A051904 -id:A051904 - cp's OEIS frontend

A157754 a(1) = 0, a(n) = lcm(A051904(n), A051903(n)) for n >= 2.

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

Offset: 1

Jaroslav Krizek, Mar 05 2009

nonn

a(n) for n >= 2 equals LCM of minimum and maximum exponents in the prime factorization of n.

a(n) for n >= 2 deviates from A072411, first different term is a(360), a(360) = 3, A072411(360) = 6.

			For n = 12 = 2^2 * 3^1 we have a(12) = lcm(2,1) = 2.
For n = 144 = 2^4 * 3^2 we have a(144) = lcm(4,2) = 4.

Cf. A000040, A003990, A006881, A033150, A120944, A000961, A000027, A072411, A051904, A051903, A158378.

Table[LCM @@ {Min@ #, Max@ #} - Boole[n == 1] &@ FactorInteger[n][[All, -1]], {n, 100}] (* Michael De Vlieger, Jul 12 2017 *)

a(n) = if(n == 1, 0, my(e = factor(n)[,2]); lcm(vecmin(e), vecmax(e))); \\ Amiram Eldar, Sep 11 2024

a(1) = 0, a(p) = 1, a(pq) = 1, a(pq...z) = 1, a(p^k) = k, for p = primes (A000040), pq = product of two distinct primes (A006881), pq...z = product of k (k > 2) distinct primes p, q, ..., z (A120944), p^k = prime powers (A000961(n) for n > 1) k = natural numbers (A000027).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A033150. - Amiram Eldar, Sep 11 2024

A158378 a(1) = 0, a(n) = gcd(A051904(n), A051903(n)) for n >= 2.

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Mar 17 2009

nonn

a(n) for n >= 2 equals GCD of minimum and maximum exponents in the prime factorization of n.

a(n) for n >= 2 it deviates from A052409(n), first different term is a(10800) = a(2^4*3^3*5^2), a(10800) = gcd(2,4) = 2, A052409(10800) = gcd(2,3,4) = 1.

			For n = 12 = 2^2 * 3^1 we have a(12) = gcd(2,1) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..65536
Index entries for sequences computed from exponents in factorization of n.

Cf. A157754, A051904, A051903, A052409.

Table[GCD @@ {Min@ #, Max@ #} - Boole[n == 1] &@ FactorInteger[n][[All, -1]], {n, 100}] (* Michael De Vlieger, Jul 12 2017 *)

A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A051904(n) = if((1==n),0,vecmin(factor(n)[, 2]));
A158378(n) = gcd(A051903(n),A051904(n)); \\ Antti Karttunen, Jul 12 2017

a(n) = if(n == 1, 0, my(e = factor(n)[,2]); gcd(vecmin(e), vecmax(e))); \\ Amiram Eldar, Sep 11 2024

For n >= 2 holds: a(n)*A157754(n) = A051904(n)*A051903(n).
a(1) = 0, a(p) = 1, a(pq) = 1, a(pq...z) = 1, a(p^k) = k, for p = primes (A000040), pq = product of two distinct primes (A006881), pq...z = product of k (k > 2) distinct primes p, q, ..., z (A120944), p^k = prime powers (A000961(n) for n > 1), k = natural numbers (A000027).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1. - Amiram Eldar, Sep 11 2024

A304776 A weakening function. a(n) = n / A007947(n)^(A051904(n) - 1) where A007947 is squarefree kernel and A051904 is minimum prime exponent.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 12, 13, 14, 15, 2, 17, 18, 19, 20, 21, 22, 23, 24, 5, 26, 3, 28, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 7, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 2, 65, 66, 67, 68, 69, 70, 71, 12, 73, 74, 75, 76, 77, 78, 79, 80, 3, 82, 83

Offset: 1

Gus Wiseman, May 18 2018

nonn

This function takes powerful numbers (A001694) to weak numbers (A052485) and leaves weak numbers unchanged.

First differs from A052410 at a(72) = 12, A052410(72) = 72.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A001597, A001694, A005117, A007947, A013929, A047966, A051903, A051904, A052485, A062759, A066636, A066638, A072774, A304768, A354090.

spr[n_]:=Module[{f,m},f=FactorInteger[n];m=Min[Last/@f];n/Times@@First/@f^(m-1)];
Array[spr,100]

A007947(n) = factorback(factorint(n)[, 1]);
A051904(n) = if((1==n),0,vecmin(factor(n)[, 2]));
A304776(n) = (n/(A007947(n)^(A051904(n)-1))); \\ Antti Karttunen, May 19 2022

a(n) = if(n == 1, 1, my(f = factor(n), p = f[, 1], e = f[, 2]); n / vecprod(p)^(vecmin(e) - 1)); \\ Amiram Eldar, Sep 12 2024

a(n) = n / A354090(n). - Antti Karttunen, May 19 2022

Sum_{k=1..n} a(k) ~ n^2 / 2. - Amiram Eldar, Sep 12 2024

Data section extended up to a(83) by Antti Karttunen, May 19 2022

A380266 a(n) is the numerator of the mean value of A051904(k) at the range k = 1..n.

Original entry on oeis.org

0, 1, 2, 1, 1, 1, 1, 5, 4, 13, 14, 5, 16, 17, 6, 11, 23, 4, 25, 13, 9, 14, 29, 5, 32, 33, 4, 37, 38, 13, 40, 45, 46, 47, 48, 25, 51, 26, 53, 27, 55, 4, 57, 29, 59, 30, 61, 31, 64, 13, 22, 67, 68, 23, 14, 71, 24, 73, 74, 5, 76, 77, 26, 21, 17, 43, 87, 22, 89, 9

Offset: 1

Amiram Eldar, Jan 18 2025

			Fractions begin with 0, 1/2, 2/3, 1, 1, 1, 1, 5/4, 4/3, 13/10, 14/11, 5/4, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ivan Niven, Averages of Exponents in Factoring Integers, Proc. Amer. Math. Soc., Vol. 22, No. 2 (1969), pp. 356-360.
Eric Weisstein's World of Mathematics, Niven's Constant.
Wikipedia, Niven's constant.

Cf. A051904, A090699, A380267 (denominators).

f[n_] := Min[FactorInteger[n][[;;, 2]]]; f[1] = 0; With[{m = 100}, Numerator[Accumulate[Array[f, m]] / Range[m]]]

lista(nmax) = {my(s = 0); print1(0, ", "); for(n = 2, nmax, s += vecmin(factor(n)[,2]);  print1(numerator(s/n), ", "));}

a(n) = numerator((Sum_{k=1..n} A051904(k))/n).

a(n)/A380267(n) = 1 + c/sqrt(n) + o(1/sqrt(n)), where c = zeta(3/2)/zeta(3) (A090699).

A380267 a(n) is the denominator of the mean value of A051904(k) at the range k = 1..n.

Original entry on oeis.org

1, 2, 3, 1, 1, 1, 1, 4, 3, 10, 11, 4, 13, 14, 5, 8, 17, 3, 19, 10, 7, 11, 23, 4, 25, 26, 3, 28, 29, 10, 31, 32, 33, 34, 35, 18, 37, 19, 39, 20, 41, 3, 43, 22, 45, 23, 47, 24, 49, 10, 17, 52, 53, 18, 11, 56, 19, 58, 59, 4, 61, 62, 21, 16, 13, 33, 67, 17, 69, 7

Offset: 1

Amiram Eldar, Jan 18 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ivan Niven, Averages of Exponents in Factoring Integers, Proc. Amer. Math. Soc., Vol. 22, No. 2 (1969), pp. 356-360.
Eric Weisstein's World of Mathematics, Niven's Constant.
Wikipedia, Niven's constant.

Cf. A051904, A380266 (numerators).

f[n_] := Min[FactorInteger[n][[;;, 2]]]; f[1] = 0; With[{m = 100}, Denominator[Accumulate[Array[f, m]] / Range[m]]]

lista(nmax) = {my(s = 0); print1(1, ", "); for(n = 2, nmax, s += vecmin(factor(n)[,2]);  print1(denominator(s/n), ", "));}

a(n) = denominator((Sum_{k=1..n} A051904(k))/n).

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.

A336063 Numbers divisible by the minimal exponent in their prime factorization (A051904).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Amiram Eldar, Jul 07 2020

nonn

The asymptotic density of this sequence is 1 (Schinzel and Šalát, 1994).

			4 = 2^2 is a term since A051904(4) = 2 is a divisor of 4.

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, chapter 3, p. 331.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Andrzej Schinzel and Tibor Šalát, Remarks on maximum and minimum exponents in factoring, Mathematica Slovaca, Vol. 44, No. 5 (1994), pp. 505-514.

Cf. A051904, A124184.
A005117 (except for 1) is subsequence.
Similar sequences: A033950, A005349, A006446, A074946, A075592, A007694, A112249, A336064.

h[1] = 0; h[n_] := Min[FactorInteger[n][[;; , 2]]]; Select[Range[2, 100], Divisible[#, h[#]] &]
Select[Range[2,100],Divisible[#,Min[FactorInteger[#][[All,2]]]]&] (* Harvey P. Dale, Aug 31 2020 *)

isok(m) = if (m>1, (m % vecmin(factor(m)[,2])) == 0); \\ Michel Marcus, Jul 08 2020

A354090 a(n) = A007947(n)^(A051904(n) - 1), where A007947 is squarefree kernel and A051904 is minimum prime exponent.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 9, 1, 1, 1, 1, 16, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 27, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, May 19 2022

nonn

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

Cf. A007947, A051904, A304776.

a[n_] := Module[{f = FactorInteger[n], e}, e = Min[f[[;; , 2]]] - 1; f[[;; , 2]] = e; Times @@ Power @@@ f]; Array[a, 100] (* Amiram Eldar, Feb 12 2023 *)

A007947(n) = factorback(factorint(n)[, 1]);
A051904(n) = if((1==n),0,vecmin(factor(n)[, 2]));
A354090(n) = (A007947(n)^(A051904(n)-1));

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

a(n) = A007947(n)^(A051904(n)-1).

a(n) = n / A304776(n).

A295417 Self-inverse permutation of natural numbers: in prime factorization of n replace each positive prime exponent e with max + min - e, where max = A051903(n) and min = A051904(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 18, 13, 14, 15, 16, 17, 12, 19, 50, 21, 22, 23, 54, 25, 26, 27, 98, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 250, 41, 42, 43, 242, 75, 46, 47, 162, 49, 20, 51, 338, 53, 24, 55, 686, 57, 58, 59, 450, 61, 62, 147, 64, 65, 66

Offset: 1

Rémy Sigrist, Nov 22 2017

nonn

This sequence was inspired by A293448.
This sequence first differs from A293448 at n = 42: a(42) = 42 whereas A293448(42) = 70.
a(A293448(n)) = A293448(a(n)) for any n > 0.
a(n) = n iff n belongs to A072774.
f(n) = f(a(n)) for any n > 0 and f in { A001221, A006530, A007947, A020639, A051903, A051904 }.
The lines visible in the logarithmic scatterplot of the sequence seems to correspond to integer sets where the function A062760 is constant (see logarithmic scatterplot in Links section).

			For n = 1620:
- 1620 = 2^2 * 3^4 * 5,
- A051903(1620) = 4 and A051904(1620) = 1,
- a(1620) = 2^(4+1-2) * 3^(4+1-4) * 5^(4+1-1) = 2^3 * 3 * 5^4 = 15000.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored logarithmic scatterplot of the first 1000000 terms (where the color is function of A062760(n))
Index entries for sequences that are permutations of the natural numbers

Cf. A001221, A006530, A007947, A020639, A051903, A051904, A062760, A072774 (fixed points), A293448.

a(n) = { my(f=factor(n)); if(#f~<=1, return(n), my(mi=vecmin(f[,2]), ma=vecmax(f[,2])); return(prod(i=1, #f~, f[i,1]^(ma+mi-f[i,2])))) }

a(n) = A007947(n)^(A051903(n) + A051904(n)) / n.

A124010 Triangle in which first row is 0, n-th row (n>1) lists the exponents of distinct prime factors ("ordered prime signature") in the prime factorization of n.

Original entry on oeis.org

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

Offset: 1

Franklin T. Adams-Watters, Nov 01 2006

A001222(n) = Sum(T(n,k), 1 <= k <= A001221(n)); A005361(n) = Product(T(n,k), 1 <= k <= A001221(n)), n>1; A051903(n) = Max(T(n,k): 1 <= k <= A001221(n)); A051904(n) = Min(T(n,k), 1 <= k <= A001221(n)); A067029(n) = T(n,1); A071178(n) = T(n,A001221(n)); A064372(n)=Sum(A064372(T(n,k)), 1 <= k <= A001221(n)). - Reinhard Zumkeller, Aug 27 2011
Any finite sequence of natural numbers appears as consecutive terms. - Paul Tek, Apr 27 2013
For n > 1: n-th row = n-th row of A067255 without zeros. - Reinhard Zumkeller, Jun 11 2013
Most often the prime signature is given as a sorted representative of the multiset of the nonzero exponents, either in increasing order, which yields A118914, or, most commonly, in decreasing order, which yields A212171. - M. F. Hasler, Oct 12 2018

			Initial values of exponents are:
1, [0]
2, [1]
3, [1]
4, [2]
5, [1]
6, [1, 1]
7, [1]
8, [3]
9, [2]
10, [1, 1]
11, [1]
12, [2, 1]
13, [1]
14, [1, 1]
15, [1, 1]
16, [4]
17, [1]
18, [1, 2]
19, [1]
20, [2, 1]
...

Reinhard Zumkeller, Rows n = 1..10000 of triangle, flattened
Index entries for sequences computed from exponents in factorization of n

Cf. A027748, A001221 (row lengths, n>1), A001222 (row sums), A027746, A020639, A064372, A067029 (first column).

Sorted rows: A118914, A212171.

a124010 n k = a124010_tabf !! (n-1) !! (k-1)
a124010_row 1 = [0]
a124010_row n = f n a000040_list where
   f 1 _      = []
   f u (p:ps) = h u 0 where
     h v e | m == 0 = h v' (e + 1)
           | m /= 0 = if e > 0 then e : f v ps else f v ps
           where (v',m) = divMod v p
a124010_tabf = map a124010_row [1..]
-- Reinhard Zumkeller, Jun 12 2013, Aug 27 2011

expts:=proc(n) local t1,t2,t3,t4,i; if n=1 then RETURN([0]); fi; if isprime(n) then RETURN([1]); fi; t1:=ifactor(n); if nops(factorset(n))=1 then RETURN([op(2,t1)]); fi; t2:=nops(t1); t3:=[]; for i from 1 to t2 do t4:=op(i,t1); if nops(t4) = 1 then t3:=[op(t3),1]; else t3:=[op(t3),op(2,t4)]; fi; od; RETURN(t3); end; # N. J. A. Sloane, Dec 20 2007
PrimeSignature := proc(n) local F, e, k; F := ifactors(n)[2]; [seq(e, e = seq(F[k][2], k = 1..nops(F)))] end:
ListTools:-Flatten([[0], seq(PrimeSignature(n), n = 1..73)]); # Peter Luschny, Jun 15 2025

row[1] = {0}; row[n_] := FactorInteger[n][[All, 2]] // Flatten; Table[row[n], {n, 1, 80}] // Flatten (* Jean-François Alcover, Aug 19 2013 *)

print1(0); for(n=2,50, f=factor(n)[,2]; for(i=1,#f,print1(", "f[i]))) \\ Charles R Greathouse IV, Nov 07 2014

A124010_row(n)=if(n,factor(n)[,2]~,[0]) \\ M. F. Hasler, Oct 12 2018

from sympy import factorint
def a(n):
    f=factorint(n)
    return [0] if n==1 else [f[i] for i in f]
for n in range(1, 21): print(a(n)) # Indranil Ghosh, May 16 2017

n = Product_k A027748(n,k)^a(n,k).

Name edited by M. F. Hasler, Apr 08 2022

cp's OEIS Frontend

A157754 a(1) = 0, a(n) = lcm(A051904(n), A051903(n)) for n >= 2.

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A158378 a(1) = 0, a(n) = gcd(A051904(n), A051903(n)) for n >= 2.

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A304776 A weakening function. a(n) = n / A007947(n)^(A051904(n) - 1) where A007947 is squarefree kernel and A051904 is minimum prime exponent.

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A380266 a(n) is the numerator of the mean value of A051904(k) at the range k = 1..n.

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A380267 a(n) is the denominator of the mean value of A051904(k) at the range k = 1..n.

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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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A336063 Numbers divisible by the minimal exponent in their prime factorization (A051904).

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