A055231 -id:A055231 - cp's OEIS frontend

A370900 Partial sums of the powerfree part function (A055231).

1, 3, 6, 7, 12, 18, 25, 26, 27, 37, 48, 51, 64, 78, 93, 94, 111, 113, 132, 137, 158, 180, 203, 206, 207, 233, 234, 241, 270, 300, 331, 332, 365, 399, 434, 435, 472, 510, 549, 554, 595, 637, 680, 691, 696, 742, 789, 792, 793, 795, 846, 859, 912, 914, 969, 976, 1033

Offset: 1

Amiram Eldar, Mar 05 2024

Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 52.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, An elementary method in the asymptotic theory of numbers, Duke Mathematical Journal, Vol. 28, No. 2 (1961), pp. 183-192.
Eckford Cohen, Some asymptotic formulas in the theory of numbers, Transactions of the American Mathematical Society, Vol. 112, No. 2 (1964), pp. 214-227.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A055231, A191622, A370901.

f[p_, e_] := If[e == 1, p, 1]; pfp[n_] := Times @@ f @@@ FactorInteger[n]; pfp[1] = 1; Accumulate[Array[pfp[#] &, 100]]

pfp(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, f[i, 1], 1));}
lista(kmax) = {my(s = 0); for(k = 1, kmax, s += pfp(k); print1(s, ", "))};

a(n) = Sum_{k=1..n} A055231(k).

a(n) = c * n^2 / 2 + O(R(n)), where c = Product_{p prime} (1 - (p^2+p-1)/(p^3*(p+1))) = 0.649606699337... (A191622), R(n) = x^(3/2) * exp(-c_1 * log(n)^(3/5) / log(log(n))^(1/5)) unconditionally, or x^(7/5) * exp(c_2 * log(n) / log(log(n))) assuming the Riemann hypothesis, and c_1 and c_2 are positive constants (Tóth, 2017).

A370901 Partial alternating sums of the powerfree part function (A055231).

Original entry on oeis.org

1, -1, 2, 1, 6, 0, 7, 6, 7, -3, 8, 5, 18, 4, 19, 18, 35, 33, 52, 47, 68, 46, 69, 66, 67, 41, 42, 35, 64, 34, 65, 64, 97, 63, 98, 97, 134, 96, 135, 130, 171, 129, 172, 161, 166, 120, 167, 164, 165, 163, 214, 201, 254, 252, 307, 300, 357, 299, 358, 343, 404, 342

Offset: 1

Amiram Eldar, Mar 05 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A055231, A191622, A370900.

Similar sequences: A068762, A068773, A307704, A357817, A362028.

f[p_, e_] := If[e == 1, p, 1]; pfp[n_] := Times @@ f @@@ FactorInteger[n]; pfp[1] = 1; Accumulate[Array[(-1)^(# + 1) * pfp[#] &, 100]]

pfp(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, f[i, 1], 1));}
lista(kmax) = {my(s = 0); for(k = 1, kmax, s += (-1)^(k+1) * pfp(k); print1(s, ", "))};

a(n) = Sum_{k=1..n} (-1)^(k+1) * A055231(k).

a(n) = (5/38) * c * n^2 + O(R(n)), where c = Product_{p prime} (1 - (p^2+p-1)/(p^3*(p+1))) = 0.649606... (A191622), R(n) = x^(3/2) * exp(-c_1 * log(n)^(3/5) / log(log(n))^(1/5)) unconditionally, or x^(7/5) * exp(c_2 * log(n) / log(log(n))) assuming the Riemann hypothesis, and c_1 and c_2 are positive constants (Tóth, 2017).

A328014 Numbers whose powerful part (A057521) is larger than their powerfree part (A055231).

Original entry on oeis.org

4, 8, 9, 12, 16, 18, 24, 25, 27, 32, 36, 40, 45, 48, 49, 50, 54, 56, 63, 64, 72, 75, 80, 81, 96, 98, 100, 108, 112, 121, 125, 128, 135, 144, 147, 150, 160, 162, 169, 175, 176, 180, 189, 192, 196, 200, 208, 216, 224, 225, 240, 242, 243, 245, 250, 252, 256, 270

Offset: 1

Amiram Eldar, Oct 01 2019

nonn

Differs from A122145(n) at n >= 25.

Cloutier et al. showed that the number of terms of this sequence below x is D0 * x^(3/4) + O(x^(2/3)*log(x)), where D0 is a constant given in A328015.

			12 is in the sequence since A057521(12) = 4 > A055231(12) = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Maurice-Étienne Cloutier, Jean-Marie De Koninck, and Nicolas Doyon, On the powerful and squarefree parts of an integer, Journal of Integer Sequences, Vol. 17 (2014), Article 14.6.6.

Cf. A057521, A055231, A328015.

funp[p_, e_] := If[e > 1, p^e, 1]; pow[n_] := Times @@ (funp @@@ FactorInteger[n]); aQ[n_] := pow[n] > n/pow[n]; Select[Range[1000], aQ]

pful(f) = prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); \\ A057521
pfree(f) = for (k=1, #f~, if (f[k, 2] > 1, f[k, 2] = 0); ); factorback(f); \\ A055231
isok(n) = my(f=factor(n)); pful(f) > pfree(f); \\ Michel Marcus, Oct 02 2019

A368825 a(1) = 1; for n > 1, a(n) = A055231(a(n-1) + n), where A055231(k) is the powerfree part of k.

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 7, 15, 3, 13, 3, 15, 7, 21, 1, 17, 34, 13, 1, 21, 42, 1, 3, 1, 26, 13, 5, 33, 62, 23, 2, 34, 67, 101, 17, 53, 10, 3, 42, 82, 123, 165, 13, 57, 102, 37, 21, 69, 118, 21, 1, 53, 106, 5, 15, 71, 1, 59, 118, 178, 239, 301, 91, 155, 55, 1, 17, 85, 154, 7, 78, 6, 79, 17, 23, 11

Offset: 1

Scott R. Shannon, Jan 07 2024

nonn

The sequence is conjectured to contain all the squarefree numbers; see A005117.
The fixed points begin 1, 7, 5759, 35435, 58403, 62051, 182363, 261763, ... , although it is likely there are infinitely more.
See A368827 for the indices where a(n) = 1.

			a(9) = 3 as a(8)+9 = 15+9 = 24 = 2*2*2*3, and the powerfree part of 24 is 3.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 1000000 terms. The green line is a(n) = n.

Cf. A368827, A368823 (multiplication), A055231, A005117, A124010.

A379579 Numerators of the partial sums of the reciprocals of the powerfree part function (A055231).

Original entry on oeis.org

1, 3, 11, 17, 91, 16, 117, 152, 187, 381, 4261, 13553, 178499, 90322, 30441, 35446, 607587, 1300259, 24875091, 25521737, 77027101, 38733998, 895731799, 932913944, 1044460379, 2097501253, 2320594123, 2352464533, 68444564327, 11443370128, 355822756173, 389249504528

Offset: 1

Amiram Eldar, Dec 26 2024

			Fractions begin with 1, 3/2, 11/6, 17/6, 91/30, 16/5, 117/35, 152/35, 187/35, 381/70, 4261/770, 13553/2310, ...

D. Suryanarayana and P. Subrahmanyam, The maximal k-full divisor of an integer, Indian J. Pure Appl. Math., Vol. 12, No. 2 (1981), pp. 175-190.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Maurice-Étienne Cloutier, Les parties k-puissante et k-libre d'un nombre, Thèse de doctorat, Université Laval, Québec (2018).
Maurice-Étienne Cloutier, Jean-Marie De Koninck, and Nicolas Doyon, On the powerful and squarefree parts of an integer, Journal of Integer Sequences, Vol. 17 (2014), Article 14.6.6.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.11, pp. 31-32.

Cf. A055231, A328013, A370900, A370901, A379580 (denominators), A379581.

f[p_, e_] := If[e==1, p, 1]; powfree[n_] := Times @@ f @@@ FactorInteger[n]; Numerator[Accumulate[Table[1/powfree[n], {n, 1, 50}]]]

powfree(n) = {my(f = factor(n)); prod(i=1, #f~, if(f[i, 2] == 1, f[i, 1], 1)); }
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / powfree(k); print1(numerator(s), ", "))};

a(n) = numerator(Sum_{k=1..n} 1/A055231(k)).

a(n)/A379580(n) = A * n^(1/2) + B * n^(1/3) + O(n^(1/5)), where A = A328013, and B = (zeta(2/3)/zeta(2)) * Product_{p prime} (1 + (p^(1/3)-1)/(p*(p^(2/3)-p^(1/3)+1))) = -2.59305556147555965163... .

A379580 Denominators of the partial sums of the reciprocals of the powerfree part function (A055231).

Original entry on oeis.org

1, 2, 6, 6, 30, 5, 35, 35, 35, 70, 770, 2310, 30030, 15015, 5005, 5005, 85085, 170170, 3233230, 3233230, 9699690, 4849845, 111546435, 111546435, 111546435, 223092870, 223092870, 223092870, 6469693230, 1078282205, 33426748355, 33426748355, 9116385915, 18232771830

Offset: 1

Amiram Eldar, Dec 26 2024

D. Suryanarayana and P. Subrahmanyam, The maximal k-full divisor of an integer, Indian J. Pure Appl. Math., Vol. 12, No. 2 (1981), pp. 175-190.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Maurice-Étienne Cloutier, Les parties k-puissante et k-libre d'un nombre, Thèse de doctorat, Université Laval, Québec (2018).
Maurice-Étienne Cloutier, Jean-Marie De Koninck, and Nicolas Doyon, On the powerful and squarefree parts of an integer, Journal of Integer Sequences, Vol. 17 (2014), Article 14.6.6.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.11, pp. 31-32.

Cf. A055231, A370900, A370901, A379579 (numerators), A379582.

f[p_, e_] := If[e==1, p, 1]; powfree[n_] := Times @@ f @@@ FactorInteger[n]; Denominator[Accumulate[Table[1/powfree[n], {n, 1, 50}]]]

powfree(n) = {my(f = factor(n)); prod(i=1, #f~, if(f[i, 2] == 1, f[i, 1], 1)); }
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / powfree(k); print1(denominator(s), ", "))};

a(n) = denominator(Sum_{k=1..n} 1/A055231(k)).

A379581 Numerators of the partial alternating sums of the reciprocals of the powerfree part function (A055231).

Original entry on oeis.org

1, 1, 5, -1, 1, -2, 1, -104, 1, -19, 1, -769, -7687, -4916, -261, -1262, -20453, -57923, -1066503, -5979161, -17475593, -8958244, -201189767, -79457304, -42275159, -87410483, -13046193, -23669663, -612055937, -1025912126, -28568429291, -128848674356, -125809879051

Offset: 1

Amiram Eldar, Dec 26 2024

			Fractions begin with 1, 1/2, 5/6, -1/6, 1/30, -2/15, 1/105, -104/105, 1/105, -19/210, 1/2310, -769/2310, ...

Amiram Eldar, Table of n, a(n) for n = 1..1000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.11, pp. 31-32.

Cf. A055231, A328013, A370900, A370901, A379579, A379582 (denominators).

f[p_, e_] := If[e==1, p, 1]; powfree[n_] := Times @@ f @@@ FactorInteger[n]; Numerator[Accumulate[Table[(-1)^(n+1)/powfree[n], {n, 1, 50}]]]

powfree(n) = {my(f = factor(n)); prod(i=1, #f~, if(f[i, 2] == 1, f[i, 1], 1)); }
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / powfree(k); print1(numerator(s), ", "))};

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A055231(k)).

a(n)/A379582(n) = A * n^(1/2) + B * n^(1/3) + O(n^(1/5)), where A = ((9-12*sqrt(2))/23) * A328013, and B = ((2^(5/3) - 3*2^(1/3) - 1)/(2^(5/3) - 2^(1/3) + 1)) * (zeta(2/3)/zeta(2)) * Product_{p prime} (1 + (p^(1/3)-1)/(p*(p^(2/3)-p^(1/3)+1))) = 1.42776088919948241359... .

A379582 Denominators of the partial alternating sums of the reciprocals of the powerfree part function (A055231).

Original entry on oeis.org

1, 2, 6, 6, 30, 15, 105, 105, 105, 210, 2310, 2310, 30030, 15015, 1001, 1001, 17017, 34034, 646646, 3233230, 9699690, 4849845, 111546435, 37182145, 37182145, 74364290, 74364290, 74364290, 2156564410, 3234846615, 100280245065, 100280245065, 100280245065, 200560490130

Offset: 1

Amiram Eldar, Dec 26 2024

Amiram Eldar, Table of n, a(n) for n = 1..1000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.11, pp. 31-32.

Cf. A055231, A370900, A370901, A379580, A379581 (numerators).

f[p_, e_] := If[e==1, p, 1]; powfree[n_] := Times @@ f @@@ FactorInteger[n]; Denominator[Accumulate[Table[(-1)^(n+1)/powfree[n], {n, 1, 50}]]]

powfree(n) = {my(f = factor(n)); prod(i=1, #f~, if(f[i, 2] == 1, f[i, 1], 1)); }
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / powfree(k); print1(denominator(s), ", "))};

a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/A055231(k)).

A140394 Numbers n, satisfying A055231(n+1) - A055231(n) = 1, and with n and n+1 not squarefree.

Original entry on oeis.org

49, 1681, 18490, 23762, 39325, 57121, 182182, 453962, 656914, 843637, 1431125, 1608574, 1609674, 1940449, 2328482, 2948406, 3203050, 3721549, 5606230, 6352825, 8984002, 10000165, 13502254, 19326874, 19740249, 21006589, 26623750, 35558770, 38067925, 46297822

Offset: 1

Michel Lagneau, Dec 19 2011

nonn

There exists an infinite number of numbers that are divisible by a square and satisfy A055231(n+1) - A055231(n) = 1 because the Fermat-Pell equation 2x^2 - y^2 = 1 admits an infinite number of solutions.

			49 is in the sequence because A055231(50) - A055231(49) = A055231(2*5^2) - A055231(7^2) = 2 - 1 = 1;
18490 is in the sequence because A055231(18491) - A055231(18490) = A055231(11*41^2) -A055231(2*5*43^2)  = 11 - 10 = 1.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 49, p. 18, Ellipses, Paris 2008.

Cf. A007913, A013929, A055231, A068781.

isA013929 := proc(n)
    n>3 and not numtheory[issqrfree](n) ;
end proc:
isA140394 := proc(n)
    isA013929(n) and isA013929(n+1) and (A055231(n+1) -A055231(n) = 1)  ;
end proc:
for n from 1 do
    if isA140394(n) then
        print(n);
    end if;
end do: # R. J. Mathar, Dec 23 2011

rad[n_] := Times @@ First /@ FactorInteger[n]; pow[n_] := Denominator[n / rad[n]^2]; aQ[n_] := !SquareFreeQ[n] && !SquareFreeQ[n + 1] && pow[n + 1] - pow[n] == 1; Select[Range[10^6], aQ] (* Amiram Eldar, Oct 01 2019 *)

a(24)-a(30) from Amiram Eldar, Oct 01 2019

A368823 a(1) = 1; for n > 1, a(n) = A055231(a(n-1) * n), where A055231(k) is the powerfree part of k.

Original entry on oeis.org

1, 2, 6, 3, 15, 10, 70, 35, 35, 14, 154, 231, 3003, 858, 1430, 715, 12155, 24310, 461890, 46189, 969969, 176358, 4056234, 676039, 676039, 104006, 104006, 7429, 215441, 6463230, 200360130, 100180065, 367326905, 43214930, 60500902, 30250451, 1119266687, 117817546, 4594884294, 11487210735

Offset: 1

Scott R. Shannon, Jan 07 2024

nonn

Terms are squarefree. - Michael De Vlieger, Jan 07 2024.

			a(4) = 3 as a(3)*4 = 6*4 = 24 = 2*2*2*3, and the powerfree part of 24 is 3.

Cf. A368825 (addition), A055231, A005117, A124010.

f[x_] := Times @@ Map[#1^(#2 Boole[#2 == 1]) & @@ # &, FactorInteger[x]]; a[n_] := f[n* a[n - 1]]; a[1] = 1; Array[a, 120] (* Michael De Vlieger, Jan 07 2024 *)

cp's OEIS Frontend

A370900 Partial sums of the powerfree part function (A055231).

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A370901 Partial alternating sums of the powerfree part function (A055231).

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A328014 Numbers whose powerful part (A057521) is larger than their powerfree part (A055231).

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A368825 a(1) = 1; for n > 1, a(n) = A055231(a(n-1) + n), where A055231(k) is the powerfree part of k.

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A379579 Numerators of the partial sums of the reciprocals of the powerfree part function (A055231).

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A379580 Denominators of the partial sums of the reciprocals of the powerfree part function (A055231).

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A379581 Numerators of the partial alternating sums of the reciprocals of the powerfree part function (A055231).

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Mathematica

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A379582 Denominators of the partial alternating sums of the reciprocals of the powerfree part function (A055231).

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