Displaying 1-10 of 13 results found.
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
COMMENTS
Dirichlet convolution of sigma with A348971.
MATHEMATICA
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 *)
PROG
(PARI)
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
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
FORMULA
a(n) = Sum_{k=1..n} A348507(gcd(n,k)).
MATHEMATICA
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 *)
PROG
(PARI)
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
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
MATHEMATICA
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 *)
PROG
(PARI)
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); };
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
COMMENTS
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.
MATHEMATICA
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 *)
PROG
(PARI)
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
CROSSREFS
Cf. A000005, A000027, A000203, A003959, A038040, A060640, A348507, A348971, A349140, A349141, A349142.
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
PROG
(PARI)
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); };
(PARI) 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); };
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
COMMENTS
Question: Is a(n) >= A305809(n) for all n?
MATHEMATICA
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 *)
PROG
(PARI)
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); };
a(n) = n' = arithmetic derivative of n: a(0) = a(1) = 0, a(prime) = 1, a(m*n) = m*a(n) + n*a(m).
(Formerly M3196)
+10
1064
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
COMMENTS
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 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 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
REFERENCES
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).
FORMULA
If n = Product p_i^e_i, a(n) = n * Sum (e_i/p_i).
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 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)
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)
EXAMPLE
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 + ...
MAPLE
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;
local a, f;
a := 0 ;
for f in ifactors(n)[2] do
a := a+ op(2, f)/op(1, f);
end do;
n*a ;
MATHEMATICA
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 *)
PROG
(PARI) 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 */ (PARI) 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 (PARI) 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 (Haskell)
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
(Magma) Ad:=func<h | h*(&+[Factorisation(h)[i][2]/Factorisation(h)[i][1]: i in [1..#Factorisation(h)]])>; [n le 1 select 0 else Ad(n): n in [0..80]]; // Bruno Berselli, Oct 22 2013 (Python)
from sympy import factorint
return sum([int(n*e/p) for p, e in factorint(n).items()]) if n > 1 else 0
(Sage)
F = [] if n == 0 else factor(n)
return n * sum(g / f for f, g in F)
(GAP)
A003415:= Concatenation([0, 0], List(List([2..10^3], Factors),
(APL, Dyalog dialect) A003415 ← { ⍺←(0 1 2) ⋄ ⍵≤1:⊃⍺ ⋄ 0=(3⊃⍺)|⍵:((⊃⍺+(2⊃⍺)×(⍵÷3⊃⍺)) ((2⊃⍺)×(3⊃⍺)) (3⊃⍺)) ∇ ⍵÷3⊃⍺ ⋄ ((⊃⍺) (2⊃⍺) (1+(3⊃⍺))) ∇ ⍵} ⍝ Antti Karttunen, Feb 18 2024
CROSSREFS
Cf. A086134 (least prime factor of n'). Cf. A086131 (greatest prime factor of n'). Cf. A027471 (derivative of 3^(n-1)). Cf. A068237 (numerator of derivative of 1/n). Cf. A068238 (denominator of derivative of 1/n). Cf. A068328 (derivative of squarefree numbers). 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. 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. 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. 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. A229501 (n divides the n-th partial sum). 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. A305809 (Dirichlet convolution square). Cf. A069359 (similar formula which agrees on squarefree numbers). Cf. A258851 (the pi-based arithmetic derivative of n). Cf. A038554 (another sequence using "derivative" in its name, but involving binary expansion of n).
a(n) = n - A003958(n), where A003958 is fully multiplicative with a(p) = (p-1).
+10
15
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
COMMENTS
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
MATHEMATICA
a[1] = 0; a[n_] := n - Times @@ ((First[#] - 1)^Last[#] & /@ FactorInteger[n]); Array[a, 60] (* Amiram Eldar, Dec 17 2018 *)
PROG
(PARI)
A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
(PARI)
A020639(n) = if(1==n, n, (factor(n)[1, 1]));
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.
+10
9
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
COMMENTS
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.
MATHEMATICA
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 *)
PROG
(PARI)
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); };
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.
+10
5
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
MATHEMATICA
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 *)
PROG
(PARI) 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); }; (PARI) 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); }
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