A001615 -id:A001615 - cp's OEIS frontend

A344705 a(n) = n + A001615(n) - sigma(n), where A001615 is the Dedekind psi-function, and sigma(n) gives the sum of divisors of n; difference between psi and the sum of proper divisors.

1, 2, 3, 3, 5, 6, 7, 5, 8, 10, 11, 8, 13, 14, 15, 9, 17, 15, 19, 14, 21, 22, 23, 12, 24, 26, 23, 20, 29, 30, 31, 17, 33, 34, 35, 17, 37, 38, 39, 22, 41, 42, 43, 32, 39, 46, 47, 20, 48, 47, 51, 38, 53, 42, 55, 32, 57, 58, 59, 36, 61, 62, 55, 33, 65, 66, 67, 50, 69, 70, 71, 21, 73, 74, 71, 56, 77, 78, 79, 38, 68, 82

Offset: 1

Antti Karttunen, May 28 2021

First negative term occurs as a(1440) = -18.

Antti Karttunen, Table of n, a(n) for n = 1..16560
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000203, A001065, A001615, A033879, A244963, A291209 (positions of zeros), A344700, A344704, A344753.

Cf. A013661, A082020.

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

A001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d));
A344705(n) = ((n + A001615(n)) - sigma(n));

a(n) = A001615(n) - A001065(n) = n - A244963(n) = n + A001615(n) - sigma(n).
a(n) = A033879(n) + A306927(n).
a(n) = n + A344753(n) - 2*A001065(n).
Sum_{k=1..n} a(k) = c * n^2 / 2 + O(n*log(n)), where c = 15/Pi^2 + 1 - Pi^2/6 = 0.874883... . - Amiram Eldar, Dec 08 2023

A345005 Odd numbers whose arithmetic derivative (A003415) is equal to Dedekind psi (A001615) applied to the same number.

Original entry on oeis.org

81, 3375, 15625, 231525, 713097, 1058841, 1500625, 4348377, 5764801, 16891497, 163555875, 209548647, 239010993, 239160735, 254205875, 267651475, 405189675, 451699875, 958403475, 1050284375, 1213014231, 1501534375, 1695809375, 1809323971, 1942143291

Offset: 1

Antti Karttunen, Jun 05 2021

Conjectured to be also the odd numbers k for which A344998(k) = A344999(k).

Odd terms in A301939, but see also A345003.

Cf. A001615, A003415, A344998, A344999.

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]));
isA345005(n) = ((n%2)&&(A003415(n)==A001615(n)));

A346470 a(n) = psi(A276086(n)), where psi is Dedekind psi function A001615, and A276086 is the prime product form of primorial base expansion of n.

Original entry on oeis.org

1, 3, 4, 12, 12, 36, 6, 18, 24, 72, 72, 216, 30, 90, 120, 360, 360, 1080, 150, 450, 600, 1800, 1800, 5400, 750, 2250, 3000, 9000, 9000, 27000, 8, 24, 32, 96, 96, 288, 48, 144, 192, 576, 576, 1728, 240, 720, 960, 2880, 2880, 8640, 1200, 3600, 4800, 14400, 14400, 43200, 6000, 18000, 24000, 72000, 72000, 216000, 56, 168, 224, 672

Offset: 0

Antti Karttunen, Jul 21 2021

Antti Karttunen, Table of n, a(n) for n = 0..2310
Index entries for sequences related to primorial base

Cf. A001615.
Other number-theoretical functions similarly applied to A276086: A267263 (omega), A276150 (bigomega), A324650 (phi), A324653 (sigma), A324655 (tau), A327860 (arithmetic derivative).
Cf. also A346471, A346475.

A346470(n) = { my(m=1, p=2, e); while(n, e = (n%p); if(e, m *= (p+1)*(p^(e-1))); n = n\p; p = nextprime(1+p)); (m); };

a(n) = A001615(A276086(n)).

A347135 a(n) = Sum_{d|n} A001615(n/d) * A069359(d).

Original entry on oeis.org

0, 1, 1, 5, 1, 12, 1, 16, 7, 16, 1, 51, 1, 20, 18, 44, 1, 68, 1, 71, 22, 28, 1, 156, 11, 32, 33, 91, 1, 167, 1, 112, 30, 40, 26, 277, 1, 44, 34, 220, 1, 215, 1, 131, 110, 52, 1, 420, 15, 140, 42, 151, 1, 300, 34, 284, 46, 64, 1, 673, 1, 68, 138, 272, 38, 311, 1, 191, 54, 295, 1, 836, 1, 80, 162, 211, 38, 359, 1, 596

Offset: 1

Antti Karttunen, Aug 23 2021

nonn

Dirichlet convolution of A001615 (Dedekind psi function) with A069359.

Dirichlet convolution of A001221 (omega, number of distinct prime factors of n) with A322577.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Wikipedia, Dirichlet convolution

Cf. A001221, A001615, A069359, A322577.

Cf. also A347132, A347133, A347134.

Table[DivisorSum[n,PrimeNu[n/#]*Sum[DirichletConvolve[j,MoebiusMu[j]^2,j,#/d] EulerPhi[d],{d,Divisors[#]}]&],{n,80}] (* Giorgos Kalogeropoulos, Oct 28 2021 *)

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
A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
A347135(n) = sumdiv(n,d,A001615(n/d)*A069359(d));

a(n) = Sum_{d|n} A001615(n/d) * A069359(d).

a(n) = Sum_{d|n} A001221(n/d) * A322577(d).

A187778 Numbers k dividing psi(k), where psi is the Dedekind psi function (A001615).

Original entry on oeis.org

1, 6, 12, 18, 24, 36, 48, 54, 72, 96, 108, 144, 162, 192, 216, 288, 324, 384, 432, 486, 576, 648, 768, 864, 972, 1152, 1296, 1458, 1536, 1728, 1944, 2304, 2592, 2916, 3072, 3456, 3888, 4374, 4608, 5184, 5832, 6144, 6912, 7776, 8748, 9216, 10368, 11664, 12288, 13122, 13824, 15552, 17496, 18432, 20736, 23328

Offset: 1

Enrique Pérez Herrero, Jan 05 2013

nonn

This sequence is closed under multiplication.
Also 1 and the numbers where psi(n)/n = 2, or n/phi(n)=3, or psi(n)/phi(n)=6.
Also 1 and the numbers of the form 2^i*3^j with i, j >= 1 (A033845).
If M(n) is the n X n matrix whose elements m(i,j) = 2^i*3^j, with i, j >= 1, then det(M(n))=0.
Numbers n such that Product_{i=1..q} (1 + 1/d(i)) is an integer where q is the number of the distinct prime divisors d(i) of n. - Michel Lagneau, Jun 17 2016

			psi(48) = 96 and 96/48 = 2 so 48 is in this sequence.

S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, p. xxiv.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..191 from Vincenzo Librandi)
R. Blecksmith, M. McCallum and J. L. Selfridge, 3-smooth representations of integers, Amer. Math. Monthly, 105 (1998), 529-543.
E. Deutsch, Tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
Eric Weisstein's World of Mathematics, Smooth Number
Wikipedia, Closure

Cf. A003586, A001615, A007694, A033950, A074946, A075592.

[6*n: n in [1..3000] | PrimeDivisors(n) subset [2, 3]]; // Vincenzo Librandi, Jan 11 2019

Select[Range[10^4],#/EulerPhi[#]==3 || #==1&]
Join[{1}, 6 Select[Range@4000, Last@Map[First, FactorInteger@#]<=3 &]] (* Vincenzo Librandi, Jan 11 2019 *)

dedekindpsi(n) = if( n<1, 0, direuler( p=2, n, (1 + X) / (1 - p*X)) [n]);
k=0; n=0; while(k<10000,n++; if( dedekindpsi(n) % n== 0, k++; print1(n, ", ")));

For n > 1, a(n) = 6 * A003586(n).

Sum_{n>0} 1/a(n)^k = 1 + Sum_{i>0} Sum_{j>0} 1/(2^i * 3^j)^k = 1 + 1/((2^k-1)*(3^k-1)).

A240112 Numbers for which the values of the Dedekind psi function (A001615) are greater than the values of the infinitary Dedekind psi function (A049417).

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 25, 28, 36, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 75, 76, 80, 81, 84, 90, 92, 98, 99, 100, 108, 112, 116, 117, 121, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 162, 164, 169, 171, 172, 175, 176, 180, 188, 192, 196, 198, 204, 207

Offset: 1

Vladimir Shevelev, Apr 01 2014

nonn

The first term of A072587 that is not in this sequence is 72.
On the set of the nonsquarefree numbers (A013929) it is complement to A240111.
The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 2, 29, 284, 2845, 28527, 285352, 2853422, 28534455, 285344362, 2853443344, ... . Apparently, the asymptotic density of this sequence exists and equals 0.2853443... . - Amiram Eldar, Feb 13 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Peter J. C. Moses)

Cf. A001615, A013929, A049417, A072587, A240111.

f1[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; f2[p_, e_] := (p+1)*p^(e-1); q[1] = False; q[n_] := Module[{fct = FactorInteger[n]}, Times @@ f2 @@@ fct > Times @@ f1 @@@ fct]; Select[Range[250], q] (* Amiram Eldar, Feb 13 2025 *)

isok(k) = {my(f = factor(k), b); prod(i=1, #f~, (f[i, 1]+1)*f[i, 1]^(f[i, 2]-1)) > prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 1+f[i, 1]^(2^(#b-k)), 1)));} \\ Amiram Eldar, Feb 13 2025

More terms from Peter J. C. Moses, Apr 02 2014

A308462 Expansion of e.g.f. exp(Sum_{k>=1} psi(k)*x^k/k), where psi() is the Dedekind psi function (A001615).

Original entry on oeis.org

1, 1, 4, 18, 114, 810, 7560, 71820, 822780, 10086300, 139532400, 2035618200, 33149655000, 562448086200, 10416443637600, 202624824402000, 4207527414090000, 91475485119018000, 2114681171586984000, 50821588411117524000, 1289125346347418580000

Offset: 0

Ilya Gutkovskiy, May 28 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..440

Cf. A001615, A301511, A318917.

nmax = 20; CoefficientList[Series[Exp[Sum[DirichletConvolve[j, MoebiusMu[j]^2, j, k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Product[1 + Boole[PrimeQ[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}]

log(a(n)/n!) ~ 2*sqrt(15*n)/Pi. - Vaclav Kotesovec, Oct 31 2024

A323327 Numbers that start an unbounded aliquot-like sequence based on Dedekind psi function (A001615).

Original entry on oeis.org

318, 330, 498, 510, 534, 546, 636, 660, 786, 798, 942, 954, 978, 990, 996, 1020, 1068, 1092, 1110, 1122, 1254, 1272, 1320, 1398, 1410, 1470, 1494, 1506, 1518, 1530, 1572, 1596, 1602, 1614, 1626, 1638, 1734, 1884, 1908, 1938, 1950, 1956, 1980, 1992, 2040, 2046

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

Let t(k) = psi(k) - k = A001615(k) - k be the sum of aliquot divisors d of k, such that k/d is squarefree. Penney & Pomerance proposed a problem to show that the aliquot-like sequence related to this function, i.e., the trajectory of an integer k under the repeated application of the map k -> t(k), can be unbounded. Since t(m^j * k) = m^j * t(k) if m|k, 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.

			318 is in the sequence since t(318) = psi(318) - 318 = 330, t(330) = 534, etc., and this repeated mapping yields an unbounded sequence.

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

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.
Eric Weisstein's World of Mathematics, Aliquot Sequence.
Wikipedia, Aliquot sequence.

Cf. A001615, A098007, A187778.

t[1]=0; t[n_] := (Times@@(1+1/Transpose[FactorInteger[n]][[1]])-1)*n; rt[n_] := Module[{f=FactorInteger[n]}, e=GCD@@f[[;;,2]]; Surd[n,e]]; divrootQ[n_, m_] := Divisible[n, rt[m]]; divQ[s_, n_] := If[n==0, 0, If[MemberQ[s, n], 1, If[ Length[Select[s, Divisible[n,#] && divrootQ[#, n/#] &]] > 0, 2, 3]]]; seqQ[n_] := Module[{n1=n}, s={}; While[divQ[s, n1] ==3, AppendTo[s, n1]; n1=t[n1]]; divQ[s, n1]] == 2; Select[Range[10000], seqQ]

A342916 a(n) = (1+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

2, 1, 1, 5, 1, 7, 1, 3, 5, 11, 1, 13, 1, 5, 2, 17, 1, 19, 1, 7, 11, 23, 1, 25, 13, 9, 7, 29, 1, 31, 1, 11, 17, 35, 3, 37, 1, 13, 5, 41, 1, 43, 1, 5, 23, 47, 1, 49, 25, 17, 13, 53, 1, 55, 7, 19, 29, 59, 1, 61, 1, 21, 2, 65, 11, 67, 1, 23, 35, 71, 1, 73, 1, 25, 19, 77, 13, 79, 1, 9, 41, 83, 1, 85, 43, 29, 11, 89, 1, 91, 23, 31

Offset: 1

Antti Karttunen, Mar 29 2021

nonn

It is conjectured that a(n) = 1 only when n is a prime, A000040. See Thomas Ordowski's May 21 2017 problem in A001615.

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

Cf. A000040, A001615, A342915, A342917.
Cf. also A160596.
After n=1 differs from A342918 for the first time at n=44, where a(44) = 5, while A342918(44) = 15.

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
A342916(n) = ((1+n)/gcd(1+n,A001615(n)));

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

Incorrect A-number in the formula corrected by Antti Karttunen, May 31 2021

A387415 Numbers k such that the odd part of (1+k) divides (1 + odd part of A001615(k)), where A001615 is Dedekind's psi-function.

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 2431, 4095, 8191, 14335, 16383, 27135, 32767, 44031, 57855, 65535, 75775, 131071, 204799, 262143, 376831, 524287, 667135, 923647, 1048575, 1441791, 1632255, 2056191, 2097151, 2315775, 2744319, 4194303, 6768639, 6815743, 8388607, 8781823, 16777215, 19922943, 24068095

Offset: 1

Antti Karttunen, Sep 01 2025

Like in many sequences of this type, the criterion seems to strongly select for numbers with a long tail of trailing 1-bits. The initial 1 is probably the only term that is not in A004767.

Cf. A000225 (subsequence), A000265, A001615.

For similar sequences, see A336700, A387410, A387418, A387419.

A000265(n) = (n>>valuation(n,2));
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)));
isA387415(n) = !((1+A000265(A001615(n)))%A000265(1+n));

cp's OEIS Frontend

A344705 a(n) = n + A001615(n) - sigma(n), where A001615 is the Dedekind psi-function, and sigma(n) gives the sum of divisors of n; difference between psi and the sum of proper divisors.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A345005 Odd numbers whose arithmetic derivative (A003415) is equal to Dedekind psi (A001615) applied to the same number.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A346470 a(n) = psi(A276086(n)), where psi is Dedekind psi function A001615, and A276086 is the prime product form of primorial base expansion of n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A347135 a(n) = Sum_{d|n} A001615(n/d) * A069359(d).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A187778 Numbers k dividing psi(k), where psi is the Dedekind psi function (A001615).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A240112 Numbers for which the values of the Dedekind psi function (A001615) are greater than the values of the infinitary Dedekind psi function (A049417).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A308462 Expansion of e.g.f. exp(Sum_{k>=1} psi(k)*x^k/k), where psi() is the Dedekind psi function (A001615).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A323327 Numbers that start an unbounded aliquot-like sequence based on Dedekind psi function (A001615).

Views

Author

Keywords

Comments

Examples