A115063 -id:A115063 - cp's OEIS frontend

A375274 Decimal expansion of the asymptotic density of the exponentially Fibonacci numbers (A115063).

9, 4, 4, 3, 3, 5, 9, 0, 5, 0, 6, 4, 0, 6, 3, 3, 2, 4, 4, 8, 0, 5, 7, 3, 1, 3, 7, 7, 5, 6, 6, 6, 8, 8, 0, 5, 6, 1, 4, 6, 3, 4, 5, 8, 3, 2, 2, 2, 0, 2, 3, 5, 5, 5, 9, 2, 3, 6, 8, 3, 7, 7, 0, 4, 5, 5, 9, 3, 9, 5, 3, 8, 4, 6, 5, 4, 4, 6, 8, 5, 8, 7, 1, 9, 4, 1, 4, 2, 8, 0, 5, 2, 0, 3, 3, 7, 9, 2, 7, 4, 7, 9, 7, 2, 4

Offset: 0

Amiram Eldar, Aug 09 2024

This constant was apparently first calculated by Juan Arias-de-Reyna and Peter J. C. Moses in 2015 (see A115063).

			0.94433590506406332448057313775666880561463458322202...

Cf. A000045, A010056, A115063, A259623.

$MaxExtraPrecision = m = 500; em = 16; f[x_] := Log[(1 - x) * (1 + Sum[x^Fibonacci[e], {e, 2, em}])]; c = Rest[CoefficientList[Series[f[x], {x, 0, m}], x] * Range[0, m]]; RealDigits[Exp[NSum[Indexed[c, k]*PrimeZetaP[k]/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 105][[1]]

c(imax) = prodeulerrat((1-1/p)*(1 + sum(i = 2, imax, 1/p^fibonacci(i))));
f(prec) = {default(realprecision, prec); my(k = 2, c1 = 0, c2 = c(k)); while(c1 != c2, k++; c1 = c2; c2 = c(k)); c1;}
f(120)

Equals Product_{p prime} (1 + Sum_{i>=2} (u(i) - u(i-1))/p^i), where u(i) = A010056(i) is the characteristic function of the Fibonacci numbers (A000045) (first formula at A115063).
Equals Product_{p prime} (1 + Sum_{i>=4} (-1)^(i+1)/p^A259623(i)).
Equals Product_{p prime} ((1 - 1/p) * (1 + Sum_{i>=2} 1/p^Fibonacci(i))).

A209061 Exponentially squarefree numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69

Offset: 1

Reinhard Zumkeller, Mar 13 2012

nonn

Numbers having only squarefree exponents in their canonical prime factorization.
According to the formula of Theorem 3 [Toth], the density of the exponentially squarefree numbers is 0.9559230158619... (A262276). - Peter J. C. Moses and Vladimir Shevelev, Sep 10 2015
From Vladimir Shevelev, Sep 24 2015: (Start)
A generalization. Let S be a finite or infinite increasing integer sequence s=s(n), s(0)=0.
Let us call a positive number N an exponentially S-number, if all exponents in its prime power factorization are in the sequence S.
Let {u(n)} be the characteristic function of S. Then, for the density h=h(S) of the exponentially S-numbers, we have the representations
h(S) = Product_{prime p} Sum_{j in S} (p-1)/p^(j+1) = Product_{p} (1 + Sum_{j>=1} (u(j) - u(j-1))/p^j). In particular, if S = {0,1}, then the exponentially S-numbers are squarefree numbers; if S consists of 0 and {2^k}A138302%20(see%20%5BShevelev%5D,%202007);%20if%20S%20consists%20of%200%20and%20squarefree%20numbers,%20then%20u(n)=%7Cmu(n)%7C,%20where%20mu(n)%20is%20the%20M%C3%B6bius%20function%20(A008683),%20we%20obtain%20the%20density%20h%20of%20the%20exponentially%20squarefree%20numbers%20(cf.%20Toth's%20link,%20Theorem%203);%20the%20calculation%20of%20h%20with%20a%20very%20high%20degree%20of%20accuracy%20belongs%20to%20_Juan%20Arias-de-Reyna">{k>=0}, then the exponentially S-numbers form A138302 (see [Shevelev], 2007); if S consists of 0 and squarefree numbers, then u(n)=|mu(n)|, where mu(n) is the Möbius function (A008683), we obtain the density h of the exponentially squarefree numbers (cf. Toth's link, Theorem 3); the calculation of h with a very high degree of accuracy belongs to _Juan Arias-de-Reyna (A262276). Note that if S contains 1, then h(S) >= 1/zeta(2) = 6/Pi^2; otherwise h(S) = 0. Indeed, in the latter case, the density of the sequence of exponentially S-numbers does not exceed the density of A001694, which equals 0. (End)
The term "exponentially squarefree number" was apparently coined by Subbarao (1972). - Amiram Eldar, May 28 2025

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Xiaodong Cao and Wenguang Zhai, Some arithmetic functions involving exponential divisors, JIS 13 (2010), Article 10.3.7.
Y.-F. S. Petermann, Arithmetical functions involving exponential divisors: note on two papers by L. Toth, Ann. Univ. Sci. Budapest, Sect. Comp. 32 (2010) 143-149.
Vladimir Shevelev, Compact integers and factorials, Acta Arithmetica 126:3 (2007), pp. 195-236.
Vladimir Shevelev, Exponentially S-numbers, arXiv:1510.05914 [math.NT], 2015.
Vladimir Shevelev, Set of all densities of exponentially S-numbers, arXiv:1511.03860 [math.NT], 2015.
Vladimir Shevelev, A fast computation of density of exponentially S-numbers, arXiv:1602.04244 [math.NT], 2016.
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175(2016), 385-395.
H. M. Stark, On the asymptotic density of the k-free integers, Proc. Amer. Soc. 17 (1966), 1211-1214.
M. V. Subbarao, On some arithmetic convolutions, in: A. A. Gioia and D. L. Goldsmith (eds.), The Theory of Arithmetic Functions: Proceedings of the Conference at Western Michigan University, April 29 - May 1, 1971, Berlin, Heidelberg: Springer Berlin Heidelberg, 2006, pp. 247-271; alternative link.
László Tóth, On certain arithmetic functions involving exponential divisors, II., Annales Univ. Sci. Budapest., Sect. Comp., 27 (2007), 155-166 and arXiv:0708.3557 [math.NT], 2007-2009.

Complement of A130897.
A005117, A004709, and A046100 are subsequences.
Cf. A001221, A001694, A008683, A008966, A036537, A115063, A138302, A197680, A262276, A262675, A268335, A270428.

a209061 n = a209061_list !! (n-1)
a209061_list = filter
   (all (== 1) . map (a008966 . fromIntegral) . a124010_row) [1..]

Select[Range@ 69, Times @@ Boole@ Map[SquareFreeQ, Last /@ FactorInteger@ #] > 0 &] (* Michael De Vlieger, Sep 07 2015 *)

is(n)=my(f=factor(n)[,2]); for(i=1,#f,if(!issquarefree(f[i]), return(0))); 1 \\ Charles R Greathouse IV, Sep 02 2015

A166234(a(n)) <> 0.
Product_{k=1..A001221(n)} A008966(A124010(n,k)) = 1.
One can prove that the principal term of Toth's asymptotics for the density of this sequence (cf. Toth's link, Theorem 3) equals also Product_{prime p}(Sum_{j in S}(p-1)/p^{j+1})*x, where S is the set of 0 and squarefree numbers. The remainder term O(x^(0.2+t)), where t>0 is arbitrarily small, was obtained by L. Toth while assuming the Riemann Hypothesis. - Vladimir Shevelev, Sep 12 2015

A197680 Numbers whose exponents in their prime power factorization are squares.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 48, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101

Offset: 1

A. Neves, Oct 17 2011

nonn

Numbers whose prime factorization has the form Product_i p_i^e_i where the e_i are all squares.
All squarefree numbers (A005117) are in the sequence. - Vladimir Shevelev, Nov 16 2015
Let h_k be the density of the subsequence of A197680 of numbers whose prime power factorization (PPF) has the form Product_i p_i^e_i where the e_i all squares <= k^2. Then for every k>1 there exists eps_k>0 such that for any x from the interval (h_k-eps_k, h_k) there is no sequence S of positive integers such that x is the density of numbers whose PPF has the form Product_i p_i^e_i where the e_i are all in S. - For a proof, see [Shevelev], second link. - Vladimir Shevelev, Nov 17 2015
Numbers with an odd number of exponential divisors (A049419). - Amiram Eldar, Nov 05 2021

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Math StackExchange, Question 73354, 2011.
Vladimir Shevelev, Exponentially S-numbers, arXiv:1510.05914 [math.NT], 2015.
Vladimir Shevelev, Set of all densities of exponentially S-numbers, arXiv:1511.03860 [math.NT], 2015.
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175 (2016), pp. 385-395.

Cf. A000290, A049419, A005117, A115063, A209061.

a:= proc(n) option remember; local k; for k from 1+
      `if`(n=1, 0, a(n-1)) while 0=mul(`if`(issqr(
       i[2]), 1, 0), i=ifactors(k)[2]) do od; k
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Jun 30 2016

Select[Range[100], Union[IntegerQ /@ Sqrt[Transpose[FactorInteger[#]][[2]]]][[1]] &] (* T. D. Noe, Oct 18 2011 *)

isok(n) = {my(f = factor(n)[,2]); #select(x->issquare(x), f) == #f; } \\ Michel Marcus, Oct 23 2015

Sum_{i<=x, i is in A197680} 1 = h*x + O(sqrt(x)*log x*e^(c*sqrt(log x)/(log(log x))), where c=4*sqrt(2.4/log 2)=7.44308... and h=Product_{prime p} (1+Sum_{i>=2} (u(i)-u(i-1))/p^i)=0.641115... where u(n) is the characteristic function of sequence A000290. The calculations of h in the formula were done independently by Juan Arias-de-Reyna and Peter J. C. Moses. For a proof of the formula, see the first Shevelev link. - Vladimir Shevelev, Nov 17 2015

Reformulation of the name by Vladimir Shevelev, Oct 14 2015

A369939 Numbers whose maximal exponent in their prime factorization is a Fibonacci number.

Original entry on oeis.org

Offset: 1

Amiram Eldar, Feb 06 2024

First differs from its subsequence A115063 at n = 2448. a(2448) = 2592 = 2^5 * 3^4 is not a term of A115063.
First differs from A209061 at n = 62.
Numbers k such that A051903(k) is a Fibonacci number.
The asymptotic density of this sequence is 1/zeta(4) + Sum_{k>=5} (1/zeta(Fibonacci(k)+1) - 1/zeta(Fibonacci(k))) = 0.94462177878047854647... .

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

Cf. A000045, A013662, A051903, A209061.
Subsequences: A005117, A062503, A062838, A113850, A115063.
Similar sequences: A368714, A369937, A369938.

fibQ[n_] := Or @@ IntegerQ /@ Sqrt[5*n^2 + {-4, 4}];
Select[Range[100], fibQ[Max[FactorInteger[#][[;; , 2]]]] &]

isfib(n) = issquare(5*n^2 - 4) || issquare(5*n^2 + 4);
is(n) = n == 1 || isfib(vecmax(factor(n)[, 2]));

A386803 Numbers without an exponent 4 in their prime factorization.

Original entry on oeis.org

Offset: 1

Amiram Eldar, Aug 03 2025

First differs from its subsequence A209061 at n = 246: a(246) = 256 = 2^8 is not a term of A209061.
First differs from its subsequences A115063 and A369939 at n = 62: a(62) = 64 = 2^6 is not a term of A115063.
The complement of this sequence is a subsequence of A336595.
These numbers were named semi-4-free integers by Suryanarayana (1971).
The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^4 + 1/p^5) = 0.95908865419555719109... (Suryanarayana, 1971).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ertan Elma and Greg Martin, Distribution of the number of prime factors with a given multiplicity, Canadian Mathematical Bulletin, Vol. 67, No. 4 (2024), pp. 1107-1122; arXiv preprint, arXiv:2406.04574 [math.NT], 2024.
D. Suryanarayana, Semi-k-free integers, Elemente der Mathematik, Vol. 26 (1971), pp. 39-40.
D. Suryanarayana and R. Sitaramachandra Rao, Distribution of semi-k-free integers, Proceedings of the American Mathematical Society, Vol. 37, No. 2 (1973), pp. 340-346.
Index entries for sequences computed from indices in prime factorization.

Subsequences: A115063, A209061, A369939.
Numbers without an exponent k in their prime factorization: A001694 (k=1), A337050 (k=2), A386799 (k=3), this sequence (k=4), A386807 (k=5).
Numbers that have exactly m exponents in their prime factorization that are equal to 4: this sequence (m=0), A386804 (m=1), A386805 (m=2), A386806 (m=3).

Select[Range[100], !MemberQ[FactorInteger[#][[;; , 2]], 4] &]

isok(k) = vecsum(apply(x -> if(x == 4, 1, 0), factor(k)[, 2])) == 0;

A384913 The number of unordered factorizations of n into exponentially Fibonacci powers of primes (A115975).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 8, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 12 2025

First differs from A384912 at n = 64.

			a(4) = 2 since 4 has 2 factorizations: 2^1 * 2^1 and 2^2, with exponents 1 and 2 that are Fibonacci numbers.

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

Cf. A000045, A003107, A115063, A115975.

Similar sequences: A000688, A046951, A050361, A050377, A188581, A188585, A322885, A370256, A384912, A384914, A384915, A384916.

fib[n_] := Boole[Or @@ IntegerQ /@ Sqrt[5*n^2 + {-4, 4}]];
s[n_] := s[n] = If[n == 0, 1, Sum[Sum[d * fib[d], {d, Divisors[j]}] * s[n-j], {j, 1, n}] / n];
f[p_, e_] := s[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

isfib(n) = issquare(5*n^2 - 4) || issquare(5*n^2 + 4);
s(n) = if(n < 1, 1, sum(j = 1, n, sumdiv(j, d, d*isfib(d)) * s(n-j))/n);
a(n) = vecprod(apply(s, factor(n)[, 2]));

Multiplicative with a(p^e) = A003107(e).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 2.05893526314055968638..., where f(x) = (1-x) / Product_{k>=2} (1-x^A000045(k)).

A375766 The maximum exponent in the prime factorization of the numbers whose exponents in their prime factorization are all Fibonacci numbers.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 5, 1, 2, 2, 2, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1

Offset: 1

Amiram Eldar, Aug 27 2024

First differs from A375768 at n = 2448.

All the terms are Fibonacci numbers by definition.

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

Cf. A000045, A051903, A115063, A375274, A375767, A375768.

fibQ[n_] := Or @@ IntegerQ /@ Sqrt[5*n^2 + {-4, 4}]; s[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, fibQ], Max[e], Nothing]]; s[1] = 0; Array[s, 100]

isfib(n) = issquare(5*n^2 - 4) || issquare(5*n^2 + 4);
lista(kmax) = {my(e, ans); print1(0, ", "); for(k = 2, kmax, e = factor(k)[,2]; ans = 1; for(i = 1, #e, if(!isfib(e[i]), ans = 0; break)); if(ans, print1(vecmax(e), ", ")));}

a(n) = A051903(A115063(n)).
a(n) = A000045(A375767(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1/zeta(2) + Sum_{k>=3} (Fibonacci(k) * (d(k) - d(k-1)))) / A375274 = 1.52546070254904121983..., where d(k) = Product_{p prime} ((1-1/p)*(1 + Sum_{i=2..k} 1/p^Fibonacci(i))) for k >= 3, and d(2) = 1/zeta(2).

A115105 Numbers of the form p^F(i)*q^F(j), where p and q are distinct primes; F(i) and F(j) are Fibonacci numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 82, 83

Offset: 1

Giovanni Teofilatto, Mar 03 2006; corrected Mar 04 2006

nonn

All the primes are in the sequence, since they are of the form p^F(0)*q^F(1) with p^F(0) = p^0 = 1.

			a(1) = 1 because 2^0*3^0 = 1;
a(3) = 3 because 3^1*5^0 = 3;
a(4) = 4 because 2^2*3^0 = 4;
a(6) = 6 because 2^1*3^1 = 6.

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

Cf. A115063.

fibQ[n_] := IntegerQ @ Sqrt[5 n^2 - 4] || IntegerQ @ Sqrt[5 n^2 + 4]; aQ[n_] :=Length[ (e=FactorInteger[n][[;;, 2]])]<3 && AllTrue[e, fibQ]; Select[Range[100], aQ] (* Amiram Eldar, Oct 06 2019 *)

2 missing terms inserted by Amiram Eldar, Oct 06 2019

cp's OEIS Frontend

A375274 Decimal expansion of the asymptotic density of the exponentially Fibonacci numbers (A115063).

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A209061 Exponentially squarefree numbers.

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A197680 Numbers whose exponents in their prime power factorization are squares.

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A369939 Numbers whose maximal exponent in their prime factorization is a Fibonacci number.

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A386803 Numbers without an exponent 4 in their prime factorization.

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A384913 The number of unordered factorizations of n into exponentially Fibonacci powers of primes (A115975).

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A375766 The maximum exponent in the prime factorization of the numbers whose exponents in their prime factorization are all Fibonacci numbers.

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