A046099 -id:A046099

A341635

a(n) = Sum_{d|n} phi(d) * mu(d) * mu(n/d).

+10
2

1, -2, -3, 1, -5, 6, -7, 0, 2, 10, -11, -3, -13, 14, 15, 0, -17, -4, -19, -5, 21, 22, -23, 0, 4, 26, 0, -7, -29, -30, -31, 0, 33, 34, 35, 2, -37, 38, 39, 0, -41, -42, -43, -11, -10, 46, -47, 0, 6, -8, 51, -13, -53, 0, 55, 0, 57, 58, -59, 15, -61, 62, -14, 0, 65

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Dirichlet inverse of A003967.

Moebius transform of A097945.

From Vaclav Kotesovec, Feb 19 2021: (Start)

Abs(a(n)) <= n.

a(n) = n iff n is in A030229. (End)

LINKS

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

FORMULA

a(n) = Sum_{k=1..n} mu(gcd(n,k)) * mu(n/gcd(n,k)).

a(1) = 1; a(n) = -Sum_{d|n, d < n} A003967(n/d) * a(d).

a(n) = Sum_{d|n} mu(n/d) * A097945(d).

Multiplicative with a(p^e) = -p if e=1, p-1 if e=2, and 0 otherwise. - Amiram Eldar, Feb 19 2021

MATHEMATICA

Table[Sum[EulerPhi[d] MoebiusMu[d] MoebiusMu[n/d], {d, Divisors[n]}], {n, 65}]

Table[Sum[MoebiusMu[GCD[n, k]] MoebiusMu[n/GCD[n, k]], {k, n}], {n, 65}]

PROG

(PARI) a(n) = sumdiv(n, d, eulerphi(d)*moebius(d)*moebius(n/d)); \\ Michel Marcus, Feb 17 2021

CROSSREFS

Cf. A000010, A003967, A007427, A007431, A008683, A030229 (fixed points), A046099 (positions of 0's), A068341, A097945, A276833.

KEYWORD

sign,mult

AUTHOR

Ilya Gutkovskiy, Feb 16 2021

STATUS

approved

A347251

a(n) = Sum_{d|n} mu(d)*mu(n/d)*d^n.

+10
2

1, -5, -28, 16, -3126, 47450, -823544, 0, 19683, 10009766650, -285311670612, -2176786432, -302875106592254, 11112685048647250, 437893920912786408, 0, -827240261886336764178, -101560344088905, -1978419655660313589123980, -100000000000001048576

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..388

FORMULA

If p is prime, a(p) = -1 - p^p.

MATHEMATICA

a[n_] := DivisorSum[n, MoebiusMu[#] * MoebiusMu[n/#] * #^n &]; Array[a, 20] (* Amiram Eldar, Aug 24 2021 *)

PROG

(PARI) a(n) = sumdiv(n, d, moebius(d)*moebius(n/d)*d^n);

CROSSREFS

Diagonal of A347227.

Cf. A007427, A008683, A046099, A067858, A321222.

KEYWORD

sign

AUTHOR

Seiichi Manyama, Aug 24 2021

STATUS

approved

A349236

Gaps between cubefree numbers: a(n) = A004709(n+1) - A004709(n).

+10
2

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,7

COMMENTS

This sequence is unbounded since by the Chinese Remainder Theorem there are arbitrarily long runs of consecutive numbers that are not cubefree.

The first occurrence of a(n) = 1, 2, ... is at n = 1, 7, 68, 1145, 18825, 15003967, ...

The asymptotic density of the occurrences of 1 in this sequence is density(A340152)/density(A004709) = A340153/A088453 = 0.8136635872...

LINKS

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

Michael J. Mossinghoff, Tomás Oliveira e Silva, and Tim Trudgian, The distribution of k-free numbers, Mathematics of Computation, Vol. 90, No. 328 (2021), pp. 907-929; arXiv preprint, arXiv:1912.04972 [math.NT], 2019-2020.

FORMULA

Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = zeta(3) (A002117).

EXAMPLE

a(1) = A004709(2) - A004709(1) = 2 - 1 = 1.

a(7) = A004709(8) - A004709(7) = 9 - 7 = 2.

MATHEMATICA

cubeFreeQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # < 3 &]; Differences @ Select[Range[100], cubeFreeQ]

PROG

(PARI)

A003557(n) = (n/factorback(factorint(n)[, 1]));

isA004709(n) = issquarefree(A003557(n));

A349236list(first_n) = { my(v=vector(first_n), k=0, e=1); for(n=2, oo, if(isA004709(n), k++; v[k] = n-e; e = n); if(#v==k, return(v))); }; \\ Antti Karttunen, Nov 11 2021

CROSSREFS

Cf. A002117, A046099, A004709, A076259, A088453, A271443, A340152, A340153.

KEYWORD

nonn

AUTHOR

Amiram Eldar, Nov 11 2021

STATUS

approved

A354235

Heinz numbers of integer partitions with at least one part divisible by 3.

+10
2

5, 10, 13, 15, 20, 23, 25, 26, 30, 35, 37, 39, 40, 45, 46, 47, 50, 52, 55, 60, 61, 65, 69, 70, 73, 74, 75, 78, 80, 85, 89, 90, 91, 92, 94, 95, 100, 103, 104, 105, 110, 111, 113, 115, 117, 120, 122, 125, 130, 135, 137, 138, 140, 141, 143, 145, 146, 148, 150

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

LINKS

Table of n, a(n) for n=1..59.

EXAMPLE

The terms together with their prime indices begin:

5: {3}

10: {1,3}

13: {6}

15: {2,3}

20: {1,1,3}

23: {9}

25: {3,3}

26: {1,6}

30: {1,2,3}

35: {3,4}

37: {12}

39: {2,6}

40: {1,1,1,3}

45: {2,2,3}

46: {1,9}

47: {15}

50: {1,3,3}

52: {1,1,6}

55: {3,5}

60: {1,1,2,3}

MATHEMATICA

Select[Range[100], MemberQ[PrimePi/@First/@If[#==1, {}, FactorInteger[#]]/3, _?IntegerQ]&]

CROSSREFS

For 4 instead of 3 we have A046101, counted by A295342.

This sequence ranks the partitions counted by A295341, compositions A335464.

For 2 instead of 3 we have A324929 (and A013929), counted by A047967.

A001222 counts prime factors with multiplicity, distinct A001221.

A004250 counts partitions with some part > 2, compositions A008466.

A004709 lists numbers divisible by no cube, counted by A000726.

A036966 lists 3-full numbers, counted by A100405.

A046099 lists non-cubefree numbers.

A056239 adds up prime indices, row sums of A112798 and A296150.

A124010 gives prime signature, sorted A118914.

A354234 counts partitions of n with at least one part divisible by k.

Cf. A000720, A003963, A008483, A018256, A062739, A064428, A117485, A181819, A339222, A353508.

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 23 2022

STATUS

approved

A375145

Numbers whose prime factorization has exactly one exponent that is larger than 2.

+10
2

8, 16, 24, 27, 32, 40, 48, 54, 56, 64, 72, 80, 81, 88, 96, 104, 108, 112, 120, 125, 128, 135, 136, 144, 152, 160, 162, 168, 176, 184, 189, 192, 200, 208, 224, 232, 240, 243, 248, 250, 256, 264, 270, 272, 280, 288, 296, 297, 304, 312, 320, 324, 328, 336, 343, 344

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Subsequence of A046099 and first differs from it at n = 35: A046099(35) = 216 = 2^3 * 3^3 is not a term of this sequence.

Numbers k such that the powerful part of k, A057521(k), is a prime power whose exponent is larger than 2 (A246549).

The asymptotic density of this sequence is (1/zeta(3)) * Sum_{p prime} 1/(p^3-1) = A286229 / A002117 = 0.16148833663564192901... .

LINKS

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

Index entries for sequences computed from exponents in factorization of n.

EXAMPLE

8 = 2^3 is a term since its prime factorization has exactly one exponent, 3, that is larger than 2.

MATHEMATICA

q[n_] := Count[FactorInteger[n][[;; , 2]], _?(# > 2 &)] == 1; Select[Range[350], q]

PROG

(PARI) is(k) = #select(x -> x > 2, factor(k)[, 2]) == 1;

CROSSREFS

Subsequence of A046099.

Cf. A002117, A057521, A190641, A246549, A286229, A375146.

KEYWORD

nonn,easy

AUTHOR

Amiram Eldar, Aug 01 2024

STATUS

approved

A074451

Non-cubefree noncubes.

+10
1

16, 24, 32, 40, 48, 54, 56, 72, 80, 81, 88, 96, 104, 108, 112, 120, 128, 135, 136, 144, 152, 160, 162, 168, 176, 184, 189, 192, 200, 208, 224, 232, 240, 243, 248, 250, 256, 264, 270, 272, 280, 288, 296, 297, 304, 312, 320, 324, 328, 336, 344, 351, 352, 360

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

FORMULA

For n > 35, a(n) < 7n. Asymptotically, a(n) ~ kn with k = zeta(3)/(zeta(3)-1) = 5.949... . - Charles R Greathouse IV, Oct 16 2015 [Corrected by Amiram Eldar, Aug 31 2024]

Sum_{n>=1} 1/a(n)^s = 1 + zeta(s) - zeta(3*s) - zeta(s)/zeta(3*s), for s > 1. - Amiram Eldar, Aug 31 2024

MATHEMATICA

With[{m = 10}, Select[Complement[Range[m^3], Range[m]^3], AnyTrue[FactorInteger[#][[;; , 2]], #1 > 2 &] &]] (* Amiram Eldar, Aug 31 2024 *)

PROG

(PARI) is(n)=my(f=factor(n)[, 2]); f%3 && vecmax(f)>2 \\ Charles R Greathouse IV, Oct 16 2015

CROSSREFS

Intersection of A046099 and A007412.

Cf. A051144, A002117.

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Sep 25 2002

STATUS

approved

A114287

Sum of the cubes of the first n noncubefree numbers.

+10
1

0, 512, 4608, 18432, 38115, 70883, 134883, 245475, 402939, 578555, 840699, 1213947, 1725947, 2257388, 2938860, 3823596, 4948460, 6208172, 7613100, 9341100, 11294225, 13391377, 15851752, 18367208, 21353192, 24865000, 28961000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..26.

FORMULA

a(n) = Sum_{k=1..n} A046099(k)^3.

a(n) ~ c * n^4, where c = (zeta(3)/(zeta(3)-1))^3/4 = 52.6373493984... . - Amiram Eldar, Feb 20 2024

EXAMPLE

a(10) = 8^3 + 16^3 + 24^3 + 27^3 + 32^3 + 40^3 + 48^3 + 54^3 + 56^3 + 64^3 = 840699.

MATHEMATICA

noncubeFreeQ[n_] := Max[FactorInteger[n][[;; , 2]]] > 2; Join[{0}, Accumulate[Select[Range[200], noncubeFreeQ]^3]] (* Amiram Eldar, Feb 20 2024 *)

CROSSREFS

Cf. A046099, A114286.

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Nov 20 2005

STATUS

approved

A175130

Indices of Fibonacci numbers that are not cubefree.

+10
1

6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 125, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 192, 198, 204, 210, 216, 222, 228, 234, 240, 246, 250, 252, 258, 264, 270, 276, 282, 288, 294, 300, 306, 312, 318, 324

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Supersequence of A037917.

Conjecture: all terms are multiples of 6 or 125. - Harvey P. Dale, Apr 28 2020

The conjecture is false. The counterexamples are 392, 784, 1183, 1210, .... . - Amiram Eldar, Oct 16 2023

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..248

Blair Kelly, Fibonacci factorizations.

FORMULA

A000045 INTERSECT A046099.

A010056(a(n)) * (1 - A212793(a(n))) = 1. - Reinhard Zumkeller, May 27 2012

EXAMPLE

Fibonacci(125) = 5^3 * 3001 * 158414167964045700001 = A000045(125) is not cubefree, which adds 125 to the sequence.

MATHEMATICA

Select[Range[350], Max[FactorInteger[Fibonacci[#]][[All, 2]]]>2&] (* Harvey P. Dale, Apr 28 2020 *)

PROG

(Haskell)

import Data.List (findIndices)

a175130 n = a175130_list !! (n-1)

a175130_list = map (+ 1) $ findIndices ((== 0) . a212793) $ tail a000045_list

-- Reinhard Zumkeller, May 27 2012

(PARI) is(n)=n>5 && vecmax(factor(fibonacci(n))[, 2])>2 \\ Charles R Greathouse IV, Nov 07 2014

CROSSREFS

Cf. A000045, A010056, A037917, A046099, A134492, A212793.

KEYWORD

nonn

AUTHOR

R. J. Mathar, Feb 16 2010

STATUS

approved

A181104

Dirichlet inverse of Ramanujan's L-series (A000594).

+10
1

1, 24, -252, 2048, -4830, -6048, 16744, 0, 177147, -115920, -534612, -516096, 577738, 401856, 1217160, 0, 6905934, 4251528, -10661420, -9891840, -4219488, -12830688, -18643272, 0, 48828125, 13865712, 0, 34291712, -128406630, 29211840

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Although it is conjectured that A000594(n) is never 0 here a(n)=0 for infinitely many n. Namely a(n)=0 iff n is not cubefree (n is in A046099).

Multiplicative because A000594 is. - Andrew Howroyd, Aug 05 2018

REFERENCES

B. Cloitre, On the order of absolute convergence of Dirichlet series and the Grand Riemann hypothesis, in preparation 2010-2011 (unpublished as of August 2018).

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..1000

FORMULA

For Re(s)>13/2 we have sum_{n>0}a(n)/n^s*sum_{n>0}A000594(n)/n^s=1. If n is squarefree then a(n)=(-1)^omega(n)*A000594(n).

MATHEMATICA

a[1] = 1; a[n_] := a[n] = -Sum[a[d]*RamanujanTau[n/d], {d, Most[Divisors[n]]}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Jun 18 2013 *)

PROG

(PARI) a(n)=if(n<2, 1/A000594(1), -1/A000594(1)*sumdiv(n, d, if(n-d, a(d)*A000594(n/d), 0)))

(PARI) seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, ramanujantau(n)))} \\ Andrew Howroyd, Aug 05 2018

CROSSREFS

Cf. A000594, A046099

KEYWORD

sign,mult

AUTHOR

Benoit Cloitre, Oct 03 2010

STATUS

approved

A267798

a(n) is the smallest prime p such that none of p + 1, p + 2,... p + n are cubefree.

+10
1

7, 79, 4373, 885623, 146447621, 309763486247, 151536553447871

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..7.

EXAMPLE

a(1) = 7 because 7 is prime and none of 7 + 1 = 8 = (2*2*2) is cubefree.

a(2) = 79 because 79 is prime and none of 79 + 1 = 80 = (2*2*2)*10, 79 + 2 = 81 = (3*3*3)*3 are cubefree.

a(3) = 4373 because 4373 is prime and none of 4373 + 1 = 4374 = (3*3*3)*162, 4373 + 2 = 4375 = (5*5*5)*35, 4373 + 3 = 4376 = (2*2*2)*547 are cubefree.

a(4) = 885623 because 885623 is prime and none of 885623 + 1 = 885624 = (2*2*2)*110703, 885623 + 2 = 885625 = (5*5*5)*7085, 885623 + 3 = 885626 = (7*7*7)*2582, 885623 + 4 = 885627 = (3*3*3)*32801 are cubefree.

MATHEMATICA

Table[SelectFirst[Prime@ Range[10^5], Function[p, AllTrue[p + Range@ n, AnyTrue[Last /@ FactorInteger@ #, # > 2 &] &]]], {n, 4}] (* Michael De Vlieger, Apr 07 2016, Version 10 *)

PROG

(PARI) isokp(p, n)=for (k=1, n, if (vecmax(factor(p+k)[, 2]) < 3, return (0)); ); 1;

a(n) = forprime(p=7, , if (isokp(p, n), return(p))) \\ Michel Marcus, Apr 07 2016

CROSSREFS

Cf. A046099, A271395.

KEYWORD

nonn,more

AUTHOR

Juri-Stepan Gerasimov, Apr 07 2016

EXTENSIONS

a(5) from Michel Marcus, Apr 07 2016

a(6)-a(7) from Giovanni Resta, Apr 12 2016

STATUS

approved