A023900 -id:A023900 - cp's OEIS frontend

A301590 Primes p such that there are no other solutions to A023900(x) = A023900(p) than a power of p.

2, 3, 5, 7, 11, 17, 19, 23, 29, 31, 43, 47, 53, 59, 67, 71, 79, 83, 101, 103, 107, 127, 131, 137, 139, 149, 151, 163, 167, 173, 179, 191, 197, 199, 211, 223, 227, 229, 239, 251, 257, 263, 269, 271, 283, 293, 307, 311, 317, 331, 347, 359, 367, 373, 379, 383, 389

Offset: 1

Michel Marcus, Mar 24 2018

nonn

In the definition, A023900(p) = 1-p. One has sign(A023900(n)) = (-1)^A001221(n), so a different solution x can only exist if x has at least 3 distinct prime factors. The smallest number of the form p*q*r such that (p-1)*(q-1)*(r-1) = P-1 for primes p, q, r, P is 2*3*7 = 42, eliminating P = 13 = A301591(1) from this sequence. This is the case whenever (P+1)/2 = p > 3 is a prime (in A005382), whence P-1 = (2-1)*(3-1)*(p-1), which eliminates all P > 5 in A005383 from this sequence. - M. F. Hasler, Aug 14 2021

			2 is a term because there are no other solutions to A023900(x) = A023900(2) = -1 than other powers of 2.
13 is not a term because A023900(42) = -12 = A023900(13). Similarly, no P > 5 in A005383 is a term because A023900(P) = 1-P = (1-2)*(1-3)*(1-p) = A023900(2*3*p) with p = (P+1)/2. - _M. F. Hasler_, Aug 14 2021

M. F. Hasler, Table of n, a(n) for n = 1..10000, Sep 01 2021

Cf. A023900, A301374.

Complement (within the primes) of A301591, which has A005383 \ {3, 5} as a subsequence. Appears to have A079151 \ {13} as subsequence.

f(n) = sumdivmult(n, d, d*moebius(d)); /* 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, ", ")); ); }

select( {is_A301590(p)=!forcomposite(k=p+1, p^2-1, A023900(k)!=1-p|| return)&& isprime(p)}, primes([1,399])) \\ M. F. Hasler, Aug 14 2021

A306146 Numbers k such that A000010(A023900(k)) = A023900(A000010(k)).

Original entry on oeis.org

1, 14, 22, 28, 44, 46, 56, 75, 88, 92, 94, 112, 118, 166, 176, 184, 188, 214, 224, 236, 332, 334, 352, 358, 368, 375, 376, 422, 428, 448, 454, 472, 526, 639, 662, 664, 668, 694, 704, 716, 718, 736, 752, 766, 844, 856, 867, 896, 908, 926, 934, 944, 958, 1006, 1052, 1075, 1094, 1126, 1142, 1174, 1179, 1324

Offset: 1

Torlach Rush, Aug 11 2018

nonn

No term is a product of an odd number of distinct prime factors (because then A023900 is negative, i.e., contains no terms from A030230).
For known terms:
- a(n) is nonsquarefree iff A000010(n) is nonsquarefree.
- If a(n) is squarefree then A000010(n) and A023900(n) are both squarefree.

			75 is a term because A000010(A023900(75)) = A023900(A000010(75)) = 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A023900, A030230.

isA306146 := proc(n)
    local a239 ;
    a239 := A023900(n) ;
    if a239 >= 1 then
        simplify( numtheory[phi](a239) = A023900(numtheory[phi](n)) );
    else
        false;
    end if;
end proc:
for n from 1 to 1000 do
    if isA306146(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Feb 14 2019

f[p_, e_] := 1 - p; d[1] = 1; d[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[1324],(d1 = d[#]) > 0 && d[EulerPhi[#]] == EulerPhi[d1] &] (* Amiram Eldar, Feb 19 2020 *)

a023900(n) = sumdivmult(n, d, d*moebius(d))
is(n) = sdm = a023900(n); if(sdm < 0, return(0), sdmphi = a023900(eulerphi(n)); eulerphi(sdm) == sdmphi) \\ David A. Corneth, Aug 17 2018

A323404 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)].

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, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 35, 39, 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, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 76, 92

Offset: 1

Antti Karttunen, Jan 13 2019

nonn

For all i, 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, A323405.

Cf. also A323373.

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
Aux323404(n) = if(1,[A003557(n), A023900(n), A063994(n)]);
v323404 = rgs_transform(vector(up_to, n, Aux323404(n)));
A323404(n) = v323404[n];

A297381 Numerator of -A023900(n)/2.

Original entry on oeis.org

-1, 1, 1, 1, 2, -1, 3, 1, 1, -2, 5, -1, 6, -3, -4, 1, 8, -1, 9, -2, -6, -5, 11, -1, 2, -6, 1, -3, 14, 4, 15, 1, -10, -8, -12, -1, 18, -9, -12, -2, 20, 6, 21, -5, -4, -11, 23, -1, 3, -2, -16, -6, 26, -1, -20, -3, -18, -14, 29, 4, 30, -15, -6, 1, -24, 10, 33, -8, -22, 12, 35, -1, 36, -18, -4, -9, -30, 12, 39, -2, 1, -20, 41, 6, -32, -21, -28, -5

Offset: 1

Mats Granvik, Dec 29 2017

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

Cf. A023900, A297382 (denominators).

Clear[n, s, nn]; nn = 64; Numerator[Table[Limit[Zeta[s]*Total[MoebiusMu[Divisors[n]]/Divisors[n]^(s - 1)], s -> 0], {n, 1, nn}]]

a(n) = numerator(-sumdiv(n, d, d*moebius(d))/2) \\ Iain Fox, Dec 29 2017

A297381(n) = numerator(-(1/2)*factorback(apply(p -> 1-p, factor(n)[, 1]))); \\ Antti Karttunen, Sep 30 2018

a(n) = numerator of -A023900(n)/2.
a(n) = numerator of lim_{s->0} zeta(s)*Sum_{d|n} A008683(d)/d^(s-1).
a(n) = numerator of lim_{N->infinity} (1/N)*Sum_{m=1..N} Sum_{k=1..m} A191898(n, k) for n > 1.
a(k) = numerators of lim_{N->infinity} (1/N)*Sum_{m=1..N} Sum_{n=1..m} A191898(n, k) for k > 1.

More terms from Antti Karttunen, Sep 30 2018

A301374 Values of A023900 which occur only at indices which are powers of a prime.

Original entry on oeis.org

-1, -2, -4, -6, -10, -16, -18, -22, -28, -30, -42, -46, -52, -58, -66, -70, -78, -82, -100, -102, -106, -126, -130, -136, -138, -148, -150, -162, -166, -172, -178, -190, -196, -198, -210, -222, -226, -228, -238, -250, -256, -262, -268, -270, -282, -292, -306

Offset: 1

Torlach Rush, Mar 19 2018

Terms are equal to A023900(p) = A023900(p^2) = A023900(p^3) = ... with p prime, but is never equal to A023900(m*p) with m not a power of p. [Corrected by M. F. Hasler, Sep 01 2021]
abs(a(n)) + 1 is prime (A301590).
For n > 1, if and only if n can't be factored into 2*m factors, m > 0, distinct factors f > 1 where f + 1 is prime then -n is a term. - David A. Corneth, Mar 25 2018
The values are of the form a(n) = 1 - p with prime p = A301590(n). These are exactly the values A023900(x) = 1 - p occurring only if x = p^j for some j >= 1. (See counterexample for p = 13 in EXAMPLE section.) - M. F. Hasler, Sep 01 2021

			a(1) = -1 = A023900(2^m), m > 0.
a(2) = -2 = A023900(3^m), m > 0.
a(3) = -4 = A023900(5^m), m > 0.
a(4) = -6 = A023900(7^m), m > 0.
a(5) = -10 = A023900(11^m), m > 0.
a(6) = -16 = A023900(17^m), m > 0.
A023900(13) = -12 is not a term as A023900(42) = -12, and 42 is the product of three prime factors.
From _David A. Corneth_, Mar 25 2018: (Start)
10 can't be factored in an even number of distinct factors f > 1 such that f + 1 is prime, so -10 is in the sequence.
12 can be factored in an even number of distinct factors f > 1; 12 = 2 * 6 and both 2 + 1 and 6 + 1 are prime, hence -12 is not a term. (End)

Cf. A000040, A001055, A023900, A301590, A301591.

Keys@ Select[Union /@ PrimeNu@ PositionIndex@ Array[DivisorSum[#, # MoebiusMu[#] &] &, 310], # == {1} &] (* Michael De Vlieger, Mar 26 2018 *)

f(n) = sumdivmult(n, d, d*moebius(d));
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(vp, ", ")););} \\ Michel Marcus, Mar 23 2018

A331178 Number of values of k, 1 <= k <= n, with A023900(k) = A023900(n), where A023900 is Dirichlet inverse of Euler totient function phi.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 4, 2, 2, 3, 2, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 2, 3, 1, 2, 1, 2, 2, 1, 1, 6, 2, 4, 1, 3, 1, 7, 1, 3, 1, 1, 1, 2, 1, 1, 4, 6, 1, 1, 1, 2, 1, 1, 1, 8, 1, 2, 3, 2, 1, 2, 1, 5, 4, 2, 1, 3, 1, 1, 1, 3, 1, 3, 1, 2, 2, 1, 2, 9, 1, 4, 2, 6, 1, 1, 1, 5, 1

Offset: 1

Antti Karttunen, Jan 11 2020

nonn

Ordinal transform of A023900.

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

Cf. A000010, A023900.

Cf. also A081373, A331175, A331179, A331180.

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
v331178 = ordinal_transform(vector(up_to, n, A023900(n)));
A331178(n) = v331178[n];

A340198 Dirichlet inverse of sequence f(n) = A319340(n)-1 = (A000010(n) + A023900(n) - 1), where A000010 is Euler Totient function phi, and A023900 is its Dirichlet inverse.

Original entry on oeis.org

1, 1, 1, 1, 1, -1, 1, -1, -2, -5, 1, -8, 1, -9, -13, -9, 1, -16, 1, -22, -21, -17, 1, -28, -14, -21, -20, -36, 1, -43, 1, -31, -37, -29, -45, -49, 1, -33, -45, -62, 1, -67, 1, -64, -64, -41, 1, -69, -34, -64, -61, -78, 1, -68, -77, -96, -69, -53, 1, -88, 1, -57, -96, -79, -93, -115, 1, -106, -85, -123, 1, -95, 1, -69

Offset: 1

Antti Karttunen, Jan 05 2021

Conversely, the Dirichlet inverse of this sequence yields a sequence which is one less than A319340, i.e., pointwise sum s(n) = A109606(n) + A023900(n).

a(9796) = 0 is the only zero among the first 2^22 terms.

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

Cf. A000010, A023900, A109606, A319340, A340090, A340094, A340197, A340367.

A023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
A319340(n) = (eulerphi(n)+A023900(n));
A340198(n) = if(1==n,1,-sumdiv(n,d,if(dA319340(n/d)-1)*A340198(d),0)));

a(1) = 1, for n > 1, a(n) = -Sum_{d|n, dA319340(n/d)-1) * a(d).

A297382 Denominator of -A023900(n)/2.

Original entry on oeis.org

2, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Mats Granvik, Dec 29 2017

Also denominator of A173557(n)/2. a(n) = 2 iff n is a power of 2, 1 otherwise. - Antti Karttunen, Sep 30 2018

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

Cf. A297381 (numerators).
Cf. A023900, A173557, A014963.
One more than A209229.

Clear[n, s, nn]; nn = 64; Denominator[Table[Limit[Zeta[s]*Total[MoebiusMu[Divisors[n]]/Divisors[n]^(s - 1)], s -> 0], {n, 1, nn}]]

A297382(n) = denominator(-(1/2)*factorback(apply(p -> 1-p, factor(n)[, 1]))); \\ Antti Karttunen, Sep 30 2018

a(n) = denominator of -A023900(n)/2.
a(n) = 1 + A209229(n). - Antti Karttunen, Sep 30 2018
a(n) = A014963(2*n). - Ridouane Oudra, Jul 03 2025

More terms from Antti Karttunen, Sep 30 2018

A328583 a(n) = A023900(A276086(n)).

Original entry on oeis.org

1, -1, -2, 2, -2, 2, -4, 4, 8, -8, 8, -8, -4, 4, 8, -8, 8, -8, -4, 4, 8, -8, 8, -8, -4, 4, 8, -8, 8, -8, -6, 6, 12, -12, 12, -12, 24, -24, -48, 48, -48, 48, 24, -24, -48, 48, -48, 48, 24, -24, -48, 48, -48, 48, 24, -24, -48, 48, -48, 48, -6, 6, 12, -12, 12, -12, 24, -24, -48, 48, -48, 48, 24, -24, -48, 48, -48, 48, 24, -24

Offset: 0

Antti Karttunen, Oct 21 2019

sign

Antti Karttunen, Table of n, a(n) for n = 0..32768
Index entries for sequences related to primorial base

Cf. A023900, A276086, A324650, A328572.

A328583(n) = { my(m=1, p=2); while(n, m *= (1-p)^!!(n%p); n = n\p; p = nextprime(1+p)); (m); };

a(n) = A023900(A276086(n)).

abs(a(n)) = A324650(n) / A328572(n).

A366450 a(n) = Sum_{k=1..n} A366362(n,k)*A023900(k)/n.

Original entry on oeis.org

1, -2, -1, -4, 1, 2, -2, -8, -3, -2, 1, 4, 4, 4, -1, -16, -2, 6, 0, -4, 2, -2, -1, 8, 5, -8, -9, 8, 0, 2, 7, -32, -1, 4, -2, 12, 3, 0, -4, -8, -8, -4, -6, -4, -3, 2, 8, 16, -14, -10, 2, -16, -6, 18, 1, 16, 0, 0, 5, 4, 12, -14, 6, -64, 4, 2, -7, 8, 1, 4, -3, 24, 4, -6, -5, 0, -2, 8, -10, -16, -27, 16, -6, -8

Offset: 1

Mats Granvik, Oct 10 2023

sign

It appears that: a(A005117(n)) = A006571(A005117(n)), verified up to n = 98. But also a(76) = A006571(76), a(116) = A006571(116) and a(152) = A006571(152). 76 = 19*2^2, 116 = 29*2^2 and 152 = 19*2^3.

Cf. A366362, A023900, A006571, A005117, A002070, A366362.

nn = 84; f = x^3 - x^2 - y^2 - y; g[n_] := DivisorSum[n, MoebiusMu[#] # &]; Monitor[Table[Sum[Sum[Sum[If[GCD[f, n] == k, 1, 0]*g[k]/n, {x, 1, n}], {y, 1, n}], {k, 1, n}], {n, 1, nn}], n]

a(n) = sum(k=1, n, my(z=sumdivmult(k, d, d*moebius(d))); sum(y=1, n, sum(x=1, n, if (gcd(x^3 - x^2 - y^2 - y, n)==k, z/n)))); \\ Michel Marcus, Oct 10 2023

a(n) = Sum_{k=1..n} A366362(n,k)*A023900(k)/n.

cp's OEIS Frontend

A301590 Primes p such that there are no other solutions to A023900(x) = A023900(p) than a power of p.

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A306146 Numbers k such that A000010(A023900(k)) = A023900(A000010(k)).

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A323404 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)].

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A297381 Numerator of -A023900(n)/2.

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A301374 Values of A023900 which occur only at indices which are powers of a prime.

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A331178 Number of values of k, 1 <= k <= n, with A023900(k) = A023900(n), where A023900 is Dirichlet inverse of Euler totient function phi.

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A340198 Dirichlet inverse of sequence f(n) = A319340(n)-1 = (A000010(n) + A023900(n) - 1), where A000010 is Euler Totient function phi, and A023900 is its Dirichlet inverse.

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A297382 Denominator of -A023900(n)/2.

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