A173557 -id:A173557 - cp's OEIS frontend

A387158 Numbers k such that A173557(k) = A173557(sigma(k)), where A173557(n) is multiplicative with a(p^e) = p-1 and sigma is the sum of divisors function.

1, 6, 26, 28, 63, 74, 120, 122, 135, 146, 270, 314, 351, 386, 416, 496, 520, 554, 626, 672, 794, 842, 875, 891, 914, 999, 1080, 1082, 1226, 1232, 1322, 1346, 1404, 1466, 1480, 1514, 1638, 1647, 1750, 1754, 1782, 1859, 1971, 1994, 2186, 2306, 2402, 2426, 2440, 2474, 2642, 2762, 2906, 2920, 3242, 3314, 3506, 3718

Offset: 1

Antti Karttunen, Aug 19 2025

Numbers k for which A173557(k) == A387157(k).

Cf. A000203, A173557, A387157.
Subsequences: A000396, A387159 (odd terms).
Cf. also A006872, A351446, A386424.

A387158Q[k_] := #[k] == #[DivisorSigma[1, k]] & [Times @@ (FactorInteger[#][[All, 1]] - 1) &];
Select[Range[10000], A387158Q] (* Paolo Xausa, Aug 20 2025 *)

A173557(n) = factorback(apply(p -> p-1,factor(n)[,1]));
is_A387158(n) = (A173557(sigma(n))==A173557(n));

A296087 Numbers n such that there is k < n for which A003557(k) = A003557(n), A048250(k) = A048250(n) and A173557(k) = A173557(n).

Original entry on oeis.org

15265, 27962, 30217, 30530, 45795, 50541, 54379, 54905, 57598, 60434, 61060, 64255, 66526, 72357, 72713, 89585, 90651, 91590, 101082, 101949, 108758, 109810, 120868, 122120, 128510, 136555, 137385, 137883, 138761, 144714, 145426, 149739, 151085, 152633, 161386, 163137, 164715, 166315, 179170, 181302, 181543, 182942

Offset: 1

Antti Karttunen, Dec 08 2017

nonn

Because Euler phi(n) = A000010(n) = A003557(n) * A173557(n), Dedekind psi(n) = A001615(n) = A003557(n) * A048250(n), and because also sigma(n) (A000203) can be computed from those three elements (see A291750), these numbers form also a subset of the positions of such duplicated occurrences of values computed for those functions. See for example A069822 and A296214.

a(11) = 61060 is the first term that is not squarefree.

			15265 is a term because A003557(15265) = 1 = A003557(15169), A048250(15265) = 19008 = A048250(15169), A173557(15265) = 11760 = A173557(15169).
27962 is a term because A003557(27962) = 1 = A003557(26355), A048250(27962) = 48384 = A048250(26355), A173557(27962) = 12000 = A173557(26355).

Antti Karttunen, Table of n, a(n) for n = 1..2876

Cf. A000010, A001615, A003557, A048250, A173557.

Subsequence of A069822 and of A296214.

search_up_to = (2^23);
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); };
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ This function from Michel Marcus, Oct 31 2017
Anotsubmitted1(n) = (1/2)*(2 + ((A003557(n)+A173557(n))^2) - A003557(n) - 3*A173557(n));
Akaikki3(n) = (1/2)*(2 + ((A048250(n)+Anotsubmitted1(n))^2) - A048250(n) - 3*Anotsubmitted1(n));
om = Map(); m = 0; i=0; for(n = 1, search_up_to, k = Akaikki3(n); if(!mapisdefined(om,k), mapput(om,k,n), i++; write("b296087.txt", i, " ", n)));

A318304 a(n) = A083254(n)/A003557(n) = (2*A173557(n) - A007947(n)).

Original entry on oeis.org

1, 0, 1, 0, 3, -2, 5, 0, 1, -2, 9, -2, 11, -2, 1, 0, 15, -2, 17, -2, 3, -2, 21, -2, 3, -2, 1, -2, 27, -14, 29, 0, 7, -2, 13, -2, 35, -2, 9, -2, 39, -18, 41, -2, 1, -2, 45, -2, 5, -2, 13, -2, 51, -2, 25, -2, 15, -2, 57, -14, 59, -2, 3, 0, 31, -26, 65, -2, 19, -22, 69, -2, 71, -2, 1, -2, 43, -30, 77, -2, 1, -2, 81, -18, 43, -2, 25

Offset: 1

Antti Karttunen, Aug 26 2018

sign

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

Cf. A000010, A003557, A007947, A083254, A173557, A318305.

Cf. A065463, A307868.

A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ From A173557
A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A318304(n) = (2*A173557(n) - A007947(n));

A083254(n) = (2*eulerphi(n)-n);
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A318304(n) = (A083254(n)/A003557(n));

a(n) = A083254(n)/A003557(n) = 2*A173557(n) - A007947(n).
a(n) = A173557(n) - A318305(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 2 * A307868 - A065463 = 0.238919... . - Amiram Eldar, Dec 07 2023

A318317 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A173557.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 3, 5, 1, 1, 5, 3, 6, 3, 2, 35, 8, 1, 9, 3, 3, 5, 11, 5, 0, 3, 1, 9, 14, 1, 15, 63, 5, 4, 6, 3, 18, 9, 6, 5, 20, 3, 21, 15, 1, 11, 23, 35, -3, 0, 8, 9, 26, 1, 10, 15, 9, 7, 29, 3, 30, 15, 3, 231, 12, 5, 33, 3, 11, 3, 35, 5, 36, 9, 0, 27, 15, 3, 39, 35, 3, 10, 41, 9, 16, 21, 14, 25, 44, 1, 18, 33, 15, 23, 18, 63, 48, -3, 5, 0, 50, 4, 51, 15, 6

Offset: 1

Antti Karttunen, Aug 24 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A173557, A318318 (denominators).

Cf. also A317925, A317935.

f[1] = 1; f[n_] := f[n] = 1/2 (Module[{fac = FactorInteger[n]}, If[n == 1, 1, Product[fac[[i, 1]] - 1, {i, Length[fac]}]]] - Sum[f[d]*f[n/d], {d, Divisors[n][[2 ;; -2]]}]); Table[Numerator[f[n]], {n, 1, 100}] (* Vaclav Kotesovec, May 10 2025 *)

up_to = 16384;
A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ From A173557
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA317937.
v318317_18 = DirSqrt(vector(up_to, n, A173557(n)));
A318317(n) = numerator(v318317_18[n]);

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A173557(n) - Sum_{d|n, d>1, d 1.
From Vaclav Kotesovec, May 10 2025: (Start)
Let f(s) = Product_{p prime} (1 - 2/(p + p^s)).
Sum_{k=1..n} A318317(k) / A318318(k) ~ n^2 * sqrt(f(2)/(4*Pi*log(n))) * (1 + (1 - gamma - f'(2)/f(2) + 6*zeta'(2)/Pi^2) / (4*log(n))), where
f(2) = A307868 = Product_{p prime} (1 - 2/(p*(p+1))) = 0.471680613612997868...
f'(2)/f(2) = Sum_{p prime} 2*p*log(p) / ((p+1)*(p^2+p-2)) = 0.7254208328519472161058521308839896283514823... and gamma is the Euler-Mascheroni constant A001620. (End)

A319341 a(n) = A000010(n) - A173557(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 3, 4, 0, 0, 2, 0, 0, 0, 7, 0, 4, 0, 4, 0, 0, 0, 6, 16, 0, 16, 6, 0, 0, 0, 15, 0, 0, 0, 10, 0, 0, 0, 12, 0, 0, 0, 10, 16, 0, 0, 14, 36, 16, 0, 12, 0, 16, 0, 18, 0, 0, 0, 8, 0, 0, 24, 31, 0, 0, 0, 16, 0, 0, 0, 22, 0, 0, 32, 18, 0, 0, 0, 28, 52, 0, 0, 12, 0, 0, 0, 30, 0, 16, 0, 22, 0, 0, 0, 30, 0, 36, 40, 36, 0, 0, 0, 36, 0

Offset: 1

Antti Karttunen, Sep 17 2018

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

Cf. A000010, A051953, A173557, A318841, A319340, A126864.

Cf. A013661, A307868.

a[n_] := EulerPhi[n] - Times @@ (FactorInteger[n][[;;, 1]] - 1); a[1] = 0; Array[a, 100] (* Amiram Eldar, Dec 21 2023 *)

A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
A319341(n) = (eulerphi(n)-A173557(n));

a(n) = A000010(n) - A173557(n).
a(n) = A318841(n) - A051953(n).
a(A005117(n)) = 0. - Ivan N. Ianakiev, Sep 18 2018
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A059956 - A307868 = 0.136246... . - Amiram Eldar, Dec 21 2023

A323366 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => A000035(i) = A000035(j) and A003557(i) = A003557(j) and A173557(i) = A173557(j).

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, 13, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 34, 38, 39, 25, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 39, 52, 36, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 56, 65, 72, 73, 74, 75, 76, 49, 77, 78, 79, 80, 81, 82, 68, 83, 56, 84, 68, 85, 86, 87, 88, 89, 90, 91, 92, 93, 60

Offset: 1

Antti Karttunen, Jan 12 2019

nonn

For all i, j:
  a(i) = a(j) => A007814(i) = A007814(j),
  a(i) = a(j) => A295887(i) = A295887(j),
  a(i) = a(j) => A323237(i) = A323237(j).

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

Cf. A000035, A003557, A007814, A173557, A291756, A295887, A323237, A323367.

Cf. also A323368.

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] = f[i, 2]-1); factorback(f); };
A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ From A173557
v323366 = rgs_transform(vector(up_to, n, [(n%2), A003557(n), A173557(n)]));
A323366(n) = v323366[n];

A333871 Sum of the iterated absolute Möbius divisor function (A173557).

Original entry on oeis.org

0, 1, 3, 1, 5, 3, 9, 1, 3, 5, 15, 3, 15, 9, 9, 1, 17, 3, 21, 5, 15, 15, 37, 3, 5, 15, 3, 9, 37, 9, 39, 1, 25, 17, 27, 3, 39, 21, 27, 5, 45, 15, 57, 15, 9, 37, 83, 3, 9, 5, 33, 15, 67, 3, 45, 9, 39, 37, 95, 9, 69, 39, 15, 1, 51, 25, 91, 17, 59, 27, 97, 3, 75, 39

Offset: 1

Amiram Eldar, Apr 08 2020

nonn

Analogous to A092693 with the absolute Möbius divisor function (A173557) instead of the Euler totient function phi (A000010).

			a(3) = A173557(3) + A173557(A173557(3)) = 2 + 1 = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Daeyeoul Kim, Umit Sarp, and Sebahattin Ikikardes, Iterating the Sum of Möbius Divisor Function and Euler Totient Function, Mathematics, Vol. 7, No. 11 (2019), pp. 1083-1094.

Cf. A092693, A173557, A333870.

f[p_, e_] := p - 1; u[1] = 1; u[n_] := Times @@ (f @@@ FactorInteger[n]); a[n_] := Plus @@ FixedPointList[u, n] - n - 1 ; Array[a, 100]

A344994 Numbers k such that A173557(k) divides nonzero A051709(k).

Original entry on oeis.org

4, 6, 8, 12, 16, 24, 27, 28, 32, 42, 48, 54, 60, 64, 96, 108, 112, 120, 126, 128, 150, 168, 176, 192, 204, 216, 240, 243, 250, 256, 294, 312, 378, 384, 396, 432, 440, 448, 456, 460, 480, 486, 496, 500, 504, 512, 540, 588, 672, 700, 768, 774, 828, 840, 864, 888, 924, 960, 972, 1000, 1014, 1024, 1080, 1134, 1176, 1216

Offset: 1

Antti Karttunen, Jun 05 2021

nonn

Cf. A051709, A173557.
Cf. A344995, A345054 (subsequences).
Cf. also A344754, A345051.

A051709(n) = ((sigma(n) + eulerphi(n)) - (2*n));
A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
isA344994(n) = { my(u=A051709(n)); ((u>0)&&(0==(u%A173557(n)))); };

A344997 a(n) = A173557(n) * A344753(n).

Original entry on oeis.org

0, 2, 4, 5, 8, 24, 12, 11, 14, 64, 20, 56, 24, 120, 144, 23, 32, 78, 36, 152, 264, 280, 44, 120, 44, 384, 44, 288, 56, 672, 60, 47, 600, 640, 624, 182, 72, 792, 816, 328, 80, 1296, 84, 680, 480, 1144, 92, 248, 90, 332, 1344, 936, 104, 240, 1360, 624, 1656, 1792, 116, 1536, 120, 2040, 888, 95, 1824, 3120, 132, 1568, 2376

Offset: 1

Antti Karttunen, Jun 05 2021

nonn

Cf. A008966, A173557, A344753.
Cf. also A344996.
Cf. A002117, A013661, A307868.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); a[1] = 0; a[n_] := Module[{fct = FactorInteger[n], p}, p = fct[[;; , 1]]; Times @@ (p - 1)*(Times @@ f @@@ fct + n*Times @@ (1 + 1/p) - 2*n)]; Array[a, 100] (* Amiram Eldar, Dec 08 2023 *)

A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
A344753(n) = sumdiv(n,d,(dA344997(n) = (A173557(n)*A344753(n));

a(n) = A173557(n) * A344753(n).
a(n) = Product(p_i - 1) * [Sum_{d|n, dA008966(n/d) * d)], where p_i are distinct primes dividing n.
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = 1/zeta(2) - 2 * A307868 + zeta(2)*zeta(3) * Product_{p prime} (1 - 2/p^2 - 1/p^3 + 1/p^4 + 3/p^5 - 2/p^6) = 0.283799589272... . - Amiram Eldar, Dec 08 2023

A295877 Restricted growth sequence transform of A173557, Product_{p|n} (p-1).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 2, 3, 5, 2, 6, 4, 7, 1, 8, 2, 9, 3, 6, 5, 10, 2, 3, 6, 2, 4, 11, 7, 12, 1, 13, 8, 14, 2, 15, 9, 14, 3, 16, 6, 17, 5, 7, 10, 18, 2, 4, 3, 19, 6, 20, 2, 16, 4, 15, 11, 21, 7, 22, 12, 6, 1, 23, 13, 24, 8, 25, 14, 26, 2, 27, 15, 7, 9, 22, 14, 28, 3, 2, 16, 29, 6, 30, 17, 31, 5, 32, 7, 27, 10, 22, 18, 27, 2, 33, 4, 13, 3, 34, 19, 35, 6, 23

Offset: 1

Antti Karttunen, Nov 29 2017

nonn

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

Cf. A173557, A295876.

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])); }
A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ This function from Michel Marcus, Oct 31 2017
write_to_bfile(1,rgs_transform(vector(up_to,n,A173557(n))),"b295877.txt");

a(2n) = a(n).

cp's OEIS Frontend

A387158 Numbers k such that A173557(k) = A173557(sigma(k)), where A173557(n) is multiplicative with a(p^e) = p-1 and sigma is the sum of divisors function.

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A296087 Numbers n such that there is k < n for which A003557(k) = A003557(n), A048250(k) = A048250(n) and A173557(k) = A173557(n).

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A318304 a(n) = A083254(n)/A003557(n) = (2*A173557(n) - A007947(n)).

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A318317 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A173557.

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A319341 a(n) = A000010(n) - A173557(n).

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A323366 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => A000035(i) = A000035(j) and A003557(i) = A003557(j) and A173557(i) = A173557(j).

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A333871 Sum of the iterated absolute Möbius divisor function (A173557).

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A344994 Numbers k such that A173557(k) divides nonzero A051709(k).

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A344997 a(n) = A173557(n) * A344753(n).

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