A023900 -id:A023900 - cp's OEIS frontend

A319340 Sum of Euler totient function and its Dirichlet inverse: a(n) = A000010(n) + A023900(n).

2, 0, 0, 1, 0, 4, 0, 3, 4, 8, 0, 6, 0, 12, 16, 7, 0, 8, 0, 12, 24, 20, 0, 10, 16, 24, 16, 18, 0, 0, 0, 15, 40, 32, 48, 14, 0, 36, 48, 20, 0, 0, 0, 30, 32, 44, 0, 18, 36, 24, 64, 36, 0, 20, 80, 30, 72, 56, 0, 8, 0, 60, 48, 31, 96, 0, 0, 48, 88, 0, 0, 26, 0, 72, 48, 54, 120, 0, 0, 36, 52, 80, 0, 12, 128, 84, 112, 50, 0, 16

Offset: 1

Antti Karttunen, Sep 17 2018

nonn

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

Cf. A000010, A023900, A051953, A318833, A319341.

A023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
A319340(n) = (eulerphi(n)+A023900(n));

a(n) = A000010(n) + A023900(n).

a(n) = A318833(n) - A051953(n).

A323915 a(n) = A023900(A005940(1+n)).

Original entry on oeis.org

1, -1, -2, -1, -4, 2, -2, -1, -6, 4, 8, 2, -4, 2, -2, -1, -10, 6, 12, 4, 24, -8, 8, 2, -6, 4, 8, 2, -4, 2, -2, -1, -12, 10, 20, 6, 40, -12, 12, 4, 60, -24, -48, -8, 24, -8, 8, 2, -10, 6, 12, 4, 24, -8, 8, 2, -6, 4, 8, 2, -4, 2, -2, -1, -16, 12, 24, 10, 48, -20, 20, 6, 72, -40, -80, -12, 40, -12, 12, 4, 120, -60, -120, -24, -240, 48, -48

Offset: 0

Antti Karttunen, Feb 22 2019

sign

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A005940, A023900, A290077, A324052, A324054, A324055, A324057, A324058.

A323915(n) = { my(m1=1, p=2); while(n, if(!(n%2), p=nextprime(1+p), if(1==(n%4), m1 *= (1-p))); n>>=1); (m1); };

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A023900(n) = sumdivmult(n, d, d*moebius(d)); \\ From A023900
A323915(n) = A023900(A005940(1+n));

a(n) = A023900(A005940(1+n)).

A130054 Inverse Moebius transform of A023900.

Original entry on oeis.org

1, 0, -1, -1, -3, 0, -5, -2, -3, 0, -9, 1, -11, 0, 3, -3, -15, 0, -17, 3, 5, 0, -21, 2, -7, 0, -5, 5, -27, 0, -29, -4, 9, 0, 15, 3, -35, 0, 11, 6, -39, 0, -41, 9, 9, 0, -45, 3, -11, 0, 15, 11, -51, 0, 27, 10, 17, 0, -57, -3, -59, 0, 15, -5, 33, 0, -65, 15, 21

Offset: 1

Gary W. Adamson, May 04 2007

Multiplicative because A023900 is. - Andrew Howroyd, Aug 03 2018

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Andrew Howroyd)

Row sums of A129691.

Cf. A126988, A000005, A023900, A062570, A001511, A018804, A000203, A007431.

[&+[d*MoebiusMu(d)*NumberOfDivisors(n div d):d in Divisors(n)]:n in [1..70]]; // Marius A. Burtea, Nov 17 2019

with(numtheory): seq(add(d*mobius(d)*tau(n/d), d in divisors(n)), n=1..60); # Ridouane Oudra, Nov 17 2019

b[n_] := Sum[d MoebiusMu[d], {d, Divisors[n]}];
a[n_] := Sum[b[n/d], {d, Divisors[n]}];
a /@ Range[1, 100] (* Jean-François Alcover, Sep 20 2019, from PARI *)
f[p_, e_] := 1-(p-1)*e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 23 2020 *)

\\ here b(n) is A023900
b(n)={sumdivmult(n, d, d*moebius(d))}
a(n)={sumdiv(n, d, b(n/d))} \\ Andrew Howroyd, Aug 03 2018

A126988 * A130054 = d(n), A000005: (1, 2, 2, 3, 2, 4, 2, 4, 3, 4, ...).
a(n) = Sum_{d|n} A023900(n/d). - Andrew Howroyd, Aug 03 2018
a(n) = Sum_{d|n} d*mu(d)*tau(n/d). - Ridouane Oudra, Nov 17 2019
From Werner Schulte, Sep 06 2020: (Start)
Multiplicative with a(p^e) = 1 - (p-1) * e for prime p and e >= 0.
Dirichlet g.f.: (zeta(s))^2 / zeta(s-1).
Dirichlet convolution with A062570 equals A001511.
Dirichlet convolution with A018804 equals A000203.
Dirichlet inverse of A007431. (End)
a(n) = 1 - Sum_{k=1..n-1} a(gcd(n,k)). - Ilya Gutkovskiy, Nov 06 2020

Name changed and terms a(11) and beyond from Andrew Howroyd, Aug 03 2018

A295886 Filter-sequence combining A003557(n) and A023900(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 21, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 34, 38, 39, 13, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 54, 71, 72, 73, 67, 74, 75, 76, 52, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 73

Offset: 1

Antti Karttunen, Dec 03 2017

nonn

For all i, j: a(i) = a(j) => A295887(i) = A295887(j).

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

Cf. A003557, A023900, A295876, A295887.

allocatemem(2^30);
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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); };
A023900(n) = sumdivmult(n, d, d*moebius(d)); \\ This function from Charles R Greathouse IV, Sep 09 2014
v295876 = rgs_transform(vector(up_to,n,A023900(n)))
A295876(n) = v295876[n];
Anotsubmitted6(n) = (1/2)*(2 + ((A003557(n)+A295876(n))^2) - A003557(n) - 3*A295876(n));
write_to_bfile(1,rgs_transform(vector(up_to,n,Anotsubmitted6(n))),"b295886.txt");

Restricted growth sequence transform of sequence a(n) = (1/2)*(2 + ((A003557(n) + A295876(n))^2) - A003557(n) - 3*A295876(n)).

A318833 a(n) = n + A023900(n).

Original entry on oeis.org

2, 1, 1, 3, 1, 8, 1, 7, 7, 14, 1, 14, 1, 20, 23, 15, 1, 20, 1, 24, 33, 32, 1, 26, 21, 38, 25, 34, 1, 22, 1, 31, 53, 50, 59, 38, 1, 56, 63, 44, 1, 30, 1, 54, 53, 68, 1, 50, 43, 54, 83, 64, 1, 56, 95, 62, 93, 86, 1, 52, 1, 92, 75, 63, 113, 46, 1, 84, 113, 46, 1, 74, 1, 110, 83, 94, 137, 54, 1, 84, 79, 122, 1, 72, 149, 128, 143, 98, 1, 82

Offset: 1

Antti Karttunen, Sep 16 2018

nonn

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

Cf. A023900, A318841.

A023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
A318833(n) = (n+A023900(n));

a(n) = n + A023900(n).

A323405 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A003557(n), A023900(n), A063994(n)] for all other numbers, except f(n) = 0 for odd primes.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 13 2019

nonn

For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A323371(i) = A323371(j),
  a(i) = a(j) => A247074(i) = A247074(j).

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

Cf. A000010, A003557, A023900, A063994, A247074, A305801, A323371, A323404.
Differs from A323370 for the first time at n=78, where a(78) = 58, while A323370(78) = 52.
Cf. also A323374.

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; };
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); };
A023900(n) = sumdivmult(n, d, d*moebius(d)); \\ From A023900
A063994(n) = { my(f=factor(n)[, 1]); prod(i=1, #f, gcd(f[i]-1, n-1)); }; \\ From A063994
Aux323405(n) = if((n>2)&&isprime(n),0,[A003557(n), A023900(n), A063994(n)]);
v323405 = rgs_transform(vector(up_to, n, Aux323405(n)));
A323405(n) = v323405[n];

A301591 Primes p that have other solutions x to A023900(x) = A023900(p) than a power of p.

Original entry on oeis.org

13, 37, 41, 61, 73, 89, 97, 109, 113, 157, 181, 193, 233, 241, 277, 281, 313, 337, 349, 353, 397, 401, 409, 421, 433, 449, 457, 461, 521, 541, 577, 593, 601, 613, 617, 641, 661, 673, 701, 733, 757, 761, 769, 821, 829, 877, 881, 929, 937, 953, 997, 1009, 1013, 1021, 1033, 1049

Offset: 1

Michel Marcus, Mar 24 2018

nonn

Contains A005383 \ {3, 5} as a subsequence, since if (p+1)/2 = q > 3 is prime, then A023900(2*3*q) = (1-2)*(1-3)*(1-q) = 1-p = A023900(p). - M. F. Hasler, Aug 14 2021

			13 is a term because A023900(42) = A023900(13), where 42 is not a power of 13.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A001055, A023900, A301374.
Complement of A301590.
A005383 \ {3,5} is a subsequence.

f(n) = sumdivmult(n, d, d*moebius(d)); /* This is A023900 */
isok(p, vp) = {for (k=p+1, p^2-1, if (f(k) == vp, return (0)); ); return (1); }
lista(nn) = {forprime(p=2, nn, vp = f(p); if (!isok(p, vp), print1(p, ", ")); ); }

A323370 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A000035(n), A003557(n), A023900(n)] for all other numbers, except f(n) = 0 for odd primes.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 13 2019

nonn

Restricted growth sequence transform of function f, defined as f(n) = 0 when n is an odd prime, and f(n) = [A000035(n), A003557(n), A023900(n)] for all other numbers.
For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A323367(i) = A323367(j),
  a(i) = a(j) => A323371(i) = A323371(j).

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

Cf. A000035, A003557, A023900, A092248, A295886, A305801, A319340, A322587, A323367, A323371.

Differs from A323405 for the first time at n=78, where a(78) = 52, while A323405(78) = 58.

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; };
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); };
A023900(n) = sumdivmult(n, d, d*moebius(d)); \\ From A023900
Aux323370(n) = if((n>2)&&isprime(n),0,[(n%2), A003557(n), A023900(n)]);
v323370 = rgs_transform(vector(up_to, n, Aux323370(n)));
A323370(n) = v323370[n];

A349373 Dirichlet convolution of Kimberling's paraphrases (A003602) with Dirichlet inverse of Euler phi (A023900).

Original entry on oeis.org

1, 0, 0, -1, -1, 0, -2, -2, -1, 0, -4, 0, -5, 0, 2, -3, -7, 0, -8, 1, 3, 0, -10, 0, -3, 0, -2, 2, -13, 0, -14, -4, 5, 0, 8, 1, -17, 0, 6, 2, -19, 0, -20, 4, 5, 0, -22, 0, -5, 0, 8, 5, -25, 0, 14, 4, 9, 0, -28, -2, -29, 0, 8, -5, 17, 0, -32, 7, 11, 0, -34, 2, -35, 0, 4, 8, 23, 0, -38, 3, -3, 0, -40, -3, 23, 0, 14, 8

Offset: 1

Antti Karttunen, Nov 15 2021

sign

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

Cf. A003602, A023900.

Cf. A347954, A347955, A347956, A349136, A349370, A349371, A349372, A349374, A349375, A349390, A349431, A349444, A349447 for Dirichlet convolutions of other sequences with A003602.

f[p_, e_] := (1 - p); d[1] = 1; d[n_] := Times @@ f @@@ FactorInteger[n]; k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, k[#] * d[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)

A003602(n) = (1+(n>>valuation(n,2)))/2;
A023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
A349373(n) = sumdiv(n,d,A003602(n/d)*A023900(d));

a(n) = Sum_{d|n} A003602(n/d) * A023900(d).

A295876 Restricted growth sequence transform of A023900, Product_{p|n} (1-p).

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 2, 3, 7, 8, 5, 9, 10, 11, 2, 12, 5, 13, 7, 14, 15, 16, 5, 4, 14, 3, 10, 17, 18, 19, 2, 20, 21, 22, 5, 23, 24, 22, 7, 25, 9, 26, 15, 11, 27, 28, 5, 6, 7, 29, 14, 30, 5, 31, 10, 32, 33, 34, 18, 35, 36, 14, 2, 37, 38, 39, 21, 40, 41, 42, 5, 43, 32, 11, 24, 44, 41, 45, 7, 3, 31, 46, 9, 47, 48, 49, 15, 50, 18, 51, 27, 44, 52, 51, 5, 53, 10

Offset: 1

Antti Karttunen, Nov 29 2017

nonn

For all i, j:
  a(i) = a(j) => A092248(i) = A092248(j).
  a(i) = a(j) => A295877(i) = A295877(j).

Antti Karttunen, Table of n, a(n) for n = 1..65536

Cf. A023900, A092248, A295877.

allocatemem(2^30);
up_to = 65536;
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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A023900(n) = sumdivmult(n, d, d*moebius(d)); \\ This function from Charles R Greathouse IV, Sep 09 2014
write_to_bfile(1,rgs_transform(vector(up_to,n,A023900(n))),"b295876.txt");

cp's OEIS Frontend

A319340 Sum of Euler totient function and its Dirichlet inverse: a(n) = A000010(n) + A023900(n).

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A323915 a(n) = A023900(A005940(1+n)).

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A130054 Inverse Moebius transform of A023900.

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A295886 Filter-sequence combining A003557(n) and A023900(n).

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A318833 a(n) = n + A023900(n).

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A323405 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A003557(n), A023900(n), A063994(n)] for all other numbers, except f(n) = 0 for odd primes.

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A301591 Primes p that have other solutions x to A023900(x) = A023900(p) than a power of p.

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A323370 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A000035(n), A003557(n), A023900(n)] for all other numbers, except f(n) = 0 for odd primes.

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