A001615 -id:A001615 - cp's OEIS frontend

A301594 Expansion of Product_{k>=1} (1 + x^k)^A001615(k), where A001615 is the Dedekind psi function.

1, 1, 3, 7, 13, 27, 55, 99, 185, 341, 604, 1064, 1863, 3181, 5411, 9123, 15167, 25051, 41083, 66715, 107703, 172735, 275034, 435484, 685753, 1073481, 1672160, 2592070, 3998278, 6140196, 9389302, 14296376, 21682534, 32759202, 49308812, 73956692, 110545113

Offset: 0

Vaclav Kotesovec, Mar 24 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A001615, A156303.

nmax = 40; CoefficientList[Series[Exp[Sum[-(-1)^j * Sum[k*Sum[MoebiusMu[d]^2 / d, {d, Divisors @ k}] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 31 2018 *)

a(n) ~ exp(3^(5/3) * (5*Zeta(3))^(1/3) * n^(2/3) / (2^(4/3) * Pi^(2/3)) - Pi^(2/3) * n^(1/3) / (2^(5/3) * 3^(2/3) * (5*Zeta(3))^(1/3)) - Pi^2 / (2160 * Zeta(3))) * (5*Zeta(3))^(1/6) / (2^(3/4) * 3^(1/6) * Pi^(5/6) * n^(2/3)).

A323328 Lexicographically earliest unbounded aliquot-like sequence based on the Dedekind psi function: a(1) = 318, a(n) = t(a(n-1)) where t(k) = A001615(k) - k.

Original entry on oeis.org

318, 330, 534, 546, 798, 1122, 1470, 2562, 3390, 4818, 5838, 7602, 9870, 17778, 17790, 24978, 27438, 30882, 30894, 34386, 40782, 52530, 82254, 82266, 82278, 106074, 111654, 111690, 176022, 266346, 266382, 266490, 480006, 480330, 674406, 740826, 833814, 834138

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

318 is the least number k whose repeated iteration of the mapping k -> A001615(k) - k yields an unbounded sequence. Since t(m^j * n) = m^j * t(n) if m|n, then if in the sequence a_0 = k, a_1 = t(k), a_2 = t(t(k))... there is a term a_{i1} = m^j * a_0 such that m|k and j > 0 then a_{i+i1} = m^j * a_i for all i and thus the sequence is unbounded. Since a(13)=9870, after 19 iterations a(32) = 27 * 9870, 27 = 3^3 and 3|9870 then a(n+19) = 27 * a(n) for n >= 13.

Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., 2009, page 71, entry 318.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Kevin Brown and Charles Vanden Eynden, Pseudo-aliquot Sequences, Solution to Problem 10323, The American Mathematical Monthly, Volume 103, No. 8 (1996), pp. 697-698.
David E. Penney and Carl Pomerance, Problem 10323, The American Mathematical Monthly, Volume 100, No. 7 (1993), p. 688.

Cf. A001615, A008892, A323327.

t[1] = 0; t[n_] := (Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]) - 1)*n; NestList[t, 318, 40]

A323329 Lesser of amicable pair m < n defined by t(n) = m and t(m) = n where t(n) = psi(n) - n and psi(n) = A001615(n) is the Dedekind psi function.

Original entry on oeis.org

1330, 2660, 3850, 5320, 6650, 7700, 10640, 11270, 13300, 14950, 15400, 18550, 19250, 21280, 22540, 26600, 29900, 30800, 33250, 37100, 38500, 42560, 45080, 53200, 59800, 61600, 66500, 73370, 74200, 74750, 77000, 78890, 85120, 90160, 92750, 96250, 106400, 119600

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

t(n) = psi(n) - n is the sum of aliquot divisors of n, d, such that n/d is squarefree. Penney & Pomerance proposed a problem to show that the "pseudo-aliquot" sequence related to this function is unbounded. This sequence lists number with pseudo-aliquot sequence of cycle 2. The sequence that is analogous to perfect numbers is A033845.

The asymptotic density of the terms relative to the positive integers is zero. See Dimitrov link. - S. I. Dimitrov, Aug 06 2025

Amiram Eldar, Table of n, a(n) for n = 1..1000
Kevin Brown and Charles Vanden Eynden, Pseudo-aliquot Sequences, Solution to Problem 10323, The American Mathematical Monthly, Volume 103, No. 8 (1996), pp. 697-698.
S. I. Dimitrov, On psi-amicable numbers and their generalizations, arXiv:2508.02318 [math.NT], 2025. See p. 2.
David E. Penney and Carl Pomerance, Problem 10323, The American Mathematical Monthly, Volume 100, No. 7 (1993), p. 688.

Cf. A001615, A002025, A033845 (Dedekind psi perfect numbers), A323327, A323328, A323330.

psi[n_] := n*Times@@(1+1/Transpose[FactorInteger[n]][[1]]); t[n_]:= psi[n] - n; s={}; Do[n=t[m]; If[n>m && t[n]==m, AppendTo[s,m]], {m, 1, 120000}]; s

A327251 Expansion of Sum_{k>=1} psi(k) * x^k / (1 - x^k)^2, where psi = A001615.

Original entry on oeis.org

1, 5, 7, 16, 11, 35, 15, 44, 33, 55, 23, 112, 27, 75, 77, 112, 35, 165, 39, 176, 105, 115, 47, 308, 85, 135, 135, 240, 59, 385, 63, 272, 161, 175, 165, 528, 75, 195, 189, 484, 83, 525, 87, 368, 363, 235, 95, 784, 161, 425, 245, 432, 107, 675, 253, 660, 273

Offset: 1

Ilya Gutkovskiy, Sep 15 2019

Inverse Moebius transform of A322577.

Dirichlet convolution of A001615 with A000027.

Cf. A000027, A001615, A018804, A060648, A322577, A347127.

nmax = 57; CoefficientList[Series[Sum[DirichletConvolve[j, MoebiusMu[j]^2, j, k] x^k/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
f[p_, e_] := p^(e - 1)*((p + 1)*e + p); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 24 2021 *)

mypsi(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615
a(n) = sumdiv(n, d, mypsi(n/d)*d); \\ Michel Marcus, Sep 15 2019

a(n) = Sum_{d|n} psi(n/d) * d.
a(p) = 2*p + 1, where p is prime.
Multiplicative with a(p^e) = p^(e-1)*((p+1)*e + p). - Antti Karttunen, Aug 24 2021

A344700 Numbers k for which A306927(k) [= A001615(k)-k] is a multiple of A344705(k) [= A001615(k)-A001065(k)], and their quotient is nonnegative.

Original entry on oeis.org

1, 6, 24, 28, 168, 496, 864, 1080, 1520, 1836, 2016, 2088, 2112, 2520, 2912, 2976, 3000, 3024, 3240, 3800, 8128, 9000, 11088, 11232, 11448, 14160, 14688, 16920, 17028, 18360, 19872, 20520, 20880, 25280, 25488, 27552, 29376, 30800, 31200, 31320, 31968, 35400, 39240, 44064, 48768, 49896, 50760, 51480, 51660, 52200, 55680

Offset: 1

Antti Karttunen, May 28 2021

nonn

Numbers k for which A344704(k) = A344705(k), i.e., numbers k such that gcd(A001615(k)-k, A001615(k)-A001065(k)) = A001615(k) - A001065(k).
Note that A306927(k) is always nonnegative, but A344705(k) = A033879(k) + A306927(k) gets also negative values. Number k is perfect only when A033879(k) = A344705(k) - A306927(k) = 0, that is, when A344705(k) = A306927(k), which necessitates that A306927(k) should be a multiple of A344705(k), and their quotient should be nonnegative (actually = +1).
In the range 1 .. 2^31 there are 782 such numbers, of which only the initial 1 is odd.

Cf. A000203, A001065, A001615, A033879, A244963, A306927, A344704, A344705, A344752 (gives the quotient A306927(k)/A344705(k) computed for these terms), A344753.
Cf. A000396 (subsequence).
Cf. also A344754, A344755.

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
isA344700(n) = { my(t=A001615(n), s=sigma(n), u = (n+t)-s); (gcd(t-n,u)==u); };
\\ Alternatively as:
isA344700(n) = { my(t=A001615(n), s=sigma(n), u = (n+t)-s); ((u>0)&&(0==((t-n)%u))); };

A203444 Numbers in range of Dedekind Psi function: A001615.

Original entry on oeis.org

1, 3, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 36, 38, 42, 44, 48, 54, 56, 60, 62, 68, 72, 74, 80, 84, 90, 96, 98, 102, 104, 108, 110, 112, 114, 120, 126, 128, 132, 138, 140, 144, 150, 152, 158, 160, 162, 164, 168, 174, 176, 180, 182, 186, 192, 194, 198, 200

Offset: 1

Enrique Pérez Herrero, Jan 02 2012

nonn

a(n) is even for n>2

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
Patrick Sole and Michel Planat, Extreme values of the Dedekind Psi function, J. Comb. Number Theory 3 (2011), no. 1, 33-38, arXiv:1011.1825v2.

Cf. A001615, A175836, A060648.
Cf. A007617.
Cf. A002202.

terms = 100; Clear[seq]; seq[k_] := seq[k] = Table[DirichletConvolve[j, MoebiusMu[j]^2, j, n], {n, 1, k terms}] // Union // PadRight[#, terms]&;
seq[k = 1]; seq[k++]; While[Print[k]; seq[k] != seq[k-1], k++];
seq[k] (* Jean-François Alcover, Dec 14 2018, after Jan Mangaldan in A001615 *)

A291167 Numbers k such that psi(k) is a perfect square where psi(k) = A001615(k).

Original entry on oeis.org

1, 3, 18, 20, 22, 27, 60, 66, 70, 72, 80, 88, 92, 94, 99, 115, 119, 162, 170, 210, 212, 214, 217, 240, 243, 252, 264, 265, 276, 280, 282, 288, 308, 310, 315, 320, 322, 345, 352, 357, 368, 376, 382, 385, 423, 497, 500, 510, 517, 527, 540, 594, 596, 612, 636, 637, 642, 648, 651, 679, 680, 710, 725, 742

Offset: 1

Altug Alkan, Aug 19 2017

The product of an even number of distinct members of A066436 is in the sequence. - Robert Israel, Aug 22 2017

			60 is a term because psi(60) = 144 is a perfect square.

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

Cf. A001615, A006532, A066436.

filter:= proc(n) issqr(n*mul(1+1/p,p=numtheory:-factorset(n))) end proc:
select(filter, [$1..1000]); # Robert Israel, Aug 22 2017

Select[Range@ 750, IntegerQ@ Sqrt[# Sum[MoebiusMu[d]^2/d, {d, Divisors@ #}]] &] (* Michael De Vlieger, Aug 19 2017 *)

a001615(n) = n*sumdivmult(n, d, issquarefree(d)/d);
is(n) = issquare(a001615(n));

A323330 Larger of amicable pair m < n defined by t(n) = m and t(m) = n where t(n) = psi(n) - n and psi(n) = A001615(n) is the Dedekind psi function.

Original entry on oeis.org

1550, 3100, 4790, 6200, 7750, 9580, 12400, 12922, 15500, 15290, 19160, 20330, 23950, 24800, 25844, 31000, 30580, 38320, 38750, 40660, 47900, 49600, 51688, 62000, 61160, 76640, 77500, 82150, 81320, 76450, 95800, 90454, 99200, 103376, 101650, 119750, 124000, 122320

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

The terms are ordered according to the order of their lesser counterparts (A323329).

Amiram Eldar, Table of n, a(n) for n = 1..1000
S. I. Dimitrov, On psi-amicable numbers and their generalizations, arXiv:2508.02318 [math.NT], 2025. See p. 2.

Cf. A001615, A002046, A033845 (Dedekind psi perfect numbers), A323327, A323328, A323329.

psi[n_] := n*Times@@(1+1/Transpose[FactorInteger[n]][[1]]); t[n_]:= psi[n] - n; s={}; Do[n=t[m]; If[n>m && t[n]==m, AppendTo[s,n]], {m, 1, 120000}]; s

A323332 The Dedekind psi function values of the powerful numbers, A001615(A001694(n)).

Original entry on oeis.org

1, 6, 12, 12, 24, 30, 36, 48, 72, 56, 96, 144, 108, 180, 216, 132, 150, 192, 288, 182, 336, 360, 432, 360, 324, 384, 576, 306, 648, 392, 380, 672, 720, 864, 672, 792, 900, 768, 552, 1152, 750, 1296, 1080, 1092, 972, 1344, 1440, 870, 1728, 2160, 992, 1584

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

The sum of the reciprocals of all the terms of this sequence is Pi^2/6 (A013661).

The asymptotic density of a sequence S that possesses the property that an integer k is a term if and only if its powerful part, A057521(k) is a term, is (1/zeta(2)) * Sum_{n>=1, A001694(n) is a term of S} 1/a(n). Examples for such sequences are the e-perfect numbers (A054979), the exponential abundant numbers (A129575), and other sequences listed in the Crossrefs section. - Amiram Eldar, May 06 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, A property of Dedekind's psi-function, Proceedings of the American Mathematical Society, Vol. 12, No. 6 (1961), p. 996.

Cf. A001615, A001694, A013661, A082695, A112526.

Sequences whose density can be calculated using this sequence: A054979, A129575, A307958, A308053, A321147, A322858, A323310, A328135, A339936, A340109, A364990, A382061, A383693, A383695, A383697.

psi[1]=1; psi[n_] := n * Times@@(1+1/Transpose[FactorInteger[n]][[1]]); psi /@ Join[{1}, Select[Range@ 1200, Min@ FactorInteger[#][[All, 2]] > 1 &]] (* after T. D. Noe at A001615 and Harvey P. Dale at A001694 *)

from math import isqrt, prod
from sympy import mobius, integer_nthroot, primefactors
def A323332(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l = n+x-squarefreepi(integer_nthroot(x,3)[0]), 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        return c+l
    a = primefactors(m:=bisection(f,n,n))
    return m*prod(p+1 for p in a)//prod(a) # Chai Wah Wu, Sep 14 2024

A342919 a(n) = A003415(n) / gcd(A001615(n), A003415(n)), where A001615 is Dedekind psi, and A003415 is the arithmetic derivative of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 1, 1, 7, 1, 2, 1, 3, 1, 4, 1, 7, 1, 2, 5, 13, 1, 11, 1, 5, 3, 2, 1, 31, 1, 5, 7, 19, 1, 5, 1, 7, 2, 17, 1, 41, 1, 2, 13, 25, 1, 7, 1, 1, 5, 2, 1, 3, 2, 23, 11, 31, 1, 23, 1, 11, 17, 2, 3, 61, 1, 2, 13, 59, 1, 13, 1, 13, 11, 2, 3, 71, 1, 11, 1, 43, 1, 31, 11, 15, 4, 35, 1, 41, 5, 2, 17, 49, 1, 17, 1, 11, 25

Offset: 1

Antti Karttunen, Mar 29 2021

nonn

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

Cf. A001615, A003415, A342001, A342458, A342459.

Cf. also A342414.

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
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342919(n) = { my(u=A003415(n)); (u/gcd(u, A001615(n))); };

a(n) = A003415(n) / A342458(n) = A003415(n) / gcd(A001615(n), A003415(n)).

a(n) = A342001(n) / A342459(n).

cp's OEIS Frontend

A301594 Expansion of Product_{k>=1} (1 + x^k)^A001615(k), where A001615 is the Dedekind psi function.

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A323328 Lexicographically earliest unbounded aliquot-like sequence based on the Dedekind psi function: a(1) = 318, a(n) = t(a(n-1)) where t(k) = A001615(k) - k.

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A323329 Lesser of amicable pair m < n defined by t(n) = m and t(m) = n where t(n) = psi(n) - n and psi(n) = A001615(n) is the Dedekind psi function.

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A327251 Expansion of Sum_{k>=1} psi(k) * x^k / (1 - x^k)^2, where psi = A001615.

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A344700 Numbers k for which A306927(k) [= A001615(k)-k] is a multiple of A344705(k) [= A001615(k)-A001065(k)], and their quotient is nonnegative.

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A203444 Numbers in range of Dedekind Psi function: A001615.

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A291167 Numbers k such that psi(k) is a perfect square where psi(k) = A001615(k).

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A323330 Larger of amicable pair m < n defined by t(n) = m and t(m) = n where t(n) = psi(n) - n and psi(n) = A001615(n) is the Dedekind psi function.

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A323332 The Dedekind psi function values of the powerful numbers, A001615(A001694(n)).

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