A001615 -id:A001615 - cp's OEIS frontend

A308443 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^(psi(k)/k), where psi() is the Dedekind psi function (A001615).

1, 1, 5, 23, 173, 1249, 13249, 130255, 1670297, 21350177, 322709021, 4933457671, 87302545285, 1551234590593, 30934738239833, 630934308253439, 14035903893341489, 320008164205036225, 7885477719156600757, 198735099970790861047, 5352424525748204265821

Offset: 0

Ilya Gutkovskiy, May 27 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..438

Cf. A000262, A001615, A060648, A156303.

nmax = 20; CoefficientList[Series[Product[1/(1 - x^k)^(DirichletConvolve[j, MoebiusMu[j]^2, j, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Sum[2^PrimeNu[d]/d, {d, Divisors[k]}] k! Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 20}]

E.g.f.: exp(Sum_{k>=1} A060648(k)*x^k/k).

A330006 Numbers m such that psi(m) > psi(k) for all k < m, where psi is the Dedekind psi function (A001615).

Original entry on oeis.org

1, 2, 3, 4, 6, 10, 12, 18, 24, 30, 42, 54, 60, 78, 84, 90, 114, 120, 150, 168, 180, 210, 270, 294, 300, 330, 390, 420, 510, 546, 570, 630, 750, 780, 840, 990, 1050, 1170, 1260, 1470, 1650, 1680, 1890, 2100, 2310, 2730, 3150, 3360, 3570, 3990, 4290, 4620, 5250

Offset: 1

Amiram Eldar, Nov 26 2019

nonn

			The first 6 values of the psi function are 1, 3, 4, 6, 6, 12. The record values, 1, 3, 4, 6, 12 are at positions 1, 2, 3, 4, 6.

Amiram Eldar, Table of n, a(n) for n = 1..808 (terms below 10^11)

Cf. A001615, A210523 (record values).

psi[1] = 1; psi[n_] := n * Times @@ (1 + 1/FactorInteger[n][[;; , 1]]); seq = {}; pmax = 0; Do[p = psi[n]; If[p > pmax, pmax = p; AppendTo[seq, n]], {n, 1, 10^5}]; seq

A001615(a(n)) = A210523(n).

A333910 Numbers k such that psi(k) is the sum of 2 squares, where psi is the Dedekind psi function (A001615).

Original entry on oeis.org

1, 3, 7, 10, 17, 18, 19, 20, 21, 22, 27, 30, 31, 36, 40, 44, 45, 46, 50, 51, 55, 57, 58, 60, 66, 67, 70, 71, 72, 73, 79, 80, 88, 89, 92, 93, 94, 97, 99, 100, 103, 106, 115, 116, 118, 119, 120, 126, 127, 132, 133, 138, 140, 144, 145, 150, 154, 160, 162, 163, 165

Offset: 1

Amiram Eldar, Apr 09 2020

nonn

			1 is a term since psi(1) = 1 = 0^2 + 1^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
William D. Banks, Florian Luca, Filip Saidak, and Igor E. Shparlinski, Values of arithmetical functions equal to a sum of two squares, Quarterly Journal of Mathematics, Vol. 56, No. 2 (2005), pp. 123-139, alternative link.

Cf. A001481, A001615, A079545, A333909, A333911.

psi[1] = 1; psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); Select[Range[200], SquaresR[2, psi[#]] > 0 &]

from itertools import count, islice
from collections import Counter
from sympy import factorint
def A333910_gen(): # generator of terms
    return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in sum((Counter(factorint(1+p))+Counter({p:e-1}) for p ,e in factorint(n).items()),start=Counter()).items()),count(1))
A333910_list = list(islice(A333910_gen(),30)) # Chai Wah Wu, Jun 27 2022

c1 * x/log(x)^(3/2) < N(x) < c2 * x/log(x)^(3/2), where N(x) is the number of terms <= x, and c1 and c2 are two positive constants (Banks et al., 2005).

A342917 a(n) = A001615(n) / gcd(1+n, A001615(n)), where A001615 is Dedekind psi, n * Product_{p|n, p prime} (1 + 1/p).

Original entry on oeis.org

1, 1, 1, 6, 1, 12, 1, 4, 6, 18, 1, 24, 1, 8, 3, 24, 1, 36, 1, 12, 16, 36, 1, 48, 15, 14, 9, 48, 1, 72, 1, 16, 24, 54, 4, 72, 1, 20, 7, 72, 1, 96, 1, 8, 36, 72, 1, 96, 28, 30, 18, 84, 1, 108, 9, 32, 40, 90, 1, 144, 1, 32, 3, 96, 14, 144, 1, 36, 48, 144, 1, 144, 1, 38, 30, 120, 16, 168, 1, 16, 54, 126, 1, 192, 54, 44, 15, 144

Offset: 1

Antti Karttunen, Mar 29 2021

The scatter plot shows two distinct "fans" separated by a gap. Why?

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

Cf. A001615, A342915, A342916.

Cf. also A160595.

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A342917(n) = { my(u=A001615(n)); (u/gcd(1+n,u)); };

a(n) = A001615(n) / A342915(n) = A001615(n) / gcd(1+n, A001615(n)).

A344703 Numbers k for which sigma(k^2) and psi(k^2) share a factor, where sigma is the sum of divisors, A000203, and psi is the Dedekind psi function, A001615.

Original entry on oeis.org

14, 21, 26, 28, 35, 38, 39, 42, 52, 56, 57, 62, 63, 65, 70, 74, 76, 77, 78, 82, 84, 86, 93, 95, 98, 99, 104, 105, 111, 112, 114, 117, 119, 122, 124, 126, 129, 130, 133, 134, 140, 143, 146, 148, 152, 154, 155, 156, 158, 161, 166, 168, 171, 172, 175, 182, 183, 185, 186, 189, 190, 194, 195, 198, 201, 203, 206, 208, 209

Offset: 1

Antti Karttunen and Peter Munn, May 27 2021

nonn

Numbers k for which A344695(k^2) > 1.
It can be shown that sigma(m) and psi(m) share a factor if m is nonsquare. (See A344695 for more detail.) So here we consider only square numbers, m = k^2.
For prime p, sigma(p^2) and psi(p^2) are coprime, since sigma(p^2) = p^2 + p + 1 = psi(p^2) + 1. So all terms are composite. We can say more, since for prime p and positive integer e, psi(p^(2*e)) = p^(2*e-1) * (p+1), whereas sigma(p^(2*e)) can be shown to be congruent to 1 modulo p and to 1 modulo p+1, so shares no factors with p^(2*e-1) * (p+1). So all terms are divisible by more than one prime.
If k is in the sequence, m*k is also present for any positive integer m coprime to k.

			Sigma (A000203) and the Dedekind psi function (A001615) are both multiplicative, so we gain insight by showing the values of these functions using their multiplicative properties:-
sigma(14^2) = sigma(2^2) * sigma(7^2) = 7 * 57 = 7 * (3*19).
psi(14^2) = psi(2^2) * psi(7^2) = 6 * 56 = (2*3) * (2^3*7).
So sigma(14^2) and psi(14^2) share factors 3 and 7, so 14 is in the sequence.
Looking in particular at the shared factor 3, we see it is present in sigma(7^2) and psi(2^2). For prime p, sigma(p^2) and psi(p^2) equate to polynomials in p, so we deduce 3 divides sigma(p^2) for p congruent to 7 modulo 3, and divides psi(p^2) for p congruent to 2 modulo 3. From this we see all products of a prime of the form 3m+1 and a prime of the form 3m+2 are in the sequence; for instance (3*4+1) * (3*1+2) = 13 * 5 = 65.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000203, A000290, A001615, A344695.

Subsequences: A344872.

filter:= proc(k) local n,F, sig, psi, t;
   n:= k^2;
   F:= map(t -> [t[1],2*t[2]], ifactors(k)[2]);
   sig:= mul((t[1]^(1+t[2])-1)/(t[1]-1),t=F);
   psi:= n*mul(1+1/t[1],t=F);
   igcd(sig,psi) > 1
end proc:
select(filter, [$1..300]); # Robert Israel, Jan 06 2024

filter[k_] := Module[{n, F, sig, psi},
   n = k^2;
   F = {#[[1]], 2 #[[2]]}& /@ FactorInteger[k];
   sig = Product[(t[[1]]^(1 + t[[2]]) - 1)/(t[[1]] - 1), {t, F}];
   psi = n*Product[1 + 1/t[[1]], {t, F}];
   GCD[sig, psi] > 1];
Select[Range[300], filter] // Quiet (* Jean-François Alcover, May 23 2024, after Robert Israel *)

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A344695(n) = gcd(sigma(n), A001615(n));
isA344703(n) = (A344695(n^2)>1);

A344704 a(n) = gcd(A001615(n)-n, sigma(n)-(A001615(n)+n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 4, 1, 2, 3, 1, 1, 3, 1, 2, 1, 2, 1, 12, 1, 2, 1, 20, 1, 6, 1, 1, 3, 2, 1, 1, 1, 2, 1, 2, 1, 6, 1, 4, 3, 2, 1, 4, 1, 1, 3, 2, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 11, 1, 1, 6, 1, 10, 3, 2, 1, 3, 1, 2, 1, 4, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 3, 4, 1, 18, 7, 4, 1, 2, 5, 12, 1, 5, 3, 1, 1, 6, 1, 2, 3

Offset: 1

Antti Karttunen, May 28 2021

nonn

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

Cf. A000203, A001615, A244963, A306927, A344700, A344705.

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A344704(n) = gcd(A001615(n)-n, sigma(n)-(A001615(n)+n));

a(n) = gcd(A306927(n), n-A244963(n)) = gcd(A001615(n)-n, sigma(n)-(A001615(n)+n)).

A353868 Numbers k such that the Carmichael function A002322(k) divides Dedekind psi A001615(k).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 14, 15, 16, 18, 20, 24, 27, 28, 30, 32, 35, 36, 40, 42, 45, 48, 52, 54, 56, 60, 63, 64, 65, 70, 72, 75, 78, 80, 81, 84, 90, 96, 98, 100, 104, 105, 108, 112, 117, 119, 120, 126, 128, 130, 135, 140, 144, 150, 156, 160, 162, 168, 175, 180, 182, 189, 190, 192, 195, 196, 200, 204, 208, 210, 216

Offset: 1

Max Alekseyev, May 08 2022

If coprime s,t are terms, then so is s*t. Also, if t is a term and prime p|t, then p*t is also a term. Squarefree terms are listed in A353869, primitive terms are listed in A353870, and their intersection forms A353871.

user142929, A definition related to pseudoprimes and the Dedekind psi function, MathOverflow, 2022.

Cf. A002322, A001615, A353869, A353870, A353871.

psi[1] = 1; psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); Select[Range[216], Divisible[psi[#], CarmichaelLambda[#]] &] (* Amiram Eldar, May 09 2022 *)

A353869 Squarefree numbers k such that the Carmichael function A002322(k) divides the Dedekind psi A001615(k).

Original entry on oeis.org

1, 2, 3, 6, 14, 15, 30, 35, 42, 65, 70, 78, 105, 119, 130, 182, 190, 195, 210, 238, 255, 357, 370, 377, 390, 418, 455, 510, 546, 570, 595, 663, 714, 754, 910, 969, 1045, 1110, 1118, 1131, 1190, 1254, 1326, 1330, 1365, 1547, 1558, 1615, 1785, 1885, 1887, 1938, 2090, 2190, 2261, 2262, 2470, 2590, 2639, 2730

Offset: 1

Max Alekseyev, May 08 2022

If s,t are terms, then so is lcm(s,t); in particular, if s,t are coprime, then s*t is also a term. Primitive squarefree terms are listed in A353871.

Intersection of A005117 and A353868.

Cf. A002322, A001615, A353870, A353871.

psi[1] = 1; psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); Select[Range[3000], SquareFreeQ[#] && Divisible[psi[#], CarmichaelLambda[#]] &] (* Amiram Eldar, May 09 2022 *)

A355045 a(n) is the least positive integer k which is a multiple of prime(n) such that for some m >= 0, psi(k) = rad(k)^m, where psi(k) = A001615(k) and rad(k) = A007947(k).

Original entry on oeis.org

18, 18, 11250, 57177414, 8696754, 10763393803185114, 501126, 23816977256250, 23981814018, 230750426250, 3730545397766934, 33914855378546706844968750, 11135545745963323734, 234030019748505421122, 246218836545018, 5018345916

Offset: 1

Vladislav Shubin, Jun 16 2022

nonn

Vladislav Shubin, Table of n, a(n) for n = 1..1000

Cf. A001615, A007947.

DedekindPsi[n_] :=
  n * Product[(1 + 1/i), {i, FactorInteger[n][[All, 1]]}];
For[s = 2, s <= 49, s++,
   If[s == 1, Print["n   =   ", 18]; s = s + 1;];
   Q = 1*Prime[s];
InitialArray = FactorInteger[If[Q != 3, 3*(Q + 1), 2]];
For[i = 1, i <= Length[InitialArray] - 1, i++,
  CurrentArray =
   FactorInteger[InitialArray[[-i, 1]] + 1] ~Join~ InitialArray;
      InitialArray =
   FactorInteger[Product[CurrentArray[[k, 1]] ^ CurrentArray[[k, 2]], {k, 1,
      Length[CurrentArray]}]];
  ];
InitialArray = InitialArray ~Join~ {{Q, 0}};
m = Max[InitialArray[[All, 2]]];
n = Product[Power[InitialArray[[k, 1]], m - InitialArray[[k, 2]] + 1], {k, 1,
    Length[InitialArray]}];
If[Q ==  3, m = m + 1];
Print["n   =   " n]];

A357820 Numerators of the partial alternating sums of the reciprocals of the Dedekind psi function (A001615).

Original entry on oeis.org

1, 2, 11, 3, 11, 5, 23, 7, 23, 65, 71, 17, 64, 491, 64, 491, 173, 505, 2651, 2581, 10639, 1151, 3593, 3523, 727, 237, 2189, 2147, 11071, 10931, 5623, 2759, 5623, 16589, 2113, 8347, 162373, 159979, 20318, 160549, 163969, 649891, 7292441, 7204661, 7292441, 7204661

Offset: 1

Amiram Eldar, Oct 14 2022

			Fractions begin with 1, 2/3, 11/12, 3/4, 11/12, 5/6, 23/24, 7/8, 23/24, 65/72, 71/72, 17/18, ...

Olivier Bordellès and Benoit Cloitre, An alternating sum involving the reciprocal of certain multiplicative functions, Journal of Integer Sequences, Vol. 16 (2013), Article 13.6.3.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A001615, A173290, A357821 (denominators).
Cf. A001620, A065463, A335707.
Similar sequence: A211177.

psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); psi[1] = 1; Numerator[Accumulate[1/Array[(-1)^(# + 1)*psi[#] &, 50]]]

f(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615
a(n) = numerator(sum(k=1, n, (-1)^(k+1)/f(k))); \\ Michel Marcus, Oct 15 2022

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/psi(k)).

a(n)/A357821(n) ~ (C/5) * (log(n) + gamma + D + 24*log(2)/5) + O(log(n)^(2/3) * log(log(n))^(4/3) / n), where C = Product_{p prime} (1 - 1/(p*(p+1))) (A065463), and D = Sum_{p prime} log(p)/(p^2+p-1) (A335707) (Bordellès and Cloitre, 2013; Tóth, 2017).

cp's OEIS Frontend

A308443 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^(psi(k)/k), where psi() is the Dedekind psi function (A001615).

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A330006 Numbers m such that psi(m) > psi(k) for all k < m, where psi is the Dedekind psi function (A001615).

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A333910 Numbers k such that psi(k) is the sum of 2 squares, where psi is the Dedekind psi function (A001615).

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A342917 a(n) = A001615(n) / gcd(1+n, A001615(n)), where A001615 is Dedekind psi, n * Product_{p|n, p prime} (1 + 1/p).

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A344703 Numbers k for which sigma(k^2) and psi(k^2) share a factor, where sigma is the sum of divisors, A000203, and psi is the Dedekind psi function, A001615.

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A344704 a(n) = gcd(A001615(n)-n, sigma(n)-(A001615(n)+n)).

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A353868 Numbers k such that the Carmichael function A002322(k) divides Dedekind psi A001615(k).

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A353869 Squarefree numbers k such that the Carmichael function A002322(k) divides the Dedekind psi A001615(k).

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A355045 a(n) is the least positive integer k which is a multiple of prime(n) such that for some m >= 0, psi(k) = rad(k)^m, where psi(k) = A001615(k) and rad(k) = A007947(k).