 0, 22, 0, 22, 2, 2, 0, 98, 1, 2, 12, 26, 0, 40, 0, 172, 2, 2, 2, 148, 0, 2, 2, 130, 0, 48, 0, 34, 28, 2, 0, 426, 1, 34, 2, 38, 0, 158, 2, 162, 2, 2, 0, 278, 0, 2, 32, 645, 2, 64, 0, 46, 2, 56, 0, 706, 0, 2, 36, 50, 2, 72, 0, 590, 91, 2, 0, 350, 2, 2, 2, 226, 0, 348

Offset: 1

Antti Karttunen, Nov 08 2021

nonn

Dirichlet convolution of A322582 with A348507.

Question: Is a(n) >= A305809(n) for all n?

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A003958, A003959, A305809, A322582, A348507, A349129.

f1[p_, e_] := (p - 1)^e; s1[1] = 0; s1[n_] := n - Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := (p + 1)^e; s2[1] = 0; s2[n_] := Times @@ f2 @@@ FactorInteger[n] - n; a[n_] := DivisorSum[n, s1[#]*s2[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A322582(n) = (n-A003958(n));
A348507(n) = (A003959(n)-n);
A349139(n) = sumdiv(n,d,A322582(d)*A348507(n/d));

a(n) = Sum_{d|n} A322582(d) * A348507(n/d).

A003415 a(n) = n' = arithmetic derivative of n: a(0) = a(1) = 0, a(prime) = 1, a(mn) = ma(n) + n*a(m).

Original entry on oeis.org

0, 0, 1, 1, 4, 1, 5, 1, 12, 6, 7, 1, 16, 1, 9, 8, 32, 1, 21, 1, 24, 10, 13, 1, 44, 10, 15, 27, 32, 1, 31, 1, 80, 14, 19, 12, 60, 1, 21, 16, 68, 1, 41, 1, 48, 39, 25, 1, 112, 14, 45, 20, 56, 1, 81, 16, 92, 22, 31, 1, 92, 1, 33, 51, 192, 18, 61, 1, 72, 26, 59, 1, 156, 1, 39, 55, 80, 18, 71

Offset: 0

N. J. A. Sloane, R. K. Guy

Can be extended to negative numbers by defining a(-n) = -a(n).
Based on the product rule for differentiation of functions: for functions f(x) and g(x), (fg)' = f'g + fg'. So with numbers, (ab)' = a'b + ab'. This implies 1' = 0. - Kerry Mitchell, Mar 18 2004
The derivative of a number x with respect to a prime number p as being the number "dx/dp" = (x-x^p)/p, which is an integer due to Fermat's little theorem. - Alexandru Buium, Mar 18 2004
The relation (ab)' = a'b + ab' implies 1' = 0, but it does not imply p' = 1 for p a prime. In fact, any function f defined on the primes can be extended uniquely to a function on the integers satisfying this relation: f(Product_i p_i^e_i) = (Product_i p_i^e_i) * (Sum_i e_i*f(p_i)/p_i). - Franklin T. Adams-Watters, Nov 07 2006
See A131116 and A131117 for record values and where they occur. - Reinhard Zumkeller, Jun 17 2007
Let n be the product of a multiset P of k primes. Consider the k-dimensional box whose edges are the elements of P. Then the (k-1)-dimensional surface of this box is 2*a(n). For example, 2*a(25) = 20, the perimeter of a 5 X 5 square. Similarly, 2*a(18) = 42, the surface area of a 2 X 3 X 3 box. - David W. Wilson, Mar 11 2011
The arithmetic derivative n' was introduced, probably for the first time, by the Spanish mathematician José Mingot Shelly in June 1911 with "Una cuestión de la teoría de los números", work presented at the "Tercer Congreso Nacional para el Progreso de las Ciencias, Granada", cf. link to the abstract on Zentralblatt MATH, and L. E. Dickson, History of the Theory of Numbers. - Giorgio Balzarotti, Oct 19 2013
a(A235991(n)) odd; a(A235992(n)) even. - Reinhard Zumkeller, Mar 11 2014
Sequence A157037 lists numbers with prime arithmetic derivative, i.e., indices of primes in this sequence. - M. F. Hasler, Apr 07 2015
Maybe the simplest "natural extension" of the arithmetic derivative, in the spirit of the above remark by Franklin T. Adams-Watters (2006), is the "pi based" version where f(p) = primepi(p), see sequence A258851. When f is chosen to be the identity map (on primes), one gets A066959. - M. F. Hasler, Jul 13 2015
When n is composite, it appears that a(n) has lower bound 2*sqrt(n), with equality when n is the square of a prime, and a(n) has upper bound (n/2)*log_2(n), with equality when n is a power of 2. - Daniel Forgues, Jun 22 2016
If n = p1*p2*p3*... where p1, p2, p3, ... are all the prime factors of n (not necessarily distinct), and h is a real number (we assume h nonnegative and < 1), the arithmetic derivative of n is equivalent to n' = lim_{h->0} ((p1+h)*(p2+h)*(p3+h)*... - (p1*p2*p3*...))/h. It also follows that the arithmetic derivative of a prime is 1. We could assume h = 1/N, where N is an integer; then the limit becomes {N -> oo}. Note that n = 1 is not a prime and plays the role of constant. - Giorgio Balzarotti, May 01 2023

			6' = (2*3)' = 2'*3 + 2*3' = 1*3 + 2*1 = 5.
Note that, for example, 2' + 3' = 1 + 1 = 2, (2+3)' = 5' = 1. So ' is not linear.
G.f. = x^2 + x^3 + 4*x^4 + x^5 + 5*x^6 + x^7 + 12*x^8 + 6*x^9 + 7*x^10 + ...

G. Balzarotti, P. P. Lava, La derivata aritmetica, Editore U. Hoepli, Milano, 2013.
E. J. Barbeau, Problem, Canad. Math. Congress Notes, 5 (No. 8, April 1973), 6-7.
L. E. Dickson, History of the Theory of Numbers, Vol. 1, Chapter XIX, p. 451, Dover Edition, 2005. (Work originally published in 1919.)
A. M. Gleason et al., The William Lowell Putnam Mathematical Competition: Problems and Solutions 1938-1964, Math. Assoc. America, 1980, p. 295.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..10000
Krassimir T. Atanassov, A formula for the n-th prime number, Comptes rendus de l'Académie bulgare des Sciences, Tome 66, No 4, 2013.
E. J. Barbeau, Remark on an arithmetic derivative, Canad. Math. Bull. vol. 4, no. 2, May 1961.
A. Buium, Home Page
A. Buium, Differential characters of Abelian varieties over p-adic fields, Invent. Math. 122 (1995), no. 2, 309-340.
A. Buium, Geometry of p-jets, Duke Math. J. 82 (1996), no. 2, 349-367.
A. Buium, Arithmetic analogues of derivations, J. Algebra 198 (1997), no. 1, 290-299.
A. Buium, Differential modular forms, J. Reine Angew. Math. 520 (2000), 95-167.
Brad Emmons and Xiao Xiao, The Arithmetic Partial Derivative, arXiv:2201.12453 [math.NT], 2022.
José María Grau and Antonio M. Oller-Marcén, Giuga Numbers and the Arithmetic Derivative, Journal of Integer Sequences, Vol. 15 (2012), #12.4.1.
R. K. Guy, Letter to N. J. A. Sloane, Apr 1975
P. Haukkanen, M. Mattila, J. K. Merikoski and T. Tossavainen, Can the Arithmetic Derivative be Defined on a Non-Unique Factorization Domain?, Journal of Integer Sequences, 16 (2013), #13.1.2. - From _N. J. A. Sloane_, Feb 03 2013
P. Haukkanen, J. K. Merikoski and T. Tossavainen, Asymptotics of partial sums of the Dirichlet series of the arithmetic derivative, Mathematical Communications 25 (2020), 107-115.
Antti Karttunen, Program in LODA-assembly
J. Kovič, The Arithmetic Derivative and Antiderivative, Journal of Integer Sequences 15 (2012), Article 12.3.8.
Michael Penn, When the derivative of a number is not zero -- The arithmetic derivative., YouTube video, 2022.
Ivars Peterson, Deriving the Structure of Numbers, Science News, March 20, 2004.
D. J. M. Shelly, Una cuestión de la teoria de los numeros, Asociation Esp. Granada 1911, 1-12 S (1911). (Abstract of ref. JFM42.0209.02 on zbMATH.org)
T. Tossavainen, P. Haukkanen, J. K. Merikoski, and M. Mattila, We can differentiate numbers, too, The College Mathematics Journal 55 (2024), no. 2, 100-108.
Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003, #03.3.4.
Linda Westrick, Investigations of the Number Derivative, Siemens Foundation competition 2003 and Intel Science Talent Search 2004.
Wikipedia, Arithmetic derivative

Cf. A086134 (least prime factor of n').
Cf. A086131 (greatest prime factor of n').
Cf. A068719 (derivative of 2n).
Cf. A068720 (derivative of n^2).
Cf. A068721 (derivative of n^3).
Cf. A001787 (derivative of 2^n).
Cf. A027471 (derivative of 3^(n-1)).
Cf. A085708 (derivative of 10^n).
Cf. A068327 (derivative of n^n).
Cf. A024451 (derivative of p#).
Cf. A068237 (numerator of derivative of 1/n).
Cf. A068238 (denominator of derivative of 1/n).
Cf. A068328 (derivative of squarefree numbers).
Cf. A068311 (derivative of n!).
Cf. A168386 (derivative of n!!).
Cf. A260619 (derivative of hyperfactorial(n)).
Cf. A260620 (derivative of superfactorial(n)).
Cf. A068312 (derivative of triangular numbers).
Cf. A068329 (derivative of Fibonacci(n)).
Cf. A096371 (derivative of partition number).
Cf. A099301 (derivative of d(n)).
Cf. A099310 (derivative of phi(n)).
Cf. A342925 (derivative of sigma(n)).
Cf. A349905 (derivative of prime shift).
Cf. A327860 (derivative of primorial base exp-function).
Cf. A369252 (derivative of products of three odd primes), A369251 (same sorted).
Cf. A068346 (second  derivative of n).
Cf. A099306 (third   derivative of n).
Cf. A258644 (fourth  derivative of n).
Cf. A258645 (fifth   derivative of n).
Cf. A258646 (sixth   derivative of n).
Cf. A258647 (seventh derivative of n).
Cf. A258648 (eighth  derivative of n).
Cf. A258649 (ninth   derivative of n).
Cf. A258650 (tenth   derivative of n).
Cf. A185232 (n-th    derivative of n).
Cf. A258651 (A(n,k) = k-th arithmetic derivative of n).
Cf. A085731 (gcd(n,n')), A083345 (n'/gcd(n,n')), A057521 (gcd(n, (n')^k) for k>1).
Cf. A342014 (n' mod n), A369049 (n mod n').
Cf. A341998 (A003557(n')), A342001 (n'/A003557(n)).
Cf. A098699 (least x such that x' = n, antiderivative of n).
Cf. A098700 (n such that x' = n has no integer solution).
Cf. A099302 (number of solutions to x' = n).
Cf. A099303 (greatest x such that x' = n).
Cf. A051674 (n such that n' = n).
Cf. A083347 (n such that n' < n).
Cf. A083348 (n such that n' > n).
Cf. A099304 (least k such that (n+k)' = n' + k').
Cf. A099305 (number of solutions to (n+k)' = n' + k').
Cf. A328235 (least k > 0 such that (n+k)' = u * n' for some natural number u).
Cf. A328236 (least m > 1 such that (m*n)' = u * n' for some natural number u).
Cf. A099307 (least k such that the k-th arithmetic derivative of n is zero).
Cf. A099308 (k-th arithmetic derivative of n is zero for some k).
Cf. A099309 (k-th arithmetic derivative of n is nonzero for all k).
Cf. A129150 (n-th derivative of 2^3).
Cf. A129151 (n-th derivative of 3^4).
Cf. A129152 (n-th derivative of 5^6).
Cf. A189481 (x' = n has a unique solution).
Cf. A190121 (partial sums).
Cf. A258057 (first differences).
Cf. A229501 (n divides the n-th partial sum).
Cf. A165560 (parity).
Cf. A235991 (n' is odd), A235992 (n' is even).
Cf. A327863, A327864, A327865 (n' is a multiple of 3, 4, 5).
Cf. A157037 (n' is prime), A192192 (n'' is prime), A328239 (n''' is prime).
Cf. A328393 (n' is squarefree), A328234 (squarefree and > 1).
Cf. A328244 (n'' is squarefree), A328246 (n''' is squarefree).
Cf. A328303 (n' is not squarefree), A328252 (n' is squarefree, but n is not).
Cf. A328248 (least k such that the (k-1)-th derivative of n is squarefree).
Cf. A328251 (k-th arithmetic derivative is never squarefree for any k >= 0).
Cf. A256750 (least k such that the k-th derivative is either 0 or has a factor p^p).
Cf. A327928 (number of distinct primes p such that p^p divides n').
Cf. A342003 (max. exponent k for any prime power p^k that divides n').
Cf. A327929 (n' has at least one divisor of the form p^p).
Cf. A327978 (n' is primorial number > 1).
Cf. A328243 (n' is a partial sum of primorial numbers and larger than one).
Cf. A328310 (maximal prime exponent of n' minus maximal prime exponent of n).
Cf. A328320 (max. prime exponent of n' is less than that of n).
Cf. A328321 (max. prime exponent of n' is >= that of n).
Cf. A328383 (least k such that the k-th derivative of n is either a multiple or a divisor of n, but not both).
Cf. A263111 (the ordinal transform of a).
Cf. A300251, A319684 (Möbius and inverse Möbius transform).
Cf. A305809 (Dirichlet convolution square).
Cf. A349133, A349173, A349394, A349380, A349618, A349619, A349620, A349621 (for miscellaneous Dirichlet convolutions).
Cf. A069359 (similar formula which agrees on squarefree numbers).
Cf. A258851 (the pi-based arithmetic derivative of n).
Cf. A328768, A328769 (primorial-based arithmetic derivatives of n).
Cf. A328845, A328846 (Fibonacci-based arithmetic derivatives of n).
Cf. A302055, A327963, A327965, A328099 (for other variants and modifications).
Cf. A038554 (another sequence using "derivative" in its name, but involving binary expansion of n).
Cf. A322582, A348507 (lower and upper bounds), also A002620.

A003415:= Concatenation([0,0],List(List([2..10^3],Factors),
i->Product(i)*Sum(i,j->1/j))); # Muniru A Asiru, Aug 31 2017
(APL, Dyalog dialect) A003415 ← { ⍺←(0 1 2) ⋄ ⍵≤1:⊃⍺ ⋄ 0=(3⊃⍺)|⍵:((⊃⍺+(2⊃⍺)×(⍵÷3⊃⍺)) ((2⊃⍺)×(3⊃⍺)) (3⊃⍺)) ∇ ⍵÷3⊃⍺ ⋄ ((⊃⍺) (2⊃⍺) (1+(3⊃⍺))) ∇ ⍵} ⍝ Antti Karttunen, Feb 18 2024

a003415 0 = 0
a003415 n = ad n a000040_list where
  ad 1 _             = 0
  ad n ps'@(p:ps)
     | n < p * p     = 1
     | r > 0         = ad n ps
     | otherwise     = n' + p * ad n' ps' where
       (n',r) = divMod n p
-- Reinhard Zumkeller, May 09 2011

Ad:=func; [n le 1 select 0 else Ad(n): n in [0..80]]; // Bruno Berselli, Oct 22 2013

A003415 := proc(n) local B,m,i,t1,t2,t3; B := 1000000000039; if n<=1 then RETURN(0); fi; if isprime(n) then RETURN(1); fi; t1 := ifactor(B*n); m := nops(t1); t2 := 0; for i from 1 to m do t3 := op(i,t1); if nops(t3) = 1 then t2 := t2+1/op(t3); else t2 := t2+op(2,t3)/op(op(1,t3)); fi od: t2 := t2-1/B; n*t2; end;
A003415 := proc(n)
        local a,f;
        a := 0 ;
        for f in ifactors(n)[2] do
                a := a+ op(2,f)/op(1,f);
        end do;
        n*a ;
end proc: # R. J. Mathar, Apr 05 2012

a[ n_] := If[ Abs @ n < 2, 0, n Total[ #2 / #1 & @@@ FactorInteger[ Abs @ n]]]; (* Michael Somos, Apr 12 2011 *)
dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Total[n*f[[2]]/f[[1]]]]]; Table[dn[n], {n, 0, 100}] (* T. D. Noe, Sep 28 2012 *)

A003415(n) = {local(fac);if(n<1,0,fac=factor(n);sum(i=1,matsize(fac)[1],n*fac[i,2]/fac[i,1]))} /* Michael B. Porter, Nov 25 2009 */

apply( A003415(n)=vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]), [0..99]) \\ M. F. Hasler, Sep 25 2013, updated Nov 27 2019

A003415(n) = { my(s=0, m=1, spf); while(n>1, spf = A020639(n); n /= spf; s += m*n; m *= spf); (s); }; \\ Antti Karttunen, Mar 10 2021

a(n) = my(f=factor(n), r=[1/(e+!e)|e<-f[,1]], c=f[,2]); n*r*c; \\ Ruud H.G. van Tol, Sep 03 2023

from sympy import factorint
def A003415(n):
    return sum([int(n*e/p) for p,e in factorint(n).items()]) if n > 1 else 0
# Chai Wah Wu, Aug 21 2014

def A003415(n):
    F = [] if n == 0 else factor(n)
    return n * sum(g / f for f, g in F)
[A003415(n) for n in range(79)] # Peter Luschny, Aug 23 2014

If n = Product p_i^e_i, a(n) = n * Sum (e_i/p_i).
a(m*p^p) = (m + a(m))*p^p, p prime: a(m*A051674(k))=A129283(m)*A051674(k). - Reinhard Zumkeller, Apr 07 2007
For n > 1: a(n) = a(A032742(n)) * A020639(n) + A032742(n). - Reinhard Zumkeller, May 09 2011
a(n) = n * Sum_{p|n} v_p(n)/p, where v_p(n) is the largest power of the prime p dividing n. - Wesley Ivan Hurt, Jul 12 2015
For n >= 2, Sum_{k=2..n} floor(1/a(k)) = pi(n) = A000720(n) (see K. T. Atanassov article). - Ivan N. Ianakiev, Mar 22 2019
From A.H.M. Smeets, Jan 17 2020: (Start)
Limit_{n -> oo} (1/n^2)*Sum_{i=1..n} a(i) = A136141/2.
Limit_{n -> oo} (1/n)*Sum_{i=1..n} a(i)/i = A136141.
a(n) = n if and only if n = p^p, where p is a prime number. (End)
Dirichlet g.f.: zeta(s-1)*Sum_{p prime} 1/(p^s-p), see A136141 (s=2), A369632 (s=3) [Haukkanen, Merikoski and Tossavainen]. - Sebastian Karlsson, Nov 25 2021
From Antti Karttunen, Nov 25 2021: (Start)
a(n) = Sum_{d|n} d * A349394(n/d).
For all n >= 1, A322582(n) <= a(n) <= A348507(n).
If n is not a prime, then a(n) >= 2*sqrt(n), or in other words, for all k >= 1 for which A002620(n)+k is not a prime, we have a(A002620(n)+k) > n. [See Ufnarovski and Åhlander, Theorem 9, point (3).]
(End)

More terms from Michel ten Voorde, Apr 11 2001

A322582 a(n) = n - A003958(n), where A003958 is fully multiplicative with a(p) = (p-1).

Original entry on oeis.org

0, 1, 1, 3, 1, 4, 1, 7, 5, 6, 1, 10, 1, 8, 7, 15, 1, 14, 1, 16, 9, 12, 1, 22, 9, 14, 19, 22, 1, 22, 1, 31, 13, 18, 11, 32, 1, 20, 15, 36, 1, 30, 1, 34, 29, 24, 1, 46, 13, 34, 19, 40, 1, 46, 15, 50, 21, 30, 1, 52, 1, 32, 39, 63, 17, 46, 1, 52, 25, 46, 1, 68, 1, 38, 43, 58, 17, 54, 1, 76, 65, 42, 1, 72, 21, 44, 31, 78, 1

Offset: 1

Antti Karttunen, Dec 17 2018

nonn

a(p*(n/p)) - (n/p) = (p-1)*a(n/p) holds for all prime divisors p of n, which can be seen by expanding the left hand side as p*(n/p) - A003958(p*(n/p)) - (n/p) = (p-1)*(n/p) - (p-1)*A003958(n/p) = (p-1)*((n/p) - A003958(n/p)) = (p-1)*a(n/p). This shows that this sequence gives a lower limit for arithmetic derivative (A003415) in the same way as A348507 gives an upper limit for it. - Antti Karttunen, Nov 07 2021

With n = Product_{i=1..k} p_i the prime factorization of n, if one constructs for each i a test with a probability of success equal to 1/p_i, and if the tests are independent, then a(n)/n is the probability that at least one of the k tests succeeds. - Luc Rousseau, Jan 14 2023

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A003415, A003958, A322581, A348507, A348928 [= gcd(n,a(n))], A348975 (difference from the arithmetic derivative).
Cf. A349139, A348980, A348981, A348982, A348983 (Dirichlet convolutions with other sequences).
Cf. A168065 (gives the arithmetic mean of this and A348507), A168066.

a[1] = 0; a[n_] := n - Times @@ ((First[#] - 1)^Last[#] & /@ FactorInteger[n]); Array[a, 60] (* Amiram Eldar, Dec 17 2018 *)

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A322582(n) = (n-A003958(n));

A020639(n) = if(1==n, n, (factor(n)[1, 1]));
A322582(n) = { my(s=0, m=1, spf); while(n>1, spf = A020639(n); n /= spf; s += m*n; m *= (spf-1)); (s); }; \\ (Compare to the similar programs given in A003415 and A348507) - Antti Karttunen, Nov 07 2021

a(n) = n - A003958(n).
From Antti Karttunen, Nov 07 2021: (Start)
a(n) = A003415(n) - A348975(n).
For all n >= 1, a(n) <= A003415(n) <= A348507(n).
For n > 1, a(n) = a(A032742(n))*(A020639(n)-1) + A032742(n). [See the comment above and compare with Reinhard Zumkeller's May 09 2011 formula for A003415]
(End)

A348970 a(n) = A003959(n) - A129283(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and A129283(n) is sum of n and its arithmetic derivative.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 7, 1, 1, 0, 8, 0, 1, 1, 33, 0, 9, 0, 10, 1, 1, 0, 40, 1, 1, 10, 12, 0, 11, 0, 131, 1, 1, 1, 48, 0, 1, 1, 54, 0, 13, 0, 16, 12, 1, 0, 164, 1, 13, 1, 18, 0, 57, 1, 68, 1, 1, 0, 64, 0, 1, 14, 473, 1, 17, 0, 22, 1, 15, 0, 204, 0, 1, 14, 24, 1, 19, 0, 230, 67, 1, 0, 80, 1, 1, 1, 96, 0, 75, 1, 28

Offset: 1

Antti Karttunen, Nov 05 2021

nonn

There are no negative terms. We prove this by induction over the prime factorization of n, showing that A348507(n) >= A003415(n) for all values of n >= 1. At n=1, both sequences have value 0, and at the primes both sequences obtain the value 1, so the base cases hold. We know that A348507(n)-(n/p) = (p+1)*A348507(n/p) for all prime factors p of n (see comment in A348507). With the arithmetic derivative we obtain respectively that A003415(n) = A003415(p*(n/p)) = A003415(p)*(n/p) + p*A003415(n/p) = (n/p) + p*A003415(n/p), for any prime factor p of n. Now A348507(p*(n/p)) >= A003415(p*(n/p)) iff A348507(p*(n/p)) - (n/p) >= A003415(p*(n/p)) - (n/p), that is, iff (p+1)*A348507(n/p) >= p*A003415(n/p), which indeed follows by the induction hypothesis, which assumes that A348507(x) >= A003415(x) for all proper divisors x of n.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003415, A003959, A129283, A211991, A348029, A348507, A348971, A348972, A348973, A348974.

Cf. A008578 (positions of zeros), A001358 (positions of ones).

d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); f[p_, e_] := (p + 1)^e; a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - n - d[n]; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348970(n) = (A003959(n) - (n+A003415(n)));

a(n) = A003959(n) - A129283(n) = A003959(n) - (n+A003415(n)).
a(n) = A348029(n) - A211991(n).
a(n) = A348507(n) - A003415(n).
For all n >= 1, a(A001358(n)) = 1.

A348971 a(n) = Product(p(p+1)^(e-1)) - Product((p-1)p^(e-1)), when n = Product(p^e), with p primes, and e their exponents.

Original entry on oeis.org

0, 1, 1, 4, 1, 4, 1, 14, 6, 6, 1, 14, 1, 8, 7, 46, 1, 18, 1, 22, 9, 12, 1, 46, 10, 14, 30, 30, 1, 22, 1, 146, 13, 18, 11, 60, 1, 20, 15, 74, 1, 30, 1, 46, 36, 24, 1, 146, 14, 40, 19, 54, 1, 78, 15, 102, 21, 30, 1, 74, 1, 32, 48, 454, 17, 46, 1, 70, 25, 46, 1, 192, 1, 38, 50, 78, 17, 54, 1, 238, 138, 42, 1, 102, 21

Offset: 1

Antti Karttunen, Nov 05 2021

Möbius transform of A348507.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A003959, A003968, A008683, A300251, A348507, A348970.

Cf. A005596, A104141.

f1[p_, e_] := p*(p + 1)^(e - 1); f2[p_, e_] := (p - 1)*p^(e - 1); a[1] = 0; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) - Times @@ f2 @@@ f; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)

A348971(n) = { my(f=factor(n),m1=1,m2=1,p); for(i=1, #f~, p = f[i, 1]; m1 *= p*(p+1)^(f[i, 2]-1); m2 *= (p-1)*p^(f[i, 2]-1)); (m1-m2); };

A348971(n) = { my(f=factor(n),p); for (i=1, #f~, p = f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f)-eulerphi(n); }

a(n) = A003968(n) - A000010(n).
a(n) = Sum_{d|n} A008683(n/d) * A348507(d).
Sum_{k=1..n} a(k) ~ c * n^2, where c = A104141 * (1/A005596 - 1) = 0.5088692487... . - Amiram Eldar, Oct 05 2023

cp's OEIS Frontend

A349140 a(n) = Sum_{d|n} d * A348507(n/d), where A348507(n) = A003959(n) - n, where A003959 is fully multiplicative with a(p) = (p+1).

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A349141 a(n) = Sum_{d|n} phi(n/d) * A348507(d), where A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).

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A349142 a(n) = Sum_{d|n} psi(n/d) * A348507(d), where psi is Dedekind psi (A001615), A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).

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A349143 a(n) = Sum_{d|n} A038040(d) * A348507(n/d), where A038040(n) = n*tau(n), A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).

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A348950 a(n) = A348507(A276086(n)), where A348507(n) = A003959(n) - n, A003959 is multiplicative with a(p^e) = (p+1)^e, and A276086 gives the prime product form of primorial base expansion of n.

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A349139 a(n) = Sum_{d|n} A322582(d) * A348507(n/d), where A322582(n) = n - A003958(n) and A348507(n) = A003959(n) - n.

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A003415 a(n) = n' = arithmetic derivative of n: a(0) = a(1) = 0, a(prime) = 1, a(m*n) = m*a(n) + n*a(m).

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A003415 a(n) = n' = arithmetic derivative of n: a(0) = a(1) = 0, a(prime) = 1, a(mn) = ma(n) + n*a(m).

A348971 a(n) = Product(p(p+1)^(e-1)) - Product((p-1)p^(e-1)), when n = Product(p^e), with p primes, and e their exponents.