A297159 -id:A297159 - cp's OEIS frontend

A324400 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) for all i, j >= 1, where f(n) = -1 if n = 2^k and k > 0, and f(n) = n for all other numbers.

1, 2, 3, 2, 4, 5, 6, 2, 7, 8, 9, 10, 11, 12, 13, 2, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 2, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 2, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78

Offset: 1

Antti Karttunen, Mar 01 2019

nonn

In the following, A stands for this sequence, A324400, and S -> T (where S and T are sequence A-numbers) indicates that for all i, j >= 1: S(i) = S(i) => T(i) = T(j).
For example, the following chains of implications hold:
  A -> A286619 -> A005811,
and
  A -> A003602 -> A286622 -> A000120,
               -> A286378 -> A002487,
               -> A323889,
               -> A000593,
               -> A001227,
among many others.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000523.
Cf. A000120, A000265, A000593, A001227, A002487, A003602, A005811, A209229, A286378, A286619, A286622, A294898, A297159, A318311, A323234, A323889, A323904, A324343, A324345 (some of the matching sequences).
Cf. also A103391, A295300, A305800, A305801, A324401.

A000523(n) = if(n<1, 0, #binary(n)-1);
A324400(n) = if(n<4,n,if(!bitand(n,n-1),2,1+n-A000523(n)));

If n <= 3, a(n) = n; and for n >= 4, if A209229(n) = 1, then a(n) = 2, otherwise a(n) = 1 + n - A000523(n).

A051709 a(n) = sigma(n) + phi(n) - 2n.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 3, 1, 2, 0, 8, 0, 2, 2, 7, 0, 9, 0, 10, 2, 2, 0, 20, 1, 2, 4, 12, 0, 20, 0, 15, 2, 2, 2, 31, 0, 2, 2, 26, 0, 24, 0, 16, 12, 2, 0, 44, 1, 13, 2, 18, 0, 30, 2, 32, 2, 2, 0, 64, 0, 2, 14, 31, 2, 32, 0, 22, 2, 28, 0, 75, 0, 2, 14, 24, 2, 36, 0

Offset: 1

Jud McCranie and Carlos Rivera

nonn

Sigma is the sum of divisors (A000203), and phi is the Euler totient function (A000010). - Michael B. Porter, Jul 05 2013
Because sigma and phi are multiplicative functions, it is easy to show that (1) if a(n)=0, then n is prime or 1 and (2) if a(n)=2, then n is the product of two distinct prime numbers. Note that a(n) is the n-th term of the Dirichlet series whose generating function is given below. Using the generating function, it is theoretically possible to compute a(n). Hence a(n)=0 could be used as a primality test and a(n)=2 could be used as a test for membership in P2 (A006881). - T. D. Noe, Aug 01 2002
It appears that a(n) - A002033(n) = zeta(s-1) * (zeta(s) - 2 + 1/zeta(s)) + 1/(zeta(s)-2). - Eric Desbiaux, Jul 04 2013
a(n) = 1 if and only if n = prime(k)^2 (n is in A001248). It seems that a(n) = k has only finitely many solutions for k >= 3. - Jianing Song, Jun 27 2021

			a(5) = sigma(5) + phi(5) - 2*5 = 6 + 4 - 10 = 0.

Antti Karttunen, Table of n, a(n) for n = 1..65537 (First 1000 terms from T. D. Noe.)
Carlos Rivera, Puzzle 76. z(n)=sigma(n) + phi(n) - 2n, The Prime Puzzles and Problems Connection.

Cf. A000010, A000203, A001065, A001248, A005843, A006881, A051612, A051953, A065387, A072780, A228498 (= a(n^2)), A297159, A324048, A344994, A344995, A344996, A345048, A345054.
Cf. A278373 (range of this sequence), A056996 (numbers not present).
Cf. also A344753, A345001 (analogous sequences).

Table[DivisorSigma[1,n]+EulerPhi[n]-2n,{n,80}] (* Harvey P. Dale, Apr 08 2015 *)

a(n)=sigma(n)+eulerphi(n)-2*n \\ Charles R Greathouse IV, Jul 05 2013

A051709(n) = -sumdiv(n,d,(dAntti Karttunen, Mar 02 2018

Dirichlet g.f.: zeta(s-1) * (zeta(s) - 2 + 1/zeta(s)). - T. D. Noe, Aug 01 2002
From Antti Karttunen, Mar 02 2018: (Start)
a(n) = A001065(n) - A051953(n).  [Difference between the sum of proper divisors of n and their Moebius-transform.]
a(n) = -Sum_{d|n, dA008683(n/d)*A001065(d). (End)
Sum_{k=1..n} a(k) = (3/(Pi^2) + Pi^2/12 - 1) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 03 2023

A323911 Sum of deficiency of n (A033879) and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 1, 0, 4, 0, 1, 4, 8, 0, -2, 0, 12, 16, 1, 0, -2, 0, 0, 24, 20, 0, -8, 16, 24, 12, 2, 0, -28, 0, 1, 40, 32, 48, -15, 0, 36, 48, -6, 0, -36, 0, 6, 16, 44, 0, -20, 36, 6, 64, 8, 0, -12, 80, -4, 72, 56, 0, -46, 0, 60, 28, 1, 96, -52, 0, 12, 88, -44, 0, -39, 0, 72, 28, 14, 120, -60, 0, -18, 37, 80, 0, -58, 128, 84, 112, 0, 0, -52, 144, 18

Offset: 1

Antti Karttunen, Feb 12 2019

sign

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

Cf. A033879, A297159, A323403, A323910, A323913.

up_to = 16384;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA033879(n) = (2*n-sigma(n));
v323910 = DirInverse(vector(up_to,n,A033879(n)));
A323910(n) = v323910[n];
A323911(n) = (A033879(n)+A323910(n));

a(n) = A033879(n) + A323910(n).

A324401 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) for all i, j >= 1, where f(n) = -1 if n is an odd prime, f(n) = -2 if n = 2^k, with k > 1, and f(n) = n for all other numbers.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 4, 6, 7, 3, 8, 3, 9, 10, 4, 3, 11, 3, 12, 13, 14, 3, 15, 16, 17, 18, 19, 3, 20, 3, 4, 21, 22, 23, 24, 3, 25, 26, 27, 3, 28, 3, 29, 30, 31, 3, 32, 33, 34, 35, 36, 3, 37, 38, 39, 40, 41, 3, 42, 3, 43, 44, 4, 45, 46, 3, 47, 48, 49, 3, 50, 3, 51, 52, 53, 54, 55, 3, 56, 57, 58, 3, 59, 60, 61, 62, 63, 3, 64, 65, 66, 67, 68, 69, 70, 3, 71, 72, 73, 3, 74, 3, 75, 76

Offset: 1

Antti Karttunen, Mar 01 2019

nonn

For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A305976(i) = A305976(j) => A001221(i) = A001221(j),
  a(i) = a(j) => A322591(i) = A322591(j),
  a(i) = a(j) => A323235(i) = A323235(j),
  a(i) = a(j) => A324399(i) = A324399(j),
  a(i) = a(j) => A297159(i) = A297159(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000523, A000720.

Cf. also A297159, A305801, A305976, A322591, A323235, A324399, A324400.

A000523(n) = if(n<1, 0, #binary(n)-1);
A324401(n) = if(n<4,n,if(isprime(n),3,if(!bitand(n,n-1),4,4+n-A000523(n)-primepi(n))));

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
Aux324401(n) = if((n>2) && (isprime(n)||!bitand(n,n-1)),-(2-(n%2)),n);
\\ Equally: Aux324401(n) = if(n<=2,n,if(isprime(n),-1,if(!bitand(n,n-1),-2,n)));
v324401 = rgs_transform(vector(up_to, n, Aux324401(n)));
A324401(n) = v324401[n];

If n <= 2, a(n) = n, for n > 2, if n is an odd prime, a(n) = 3, if n = 2^k, with k >= 2, a(n) = 4, otherwise a(n) = 4 + n - A000523(n) - A000720(n).

A323904 Lexicographically earliest sequence such that a(i) = a(j) => A033879(i) = A033879(j) and A083254(i) = A083254(j), for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 2, 7, 8, 9, 10, 11, 12, 13, 2, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 2, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 38, 50, 51, 52, 53, 54, 55, 56, 57, 35, 2, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Antti Karttunen, Feb 10 2019

nonn

Restricted growth sequence transform of the ordered pair [A033879(n), A083254(n)], the deficiency of n, and its Möbius transform.

For all i, j: a(i) = a(j) => A297159(i) = A297159(j).

			For n=39, we have A033879(39) = 2*39 - A000203(39) = 22, and A083254(39) = 2*A000010(39)-39 = 9. For n=63 the results are same, with A033879(63) = 22 and A083254(63) = 9, thus a(39) and a(63) are allotted the same number by the restricted growth sequence transform, which in this case is 35, thus a(39) = a(63) = 35.
For n=42 and 54, we have A033879(42) = -12, A083254(42) = -18 and A033879(54) = -12, A083254(54) = -18, thus a(42) = a(54) (= 38).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A000203, A033879, A083254.

Cf. also A318310, A323892, A323897, A323898.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A033879(n) = (2*n-sigma(n));
A083254(n) = (2*eulerphi(n)-n);
A323904aux(n) = [A033879(n), A083254(n)];
v323904 = rgs_transform(vector(up_to,n,A323904aux(n)));
A323904(n) = v323904[n];

a(2^n) = 2 for all n >= 1.

A324048 a(n) = A000203(n) - A083254(n) = n + sigma(n) - 2*phi(n).

Original entry on oeis.org

0, 3, 3, 7, 3, 14, 3, 15, 10, 20, 3, 32, 3, 26, 23, 31, 3, 45, 3, 46, 29, 38, 3, 68, 16, 44, 31, 60, 3, 86, 3, 63, 41, 56, 35, 103, 3, 62, 47, 98, 3, 114, 3, 88, 75, 74, 3, 140, 22, 103, 59, 102, 3, 138, 47, 128, 65, 92, 3, 196, 3, 98, 95, 127, 53, 170, 3, 130, 77, 166, 3, 219, 3, 116, 119, 144, 53, 198, 3, 202, 94, 128, 3

Offset: 1

Antti Karttunen, Feb 13 2019

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

Cf. A000010, A000203, A001065, A051709, A051612, A051953, A083254, A297159.

a[n_] := n + DivisorSigma[1, n] - 2 * EulerPhi[n]; Array[a, 100] (* Amiram Eldar, Dec 04 2023 *)

A083254(n) = (2*eulerphi(n)-n);
A324048(n) = (sigma(n) - A083254(n));

a(n) = {my(f = factor(n)); n + sigma(f) - 2*eulerphi(f);} \\ Amiram Eldar, Dec 04 2023

a(n) = A000203(n) - A083254(n) = n + A000203(n) - 2*A000010(n).
a(n) = A051612(n) + A051953(n).
a(n) = A297159(n) + 2*A001065(n).
Sum_{k=1..n} a(k) = (Pi^2/12 - 6/Pi^2 + 1/2) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 04 2023

cp's OEIS Frontend

A324400 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) for all i, j >= 1, where f(n) = -1 if n = 2^k and k > 0, and f(n) = n for all other numbers.

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A051709 a(n) = sigma(n) + phi(n) - 2n.

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A323911 Sum of deficiency of n (A033879) and its Dirichlet inverse.

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A324401 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) for all i, j >= 1, where f(n) = -1 if n is an odd prime, f(n) = -2 if n = 2^k, with k > 1, and f(n) = n for all other numbers.

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A323904 Lexicographically earliest sequence such that a(i) = a(j) => A033879(i) = A033879(j) and A083254(i) = A083254(j), for all i, j >= 1.

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A324048 a(n) = A000203(n) - A083254(n) = n + sigma(n) - 2*phi(n).

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