A173557 -id:A173557 - cp's OEIS frontend

A318318 Denominators of rational valued sequence whose Dirichlet convolution with itself yields A173557.

1, 2, 1, 8, 1, 2, 1, 16, 2, 1, 1, 8, 1, 2, 1, 128, 1, 4, 1, 4, 1, 2, 1, 16, 1, 1, 2, 8, 1, 1, 1, 256, 1, 1, 1, 16, 1, 2, 1, 8, 1, 2, 1, 8, 1, 2, 1, 128, 2, 1, 1, 4, 1, 4, 1, 16, 1, 1, 1, 4, 1, 2, 2, 1024, 1, 2, 1, 1, 1, 1, 1, 32, 1, 1, 1, 8, 1, 1, 1, 64, 8, 1, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 256, 1, 4, 2, 1, 1, 1, 1, 8, 1

Offset: 1

Antti Karttunen, Aug 24 2018

Not multiplicative because A318317 contains zeros.
Differs from A317926 at n = 200, 400, 600, 800, 900, 1200, 1400, 1600, 1800, 2200, 2400, 2700, 2800, 3200, 3600, 3800, 4050, 4200, 4400, 4600, 4800, 4900, 5200, ..., which seem to be a subsequence of positions of zeros in A318317.
Here a(200) = 1, while A317926(200) = 2.

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

Cf. A173557, A318317 (numerators).

Cf. also A317926.

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[Denominator[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(nA173557)); 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)));
A318318(n) = denominator(v318317_18[n]);

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A173557(n) - Sum_{d|n, d>1, d 1.

A319993 a(n) = A319997(n) / A173557(n).

Original entry on oeis.org

1, -1, 1, 0, 1, -1, 1, 0, 3, -1, 1, 0, 1, -1, 1, 0, 1, -3, 1, 0, 1, -1, 1, 0, 5, -1, 9, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 3, -1, 1, 0, 7, -5, 1, 0, 1, -9, 1, 0, 1, -1, 1, 0, 1, -1, 3, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 5, 0, 1, -1, 1, 0, 27, -1, 1, 0, 1, -1, 1, 0, 1, -3, 1, 0, 1, -1, 1, 0, 1, -7, 3, 0, 1, -1, 1, 0, 1

Offset: 1

Antti Karttunen, Nov 08 2018

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

Cf. A003557, A173557, A319997, A319999.

A319993(n) =  { my(f=factor(n)); prod(i=1,#f~,if(2==f[i,1],-(1==f[i,2]),(f[i,1]^(f[i,2]-1)))); };

A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
A319997(n) = sumdiv(n,d,(d%2)*moebius(n/d)*d);
A319993(n) = (A319997(n)/A173557(n));

Multiplicative with a(2^1) = -1, a(2^e) = 0 for e > 1, and a(p^e) = p^(e-1) when p is an odd prime.
a(n) = A319997(n) / A173557(n).
a(2n) = A003557(2n) - 2*A003557(n), a(2n+1) = A003557(2n+1).

A319999 Filter sequence combining A173557(n) with A319993(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 08 2018

nonn

Restricted growth sequence transform of ordered pair [A173557(n), A319993(n)].

For all i, j: a(i) = a(j) => A319997(i) = A319997(j).

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

Cf. A173557, A319993, A319997.

Cf. also A295886, A295887.

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; };
A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
A319993(n) =  { my(f=factor(n)); prod(i=1,#f~,if(2==f[i,1],-(1==f[i,2]),(f[i,1]^(f[i,2]-1)))); };
v319999 = rgs_transform(vector(up_to,n,[A173557(n),A319993(n)]));
A319999(n) = v319999[n];

A331179 Number of values of k, 1 <= k <= n, with A173557(k) = A173557(n), where A173557(n) = Product_{p-1 | p is prime and divisor of n}.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 1, 5, 1, 5, 1, 3, 2, 2, 1, 6, 4, 3, 7, 3, 1, 2, 1, 6, 1, 2, 1, 8, 1, 2, 2, 5, 1, 4, 1, 3, 3, 2, 1, 9, 4, 6, 1, 5, 1, 10, 2, 5, 2, 2, 1, 4, 1, 2, 6, 7, 1, 2, 1, 3, 1, 3, 1, 11, 1, 3, 5, 3, 2, 4, 1, 7, 12, 3, 1, 7, 1, 2, 1, 4, 1, 6, 2, 3, 3, 2, 3, 13, 1, 6, 3, 8, 1, 2, 1, 8, 2

Offset: 1

Antti Karttunen, Jan 11 2020

nonn

Ordinal transform of A173557.

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

Cf. A173557.

Cf. also A081373, A331175, A331178.

A173557[n_] := If[n == 1, 1, Times @@ (FactorInteger[n][[All, 1]] - 1)];
Module[{b}, b[_] = 0;
a[n_] := With[{t = A173557[n]}, b[t] = b[t] + 1]];
Array[a, 105] (* Jean-François Alcover, Jan 12 2022 *)

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; };
A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
v331179 = ordinal_transform(vector(up_to, n, A173557(n)));
A331179(n) = v331179[n];

A333870 The number of iterations of the absolute Möbius divisor function (A173557) required to reach from n to 1.

Original entry on oeis.org

0, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 2, 1, 2, 2, 3, 2, 3, 3, 4, 2, 2, 3, 2, 3, 4, 2, 3, 1, 3, 2, 3, 2, 3, 3, 3, 2, 3, 3, 4, 3, 2, 4, 5, 2, 3, 2, 2, 3, 4, 2, 3, 3, 3, 4, 5, 2, 3, 3, 3, 1, 3, 3, 4, 2, 4, 3, 4, 2, 3, 3, 2, 3, 3, 3, 4, 2, 2, 3, 4, 3, 2, 4, 4

Offset: 1

Amiram Eldar, Apr 08 2020

nonn

Apparently, the least number that reaches 1 after k iterations is A082449(k-1) (checked numerically for 1 <= k <= 17).

			a(3) = 2 since there are 2 iterations from 3 to 1: A173557(3) = 2 and A173557(2) = 1.

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. A003434, A082449, A173557.

f[p_, e_] := p - 1; u[1] = 1; u[n_] := Times @@ (f @@@ FactorInteger[n]);a[n_] := Length @ FixedPointList[u, n] - 2; Array[a, 100]

A333874 Numbers k such that A173557(k) = A173557(k+1).

Original entry on oeis.org

1, 168, 194, 350, 1368, 1628, 3705, 5186, 5328, 6929, 7475, 25545, 26047, 26864, 28251, 34936, 37248, 56724, 65675, 81732, 82368, 87308, 87367, 88450, 91539, 132308, 164691, 166624, 244215, 265524, 280818, 281897, 388245, 465651, 501024, 577524, 806895, 859901

Offset: 1

Amiram Eldar, Apr 08 2020

nonn

Kim et al. (2019) conjectured that A173557(k) = A173557(k+1) is divisible by 12 for all the terms k > 1.

			1 is a term since A173557(1) = A173557(2) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..712 (terms below 2*10^10)
Daeyeoul Kim, Umit Sarp, and Sebahattin Ikikardes, Certain combinatoric convolution sums arising from Bernoulli and Euler Polynomials, Miskolc Mathematical Notes, No. 20, Vol. 1 (2019): pp. 311-330.

Cf. A001274, A173557, A333875.

f[p_, e_] := p - 1; u[1] = 1; u[n_] := Times @@ (f @@@ FactorInteger[n]); s = {}; u1 = 1; Do[u2 = u[n]; If[u1 == u2, AppendTo[s, n-1]]; u1 = u2, {n, 2, 10^5}]; s

A344995 Numbers k such that A051709(k)/A173557(k) is a positive natural number and a divisor of k.

Original entry on oeis.org

4, 6, 12, 28, 42, 312, 456, 496, 588, 828, 1080, 1216, 1242, 1377, 1560, 4560, 5964, 7320, 7480, 7584, 8128, 11400, 13728, 14784, 23760, 33462, 59400, 59520, 124020, 147840, 188600, 277648, 321000, 543552, 1288224, 1510272, 1596048, 1964544, 2038140, 3323736, 3611520, 3780672, 3909816, 6137440, 9034032, 10783890

Offset: 1

Antti Karttunen, Jun 05 2021

nonn

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Cf. A051709, A173557.
Subsequence of A344994.
Cf. also A344755, A345002.

A051709(n) = ((sigma(n) + eulerphi(n)) - (2*n));
A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
isA344995(n) = { my(u=A051709(n),t=A173557(n),r=u/t); ((u>0)&&(1==denominator(r)&&!(n%r))); };

A345052 a(n) = A003557(n) * A048250(n) * A173557(n).

Original entry on oeis.org

1, 3, 8, 6, 24, 24, 48, 12, 24, 72, 120, 48, 168, 144, 192, 24, 288, 72, 360, 144, 384, 360, 528, 96, 120, 504, 72, 288, 840, 576, 960, 48, 960, 864, 1152, 144, 1368, 1080, 1344, 288, 1680, 1152, 1848, 720, 576, 1584, 2208, 192, 336, 360, 2304, 1008, 2808, 216, 2880, 576, 2880, 2520, 3480, 1152, 3720, 2880, 1152, 96

Offset: 1

Antti Karttunen, Jun 08 2021

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

Cf. A000010, A003557, A007434, A048250, A065484, A173557, A185197.

f[p_, e_] := (p^2 - 1)*p^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 16 2022 *)

A345052(n) = { my(f=factor(n)); prod(i=1, #f~, ((f[i,1]^2)-1)*(f[i,1]^(f[i, 2]-1))); };

Multiplicative with a(p^e) = (p^2 - 1) * p^(e-1).
a(n) = A007434(n) / A003557(n) = A003557(n) * A048250(n) * A173557(n).
From Amiram Eldar, Oct 16 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^3, where c = 2/Pi^2 = 0.202642... (A185197).
Sum_{n>=1} 1/a(n) = A065484.
a(n) = A000010(n) * A048250(n). (End)

A345054 Odd numbers k such that A173557(k) divides nonzero A051709(k).

Original entry on oeis.org

27, 243, 1377, 2187, 3125, 19683, 28125, 55233, 68445, 177147, 195625, 203125, 239805, 253125, 453125, 823543, 907137, 1323297, 1378125, 1464561, 1594323, 1953125, 2278125, 3341637, 3572829, 5255361, 5877117, 9034497, 9819837, 11701053, 14348907, 17453125, 19460393, 20503125, 22209633, 26010621, 30074733, 44910045

Offset: 1

Antti Karttunen, Jun 06 2021

nonn

Question: Are there any common terms with A345051?

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Odd terms in A344994.

Cf. A051709, A173557, A345051.

A051709(n) = ((sigma(n) + eulerphi(n)) - (2*n));
A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
isA345054(n) = if(!(n%2),0,my(u=A051709(n)); ((u>0)&&(0==(u%A173557(n)))));

A387159 Odd 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.

Original entry on oeis.org

1, 63, 135, 351, 875, 891, 999, 1647, 1859, 1971, 4239, 5211, 7479, 8451, 10719, 11367, 12339, 14607, 16317, 16551, 17847, 18171, 19791, 20439, 22103, 23679, 26919, 27951, 29511, 31131, 31407, 31487, 32427, 32751, 33399, 35667, 37287, 39231, 43767, 44739, 47331, 50571, 52191, 53811, 54459, 57319, 57699, 63207, 66771

Offset: 1

Antti Karttunen, Aug 19 2025

nonn

Odd numbers k for which A173557(k) == A387157(k).

Odd terms of A387158.
Cf. A000203, A173557, A387157.
Cf. also A351443, A353679, A386425.

A387159Q[k_] := OddQ[k] && #[k] == #[DivisorSigma[1, k]] & [Times @@ (FactorInteger[#][[All, 1]] - 1) &];
Select[Range[100000], A387159Q] (* Paolo Xausa, Aug 20 2025 *)

A173557(n) = factorback(apply(p -> p-1,factor(n)[,1]));
is_A387159(n) = (n%2 && (A173557(sigma(n))==A173557(n)));

cp's OEIS Frontend

A318318 Denominators of rational valued sequence whose Dirichlet convolution with itself yields A173557.

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A319993 a(n) = A319997(n) / A173557(n).

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A319999 Filter sequence combining A173557(n) with A319993(n).

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A331179 Number of values of k, 1 <= k <= n, with A173557(k) = A173557(n), where A173557(n) = Product_{p-1 | p is prime and divisor of n}.

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A333870 The number of iterations of the absolute Möbius divisor function (A173557) required to reach from n to 1.

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A333874 Numbers k such that A173557(k) = A173557(k+1).

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A344995 Numbers k such that A051709(k)/A173557(k) is a positive natural number and a divisor of k.

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A345052 a(n) = A003557(n) * A048250(n) * A173557(n).

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A345054 Odd numbers k such that A173557(k) divides nonzero A051709(k).