A211991 -id:A211991 - cp's OEIS frontend

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.

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.

A343226 a(n) = gcd(sigma(n), n+A003415(n)), where A003415 is the arithmetic derivative, and sigma is the sum of divisors of n.

Original entry on oeis.org

1, 3, 4, 1, 6, 1, 8, 5, 1, 1, 12, 28, 14, 1, 1, 1, 18, 39, 20, 2, 1, 1, 24, 4, 1, 1, 2, 4, 30, 1, 32, 7, 1, 1, 1, 1, 38, 1, 1, 18, 42, 1, 44, 4, 6, 1, 48, 4, 3, 1, 1, 2, 54, 15, 1, 4, 1, 1, 60, 8, 62, 1, 2, 1, 1, 1, 68, 14, 1, 3, 72, 3, 74, 1, 2, 4, 1, 1, 80, 2, 1, 1, 84, 16, 1, 1, 1, 12, 90, 3, 1, 4, 1, 1, 1, 4

Offset: 1

Antti Karttunen, Apr 15 2021

nonn

a(n) = n+1 iff n is prime (A000040). - Bernard Schott, Jun 01 2021

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A003415, A129283, A211991, A343227.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A129283(n) = (n+A003415(n));
A343226(n) = gcd(sigma(n), A129283(n));

a(n) = gcd(A000203(n), A129283(n)) = gcd(A000203(n), A211991(n)).
a(n) = A000203(n) / A343227(n).
a(n) = 1 if n is squarefree semiprime (A006881). - Bernard Schott, Jun 02 2021

A212127 Numbers n whose arithmetic derivative equals the sum of its proper divisors.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 220, 223, 227, 229, 233, 239, 241, 251, 257

Offset: 1

Omar E. Pol, Dec 18 2012

nonn

Numbers n such that A003415(n) = A001065(n). Also numbers n such that A211991(n) = 0. By definition, all prime numbers are in the sequence. Nonprime numbers in the sequence are 1, 12, 18, 220,...

			The arithmetic derivative of 12 is equal to 16 (see A003415). On the other hand the sum of proper divisors of 12 is equal to 16 since 1+2+3+4+6 = 16, so 12 is in the sequence.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A000040, A000203, A001065, A003415, A211991.

with(numtheory);
A212127:=proc(i)
local n, p;
for n from 1 to i do
  if sigma(n)/n-1=add(op(2,p)/op(1,p),p=ifactors(n)[2]) then print(n);
fi; od; end:
A212127(1000);  # Paolo P. Lava, Jan 04 2012

dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; Select[Range[300], dn[#] == DivisorSigma[1, #] - # &] (* T. D. Noe, Dec 27 2012 *)

A344586 Numbers k for which A003415(k) >= A001065(k), where A003415 gives the arithmetic derivative, and A001065 is the sum of proper divisors.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 36, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 100, 101, 103, 104, 107, 108, 109, 112, 113, 116, 117, 120, 121, 124, 125, 127, 128, 131

Offset: 1

Antti Karttunen, May 24 2021

nonn

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

Cf. A001065, A003415.
Cf. A212127, A212128 (subsequences), A344585 (complement).
Positions of nonnegative terms in A211991.
Differs from A212165 for the first time at n=121, where a(121) = 220, while A212165(121) = 223.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA344586(n) = (A003415(n) >= (sigma(n)-n));

A212128 Nonprimes whose arithmetic derivative equals the sum of its proper divisors.

Original entry on oeis.org

1, 12, 18, 220, 396, 287532

Offset: 1

Omar E. Pol, Dec 19 2012

a(7) > 10^12. - Giovanni Resta, Mar 11 2014

			The arithmetic derivative of 12 is equal to 16 (see A003415). On the other hand the sum of proper divisors of 12 is equal to 16 since 1+2+3+4+6 = 16, so 12 is in the sequence.

Cf. A018252, A000203, A001065, A003415, A211991, A212127.

dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; Select[Range[300000], ! PrimeQ[#] && dn[#] == DivisorSigma[1, #] - # &] (* T. D. Noe, Dec 27 2012 *)

A344178 Difference between the arithmetic derivative of n and the cototient of n: a(n) = A003415(n) - A051953(n).

Original entry on oeis.org

0, 0, 0, 2, 0, 1, 0, 8, 3, 1, 0, 8, 0, 1, 1, 24, 0, 9, 0, 12, 1, 1, 0, 28, 5, 1, 18, 16, 0, 9, 0, 64, 1, 1, 1, 36, 0, 1, 1, 44, 0, 11, 0, 24, 18, 1, 0, 80, 7, 15, 1, 28, 0, 45, 1, 60, 1, 1, 0, 48, 0, 1, 24, 160, 1, 15, 0, 36, 1, 13, 0, 108, 0, 1, 20, 40, 1, 17, 0, 128, 81, 1, 0, 64, 1, 1, 1, 92, 0, 57, 1, 48, 1, 1, 1, 208

Offset: 1

Antti Karttunen, May 23 2021

nonn

Question: Are all terms nonnegative? See also A211991 and A344584.
From Bernard Schott, May 25 2021: (Start)
Answer: Yes, can be proved when n = Product_{i=1..k} p_i^e_i with n' = n * Sum_{i=1..k} (e_i/p_i) and cototient(n) = n * (1 - Product_{i=1..k} (1 - 1/p_i)).
a(n) = 0 iff n is in A008578 (1 together with the primes).
a(n) = 1 iff n is in A006881 (squarefree semiprimes) (End).

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

Cf. A000010, A003415, A051953, A168036, A344584 (inverse Möbius transform).
Cf. also A211991.
Cf. A006881, A008578.

Array[If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] - # + EulerPhi[#] &, 96] (* Michael De Vlieger, May 24 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A344178(n) = A003415(n) - (n-eulerphi(n));

a(n) = A003415(n) - A051953(n) = A168036(n) + A000010(n).

A344584 Difference between the inverse Möbius transform of the arithmetic derivative of n and the sum of the proper divisors of n: a(n) = A319684(n) - A001065(n).

Original entry on oeis.org

0, 0, 0, 2, 0, 1, 0, 10, 3, 1, 0, 11, 0, 1, 1, 34, 0, 13, 0, 15, 1, 1, 0, 47, 5, 1, 21, 19, 0, 12, 0, 98, 1, 1, 1, 59, 0, 1, 1, 67, 0, 14, 0, 27, 22, 1, 0, 151, 7, 21, 1, 31, 0, 76, 1, 87, 1, 1, 0, 82, 0, 1, 28, 258, 1, 18, 0, 39, 1, 16, 0, 203, 0, 1, 26, 43, 1, 20, 0, 219, 102, 1, 0, 104, 1, 1, 1, 127, 0, 99, 1, 51, 1, 1, 1, 423

Offset: 1

Antti Karttunen, May 24 2021

nonn

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

Cf. A000203, A001065, A003415, A168036, A211991, A319683, A319684.

Inverse Möbius transform of A344178.

Block[{a}, a[1] = 0; a[n_] := a[n] = If[n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[n]]]; Array[DivisorSum[#, a[#] &] - DivisorSigma[1, #] + # &, 96]] (* Michael De Vlieger, May 24 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A319684(n) = sumdiv(n, d, A003415(d));
A344584(n) = (A319684(n) - (sigma(n)-n));

a(n) = A319684(n) - A001065(n) = A211991(n) + A319683(n).
a(n) = Sum_{d|n} A344178(d).
a(n) = n + Sum_{d|n} A168036(d).

A344585 Numbers whose sum of proper divisors (A001065) is greater than their arithmetic derivative (A003415).

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 38, 39, 42, 46, 51, 55, 57, 58, 60, 62, 65, 66, 69, 70, 74, 77, 78, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 102, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 126, 129, 130, 132, 133, 134, 138, 140, 141, 142, 143, 145, 146, 150, 154, 155, 156, 158, 159, 161, 165, 166, 170

Offset: 1

Antti Karttunen, May 24 2021

nonn

Differs from A212168 for the first time at n=100, where a(100) = 221, while A212168(100) = 220. See also comments in A211991.

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

Cf. A001065, A003415, A212168, A344586 (complement).

Positions of negative terms in A211991.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA344585(n) = (A003415(n) < (sigma(n)-n));

A345059 a(n) = A129283(n) / gcd(sigma(n), A129283(n)), where A129283(n) is the sum of n and its arithmetic derivative.

Original entry on oeis.org

1, 1, 1, 8, 1, 11, 1, 4, 15, 17, 1, 1, 1, 23, 23, 48, 1, 1, 1, 22, 31, 35, 1, 17, 35, 41, 27, 15, 1, 61, 1, 16, 47, 53, 47, 96, 1, 59, 55, 6, 1, 83, 1, 23, 14, 71, 1, 40, 21, 95, 71, 54, 1, 9, 71, 37, 79, 89, 1, 19, 1, 95, 57, 256, 83, 127, 1, 10, 95, 43, 1, 76, 1, 113, 65, 39, 95, 149, 1, 128, 189, 125, 1, 13, 107

Offset: 1

Antti Karttunen, Jun 07 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12800
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A003415, A129283, A211991, A343226, A343227.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A129283(n) = (n+A003415(n));
A345059(n) = { my(u=A129283(n)); (u/gcd(u,sigma(n))); };

a(n) = A129283(n) / A343226(n) = A129283(n) / gcd(A000203, A129283(n)).

cp's OEIS Frontend

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.

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A343226 a(n) = gcd(sigma(n), n+A003415(n)), where A003415 is the arithmetic derivative, and sigma is the sum of divisors of n.

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A212127 Numbers n whose arithmetic derivative equals the sum of its proper divisors.

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A344586 Numbers k for which A003415(k) >= A001065(k), where A003415 gives the arithmetic derivative, and A001065 is the sum of proper divisors.

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A212128 Nonprimes whose arithmetic derivative equals the sum of its proper divisors.

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A344178 Difference between the arithmetic derivative of n and the cototient of n: a(n) = A003415(n) - A051953(n).

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A344584 Difference between the inverse Möbius transform of the arithmetic derivative of n and the sum of the proper divisors of n: a(n) = A319684(n) - A001065(n).

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A344585 Numbers whose sum of proper divisors (A001065) is greater than their arithmetic derivative (A003415).

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