A051903 -id:A051903 - cp's OEIS frontend

A375932 The largest unitary k-free divisor of n where k = A051903(n) is the maximum exponent in the prime factorization of n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 5, 1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 11, 5, 1, 1, 3, 1, 2, 1, 13, 1, 2, 1, 7, 1, 1, 1, 15, 1, 1, 7, 1, 1, 1, 1, 17, 1, 1, 1, 9, 1, 1, 3, 19, 1, 1, 1, 5, 1, 1, 1, 21, 1

Offset: 1

Amiram Eldar, Sep 03 2024

The product of the prime powers in the prime factorization of n that have an exponent that is smaller than the maximum exponent in this factorization.

			60 = 2^2 * 3 * 5, and the maximum exponent in the prime factorization of 60 is 2, which is the exponent of its prime factor 2. Therefore a(60) = 3 * 5 = 15.

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

Cf. A005117, A049606, A051903, A072774, A375931, A375933.

a[n_] := Module[{f = FactorInteger[n], p, e, i, m}, p = f[[;; , 1]]; e = f[[;; , 2]]; m = Max[e]; i = Position[e, m] // Flatten; n / (Times @@ p[[i]])^m]; Array[a, 100]

a(n) = {my(f = factor(n), p = f[,1], e = f[,2], m); if(n == 1, 1, m = vecmax(e); prod(i = 1, #p, if(e[i] < m, p[i]^e[i], 1)));}

If n = Product_{i} p_i^e_i (where p_i are distinct primes) then a(n) = Product_{i} p_i^(e_i * [e_i < max_{j} e_j]), where [] is the Iverson bracket.
a(n) = n / A375931(n).
a(n) = 1 if and only if n is a power of a squarefree number (A072774).
A051903(a(n)) = A375933(n).
a(n!) = A049606(n) for n != 3.

A328385 If n is of the form p^p, a(n) = n, otherwise a(n) is the first number found by iterating the map x -> A003415(x) that is different from n and either a prime, or whose degree (A051903) differs from the degree of n.

Original entry on oeis.org

0, 1, 1, 4, 1, 5, 1, 12, 6, 7, 1, 16, 1, 9, 8, 32, 1, 21, 1, 24, 7, 13, 1, 44, 10, 8, 27, 32, 1, 31, 1, 80, 9, 19, 12, 96, 1, 7, 16, 68, 1, 41, 1, 48, 39, 25, 1, 608, 14, 39, 20, 56, 1, 81, 16, 92, 13, 31, 1, 96, 1, 9, 51, 640, 18, 61, 1, 72, 8, 59, 1, 156, 1, 16, 55, 80, 18, 71, 1, 3424, 108, 43, 1, 128, 13, 45, 32, 140, 1, 123, 20, 96, 19

Offset: 1

Antti Karttunen, Oct 14 2019

nonn

			For n = 3, 3 is a prime, thus a(3) = 1.
For n = 4, A003415(4) = 4, thus as it is among the fixed points of A003415 and a(4) = 4.
For n = 8 = 2^3, its "degree" is A051903(33) = 3, but A003415(8) = 12 = 2^2 * 3, with degree 2, thus a(8) = 12.
For n = 21 = 3*7, A051903(21) = 1, the first derivative A003415(21) = 10 = 2*5 is of the same degree as A051903(10) = 1, but then continuing, we have A003415(10) = 7, which is a prime, thus a(21) = 7.
For n = 33 = 3*11, A051903(33) = 1, A003415(33) = 14 = 2*7, is of the same degree, but on the second iteration, A003415(14) = 9 = 3^2, with A051903(9) = 2, different from the initial degree, thus a(33) = 9.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003415, A051674, A051903, A157037.
Cf. A328384 (the number of iterations needed to reach such a number).
Cf. also A327968, A327975, A327977, A328252, A328305, A328310, A328311, A328320, A328321.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A328385(n) = { my(d=A051903(n), u=A003415(n)); while(u && (u!=n) && !isprime(u) && A051903(u)==d, n = u; u = A003415(u)); (u); };

a(1) = 0 [as here the degrees of 0 and 1 are considered different].
a(p) = 1 for all primes.
a(A051674(n)) = A051674(n).
a(A157037(n)) = A003415(A157037(n)), a prime.
a(A328252(n)) = A003415(A328252(n)), a squarefree number.
a(n) = A003415^(k)(n), when k = abs(A328384(n)). [Taking the abs(A328384(n))-th arithmetic derivative of n gives a(n)]

A329339 a(n) = 2^A051903(n)Sum_{k=0..n-1} 2^(A268336(n)k): upper bound for A329000(n).

Original entry on oeis.org

1, 6, 42, 60, 139810, 126, 139620524162, 2040, 349524, 699050, 2537779500750160131246576896002, 16380, 44612382091907903486070965589630128805126146, 1256584717458, 153722867280912930, 1048560, 231587712222682663714935471840371426842813815977643091627066215779128553111554, 1048572

Offset: 1

M. F. Hasler, Nov 13 2019

nonn

This corresponds to the upper bound for A329000 as explained in the "FORMULA" for A329126.

Differs from A329000(n) for n = 10, 14, 15, ...

Cf. A329000, A329126, A329338 (a(n) written in binary), A067029, A051903.

apply( A329339(n)={my(m=2^(lcm(lcm(znstar(n)[2]),n)/n)); (m^n-1)\(m-1)<1,vecmax(factor(n)[,2]))}, [1..20])

a(n) = 2^A051903(n)*(m^n-1)/(m-1) with m = 2^A268336(n).

A329885 a(n) = A051903(n) mod A002322(n), where A051903 gives the maximal prime exponent of n, and A002322 is Carmichael's lambda (also known as psi).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 0, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 0, 4, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 5, 1, 2, 2, 2, 1, 1, 1, 3, 1

Offset: 1

Antti Karttunen, Dec 11 2019

nonn

This differs from A051903 at n = 2, 4, 8, 12, 16, 24, 48, 80, 240. Are there any other such n? (None other found <= 201326592.)

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A002322, A051903, A327295.

A002322(n) = lcm(znstar(n)[2]); \\ From A002322
A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A329885(n) = (A051903(n)%A002322(n));

A342003 Maximal exponent in the prime factorization of the arithmetic derivative of n: a(n) = A051903(A003415(n)).

Original entry on oeis.org

0, 0, 2, 0, 1, 0, 2, 1, 1, 0, 4, 0, 2, 3, 5, 0, 1, 0, 3, 1, 1, 0, 2, 1, 1, 3, 5, 0, 1, 0, 4, 1, 1, 2, 2, 0, 1, 4, 2, 0, 1, 0, 4, 1, 2, 0, 4, 1, 2, 2, 3, 0, 4, 4, 2, 1, 1, 0, 2, 0, 1, 1, 6, 2, 1, 0, 3, 1, 1, 0, 2, 0, 1, 1, 4, 2, 1, 0, 4, 3, 1, 0, 2, 1, 2, 5, 2, 0, 1, 2, 5, 1, 2, 3, 4, 0, 1, 2, 2, 0, 1, 0, 2, 1

Offset: 2

Antti Karttunen, Mar 01 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 2..65537

Cf. A003415, A051903, A328310, A328311, A328320, A328321, A341994, A341995, A342004.

Cf. A000040 (indices of zeros), A328234 (of ones), A328393 (of the terms < 2).

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A342003(n) = A051903(A003415(n));

a(n) = A051903(A003415(n)).
a(n) = A051903(n) + A328310(n).
a(n) = 1 iff A341994(n) = 1.

A359071 Numerators of the partial sums of the reciprocals of the maximal exponent in prime factorization of the positive integers (A051903).

Original entry on oeis.org

1, 2, 5, 7, 9, 11, 35, 19, 22, 25, 53, 59, 65, 71, 145, 157, 163, 175, 181, 193, 205, 217, 221, 227, 239, 81, 83, 87, 91, 95, 479, 499, 519, 539, 549, 569, 589, 609, 1847, 1907, 1967, 2027, 2057, 2087, 2147, 2207, 1111, 563, 1141, 1171, 593, 608, 613, 628, 211

Offset: 2

Amiram Eldar, Dec 15 2022

			Fractions begin with 1, 2, 5/2, 7/2, 9/2, 11/2, 35/6, 19/3, 22/3, 25/3, 53/6, 59/6, ...

Amiram Eldar, Table of n, a(n) for n = 2..10000
Wolfgang Schwarz and Jürgen Spilker, A remark on some special arithmetical functions, in: E. Laurincikas , E. Manstavicius and V. Stakenas (eds.), Analytic and Probabilistic Methods in Number Theory, Proceedings of the Second International Conference in Honour of J. Kubilius, Palanga, Lithuania, 23-27 September 1996, New Trends in Probability and Statistics, Vol. 4, VSP BV & TEV Ltd. (1997), pp. 221-245.
D. Suryanarayana and R. Chandra Rao, On the maximum and minimum exponents in factoring integers, Archiv der Mathematik, Vol. 28, No. 1 (1977), pp. 261-269.
Index entries for sequences computed from exponents in factorization of n.

Cf. A051903, A129132, A242977, A359072 (denominators).

f[n_] := Max[FactorInteger[n][[;; , 2]]]; f[1] = 0; Numerator[Accumulate[Table[1/f[n], {n, 2, 100}]]]

a(n) = numerator(Sum_{k=2..n} 1/A051903(k)).

a(n)/A359072(n) = c_1 * n + O(n^(1/2)*exp(-c_2*log(n)^(3/5)/log(log(n))^(1/5))), where c_1 = A242977 and c_2 is a constant, 0 < c_2 < 1/2^(8/5) (Suryanarayana and R. Chandra Rao, 1977).

A359072 Denominators of the partial sums of the reciprocals of the maximal exponent in prime factorization of the positive integers (A051903).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 6, 3, 3, 3, 6, 6, 6, 6, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 4, 4, 4, 4, 4, 20, 20, 20, 20, 20, 20, 20, 20, 60, 60, 60, 60, 60, 60, 60, 60, 30, 15, 30, 30, 15, 15, 15, 15, 5, 5, 5, 5, 10, 10, 10, 5, 30, 30, 30, 30, 15, 15, 15, 15, 5, 5

Offset: 2

Amiram Eldar, Dec 15 2022

Amiram Eldar, Table of n, a(n) for n = 2..10000
Wolfgang Schwarz and Jürgen Spilker, A remark on some special arithmetical functions, in: E. Laurincikas, E. Manstavicius and V. Stakenas (eds.), Analytic and Probabilistic Methods in Number Theory, Proceedings of the Second International Conference in Honour of J. Kubilius, Palanga, Lithuania, 23-27 September 1996, New Trends in Probability and Statistics, Vol. 4, VSP BV & TEV Ltd. (1997), pp. 221-245.
D. Suryanarayana and R. Chandra Rao, On the maximum and minimum exponents in factoring integers, Archiv der Mathematik, Vol. 28, No. 1 (1977), pp. 261-269.
Index entries for sequences computed from exponents in factorization of n.

Cf. A051903, A129132, A359071 (numerators).

f[n_] := Max[FactorInteger[n][[;; , 2]]]; f[1] = 0; Denominator[Accumulate[Table[1/f[n], {n, 2, 100}]]]

a(n) = denominator(Sum_{k=2..n} 1/A051903(k)).

A093770 Non-perfect powers k for which q = A051903(k)/A051904(k) is an integer, A051904(k) > 1.

Original entry on oeis.org

10800, 16200, 18000, 21168, 31752, 40500, 43200, 45000, 49392, 52272, 67500, 72000, 73008, 78408, 84672, 98000, 109512, 111132, 124848, 137200, 145800, 155952, 172800, 172872, 187272, 191664, 197568, 209088, 228528, 233928, 242000

Offset: 1

Labos Elemer, Apr 16 2004

nonn

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

Cf. A001597, A051903, A051904, A093769.

ffi[x_] :=Flatten[FactorInteger[x]] ep[x_] :=Table[Part[ffi[x], 2*w], {w, 1, lf[x]}] lf[x_] :=Length[FactorInteger[x]] Do[s1=Min[ep[n]];s2=Max[ep[n]]; If[ !Equal[s1, 1]&&IntegerQ[q=(s2/s1)]&& Equal[Union[Table[IntegerQ[Part[ep[n], j]/s1], {j, 1, lf[n]}]], {False, True}], Print[n]], {n, 2, 10000000}]

Solutions to integer values of q = A051903(x)/A051904(x), when A051904(x) > 1.

A304768 Augmented integer conjugate of n. a(n) = (1/n) * A007947(n)^(1 + A051903(n)) where A007947 is squarefree kernel and A051903 is maximum prime exponent.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 18, 13, 14, 15, 2, 17, 12, 19, 50, 21, 22, 23, 54, 5, 26, 3, 98, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 250, 41, 42, 43, 242, 75, 46, 47, 162, 7, 20, 51, 338, 53, 24, 55, 686, 57, 58, 59, 450, 61, 62, 147, 2, 65, 66, 67

Offset: 1

Gus Wiseman, May 18 2018

nonn

Image is the weak numbers A052485, on which n -> a(n) is an involution whose fixed points are the squarefree numbers A005117.

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

Cf. A001597, A001694, A005117, A007916, A007947, A013929, A051903, A052410, A062759, A066638, A072774, A087320, A303554.

acj[n_]:=Module[{f,m},f=FactorInteger[n];m=Max[Last/@f];Times@@Table[p[[1]]^(m-p[[2]]+1),{p,f}]];
Array[acj,100]

a(n) = {if(n==1, 1, my(f = factor(n), e = vecmax(f[,2]) + 1); prod(i = 1, #f~, f[i,1]^e) / n);} \\ Amiram Eldar, Feb 12 2023

If n = Product_{i = 1..k} prime(x_i)^y_i, then a(n) = Product_{i = 1..k} prime(x_i)^(max{y_1,...,y_k} - y_i + 1).

A369645 Numbers k for which the difference A051903(k) - A328114(k) reaches a new maximum in range 1..k, where A051903 is the maximal exponent in the prime factorization of n, and A328114 is the maximal digit in the primorial base expansion of n.

Original entry on oeis.org

1, 2, 8, 32, 256, 2560, 30720, 32768, 4194304, 20971520, 58720256, 234881024, 536870912, 1342177280

Offset: 1

Antti Karttunen, Feb 01 2024

			           k   factorization   max.exp.  in primorial  max digit  diff
                                             base
           1                       0,            1,       1,      -1
           2 = 2^1,                1,           10,       1,       0
           8 = 2^3,                3,          110,       1,       2
          32 = 2^5,                5,         1010,       1,       4
         256 = 2^8,                8,        11220,       2,       6
        2560 = 2^9 * 5^1,          9,       111120,       2,       7
       30720 = 2^11 * 3^1 * 5^1,  11,      1032000,       3,       8
       32768 = 2^15,              15,      1120110,       2,      13
     4194304 = 2^22,              22,     83876020,       8,      14
    20971520 = 2^22 * 5^1,        22,    231462310,       6,      16
    58720256 = 2^23 * 7^1,        23,    610501410,       6,      17
   234881024 = 2^25 * 7^1,        25,   1141710210,       7,      18
   536870912 = 2^29,              29,   296AA71010,      10,      19
  1342177280 = 2^28 * 5^1,        28,   6071712310,       7,      21.
On the penultimate row, letter "A" in the primorial base expansion stands for ten (10 in decimal), as 2^29 = 0*prime(0)# + 1*prime(1)# + 0*prime(2)# + 1*prime(3)# + 7*prime(4)# + 10*prime(5)# + 10*prime(6)# + 6*prime(7)# + 9*prime(8)# + 2*prime(9)#, where prime(n)# = A002110(n).

Index entries for sequences related to primorial base

Positions of records for -A350074(n).
Cf. A002110, A049345, A051903, A328114.
Cf. also A369646, A369647.
After the initial 1, subsequence of A351038, after the two initial terms, subsequence of A350075.

A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A328114(n) = { my(s=0, p=2); while(n, s = max(s, (n%p)); n = n\p; p = nextprime(1+p)); (s); };
A350074(n) = (A328114(n) - A051903(n));
m=A350074(1); print1(1,", "); for(n=2,oo,x=A350074(n); if(x

cp's OEIS Frontend

A375932 The largest unitary k-free divisor of n where k = A051903(n) is the maximum exponent in the prime factorization of n.

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A328385 If n is of the form p^p, a(n) = n, otherwise a(n) is the first number found by iterating the map x -> A003415(x) that is different from n and either a prime, or whose degree (A051903) differs from the degree of n.

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A329339 a(n) = 2^A051903(n)*Sum_{k=0..n-1} 2^(A268336(n)*k): upper bound for A329000(n).

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A329885 a(n) = A051903(n) mod A002322(n), where A051903 gives the maximal prime exponent of n, and A002322 is Carmichael's lambda (also known as psi).

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A342003 Maximal exponent in the prime factorization of the arithmetic derivative of n: a(n) = A051903(A003415(n)).

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A359071 Numerators of the partial sums of the reciprocals of the maximal exponent in prime factorization of the positive integers (A051903).

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A359072 Denominators of the partial sums of the reciprocals of the maximal exponent in prime factorization of the positive integers (A051903).

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A093770 Non-perfect powers k for which q = A051903(k)/A051904(k) is an integer, A051904(k) > 1.

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A304768 Augmented integer conjugate of n. a(n) = (1/n) * A007947(n)^(1 + A051903(n)) where A007947 is squarefree kernel and A051903 is maximum prime exponent.

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A329339 a(n) = 2^A051903(n)Sum_{k=0..n-1} 2^(A268336(n)k): upper bound for A329000(n).