A051903 -id:A051903 - cp's OEIS frontend

A380264 a(n) is the numerator of the mean value of A051903(k) at the range k = 1..n.

0, 1, 2, 1, 1, 1, 1, 5, 4, 13, 14, 4, 17, 9, 19, 23, 24, 13, 27, 29, 10, 31, 32, 35, 37, 19, 41, 43, 44, 3, 46, 51, 52, 53, 54, 14, 57, 29, 59, 31, 63, 32, 65, 67, 23, 35, 71, 25, 11, 79, 80, 41, 83, 43, 87, 45, 91, 46, 93, 19, 96, 97, 11, 105, 106, 107, 108, 55

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, 4/3, ...

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. A033150, A051903, A129132, A359071, A359072, A380265 (denominators).

f[n_] := Max[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 += vecmax(factor(n)[,2]);  print1(numerator(s/n), ", "));}

a(n) = numerator((Sum_{k=1..n} A051903(k))/n).
a(n)/A380265(n) = A129132(n)/n.
Limit_{n->oo} a(n)/A380265(n) = c, where c is Niven's constant (A033150).
abs(a(n)/A380265(n) - c) <= 3*log_2(n)/sqrt(n).

A380265 a(n) is the denominator of the mean value of A051903(k) at the range k = 1..n.

Original entry on oeis.org

1, 2, 3, 1, 1, 1, 1, 4, 3, 10, 11, 3, 13, 7, 15, 16, 17, 9, 19, 20, 7, 22, 23, 24, 25, 13, 27, 28, 29, 2, 31, 32, 33, 34, 35, 9, 37, 19, 39, 20, 41, 21, 43, 44, 15, 23, 47, 16, 7, 50, 51, 26, 53, 27, 55, 28, 57, 29, 59, 12, 61, 62, 7, 64, 65, 66, 67, 34, 23, 5, 71

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. A051903, A380264 (numerators).

f[n_] := Max[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 += vecmax(factor(n)[,2]);  print1(denominator(s/n), ", "));}

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

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

Original entry on oeis.org

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

A275803 a(n) = A051903(A275725(n)); maximal cycle sizes of finite permutations listed in the order A060117 / A060118.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 2, 3, 4, 3, 4, 2, 3, 3, 4, 2, 4, 2, 3, 2, 4, 3, 4, 2, 2, 2, 3, 2, 3, 3, 3, 4, 5, 4, 5, 3, 4, 4, 5, 3, 5, 3, 4, 3, 5, 4, 5, 2, 2, 3, 4, 3, 4, 3, 3, 4, 5, 4, 5, 2, 3, 4, 5, 3, 5, 2, 3, 3, 5, 4, 5, 2, 3, 3, 4, 2, 4, 2, 3, 4, 5, 3, 5, 3, 4, 4, 5, 3, 5, 2, 4, 3, 5, 3, 5, 2, 3, 2, 4, 3, 4, 2, 3, 3, 5, 4, 5, 2, 4, 3, 5, 3, 5, 3, 4, 3, 5, 4, 5, 2

Offset: 0

Antti Karttunen, Aug 10 2016

			For n=27, which in factorial base (A007623) is "1011" and encodes (in A060118-order) permutation "23154" with one 3-cycle and one 2-cycle, the longest cycle has three elements, thus a(27) = 3.

Antti Karttunen, Table of n, a(n) for n = 0..40320
Index entries for sequences related to factorial base representation

Cf. A007623, A060117, A060118.
Cf. A051903, A275725.
Cf. A261220 (gives the positions of 1 and 2's).
Differs from A060131 for the first time at n=27, where a(27) = 3, while A060131(27) = 6.

(define (A275803 n) (A051903 (A275725 n)))

a(n) = A051903(A275725(n)).

A327295 Numbers k such that e(k) > 1 and k == e(k) (mod lambda(k)), where e(k) = A051903(k) is the maximal exponent in prime factorization of k.

Original entry on oeis.org

4, 12, 16, 48, 80, 112, 132, 208, 240, 1104, 1456, 1892, 2128, 4144, 5852, 12208, 17292, 18544, 21424, 25456, 30160, 45904, 78736, 97552, 106384, 138864, 153596, 154960, 160528, 289772, 311920, 321904, 399212, 430652, 545584, 750064, 770704, 979916, 1037040, 1058512

Offset: 1

Thomas Ordowski, Dec 05 2019

nonn

The condition e(k) > 1 excludes primes and Carmichael numbers.
Numbers n such that e(k) > 1 and b^k == b^e(k) (mod k) for all b.
These are numbers k such that A276976(k) = e(k) > 1.
Are there infinitely many such numbers? Are all such numbers even?
A number k is a term if and only if k is e(k)-Knödel number with e(k) > 1. So they may have the name nonsquarefree e(k)-Knodel numbers k.
It seems that if k is in this sequence, then e(k) = A007814(k) and k/2^e(k) is squarefree.
Conjecture: there are no composite numbers m > 4 such that m == e(m) (mod phi(m)). By Lehmer's totient conjecture, there are no such squarefree numbers.
Problem: are there odd numbers n such that e(n) > 1 and n == e(n) (mod ord_{n}(2)), where ord_{n}(2) = A002326((n-1)/2)? These are odd numbers n such that 2^n == 2^e(n) (mod n) with e(n) > 1.
Numbers k for which A051903(k) > 1 and A219175(k) = A329885(k). - Antti Karttunen, Dec 11 2019

			The number 4 = 2^2 is a term, because e(4) = A051903(4) = 2 > 1 and 4 == 2 (mod lambda(4)), where lambda(4) = A002322(4) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..500

Cf. A000010, A002322, A002326, A002997, A050990, A050992, A051903, A068494, A219175, A270096, A276976, A324050, A327979, A329885.

A proper subset of A013929.

Select[Range[10^5], (e = Max @@ Last /@ FactorInteger[#]) > 1 && Divisible[# -e, CarmichaelLambda[#]] &] (* Amiram Eldar, Dec 05 2019 *)

isok(n) = ! issquarefree(n) && (Mod(n, lcm(znstar(n)[2])) == vecmax(factor(n)[, 2])); \\ Michel Marcus, Dec 05 2019

More terms from Amiram Eldar, Dec 05 2019

A328384 If n is of the form p^p, a(n) = 0, otherwise a(n) gives the number of iterations of x -> A003415(x) needed to reach the first number different from n which is either a prime, or whose degree (A051903) differs from the degree of n. If the degree of the final number is <= that of n, then a(n) = -1 * iteration count.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 15 2019

sign

The records -1, 0, 1, 2, 3, 5, 8, 10, 11, 13, ... occur at n = 1, 4, 12, 26, 36, 80, 108, 4887, 18688, 22384, ...

			For n = 21 = 3*7, A051903(21) = 1, A003415(21) = 10 = 2*5, is of the same degree as A051903(10) = 1, but then A003415(10) = 7, which is a prime, having degree <= of the starting value (as we have A051903(7) <= A051903(21)), thus a(21) = -1 * 2 = -2.
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, which is larger than the initial degree, thus a(33) = +2.

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

Cf. A003415, A051903, A051674, A157037, A258651, A328252, A328310, A328320.
Cf. A328385 (the number found in the iteration).
Cf. A256750, A328248, A328383 for similar counts.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A051903(n) = if((n<=1),n-1,vecmax(factor(n)[, 2]));
A328384(n) = { my(d=A051903(n), u=A003415(n), k=1); if(u==n,return(0)); while(u && (u!=n) && !isprime(u) && A051903(u)==d, k++; n = u; u = A003415(u)); if(A051903(u)<=d,-k,k); };

a(1) = -1 as 0 is here considered having a smaller degree than 1.
a(p) = -1 for all primes.
a(A051674(n)) = 0.
a(A157037(n)) = -1.
a(A328252(n)) = -1.
a(A328320(n)) = -1.

A329338 a(n) = {1{0}^(A268336(n)-1)}^(n-1){1}{0}^A051903(n): upper bound for A329126(n).

Original entry on oeis.org

1, 110, 101010, 111100, 100010001000100010, 1111110, 10000010000010000010000010000010000010, 11111111000, 1010101010101010100, 10101010101010101010, 100000000010000000001000000000100000000010000000001000000000100000000010000000001000000000100000000010

Offset: 1

M. F. Hasler, Nov 13 2019

nonn

This is the upper bound for A329126 as explained in the "FORMULA" there.

It is sharp for all n except 10, 14, 15, ...

Cf. A329126, A329000, A329339 (this converted from binary to decimal), A268336, A051903.

apply( {A329338(n)=my(k=lcm(lcm([p-1|p<-factor(n)[,1]]), n)/n); fromdigits(concat(vector(n, i, Vec(1, if(i1, vecmax(factor(n)[,2])+1)))))}, [1..16])

a(n) = A007088(A329339(n)), where A007088 = binary numbers and A329339(n) = 2^A051903(n)*(m^n-1)/(m-1) with m = 2^A268336(n).

A350076 Numbers k for which the maximal digit in their primorial base expansion (A328114) is greater than or equal to the maximal exponent in the prime factorization of k (A051903).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92

Offset: 1

Antti Karttunen, Feb 01 2022

Numbers k for which the maximal prime exponent of A276086(k) is greater than or equivalent to the maximal prime exponent of k, A051903(k).

Index entries for sequences related to primorial base

Cf. A051903, A276086, A328114, A350070 (characteristic function), A350075 (complement).
Positions of nonnegative terms in A350074.
Cf. also A342006.

A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA350076(n) = (A051903(A276086(n)) >= A051903(n));

A351076 Numbers k such that the maximal exponent in the prime factorization of A327860(k) is greater than or equal to A051903(k), the maximal exponent in the prime factorization of k.

Original entry on oeis.org

1, 3, 5, 7, 8, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 72, 73, 74, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98

Offset: 1

Antti Karttunen, Feb 01 2022

nonn

Numbers k for which A328391(k) >= A051903(k).

Index entries for sequences related to primorial base

Cf. A003415, A276086, A051903, A327860, A328391, A351075 (complement), A351077 (the characteristic function).
Positions of nonnegative terms in A351074.
Cf. also A350076.

A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
isA351076(n) = (A051903(A327860(n)) >= A051903(n));

cp's OEIS Frontend

A380264 a(n) is the numerator of the mean value of A051903(k) at the range k = 1..n.

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A380265 a(n) is the denominator of the mean value of A051903(k) at the range k = 1..n.

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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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A275803 a(n) = A051903(A275725(n)); maximal cycle sizes of finite permutations listed in the order A060117 / A060118.

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A327295 Numbers k such that e(k) > 1 and k == e(k) (mod lambda(k)), where e(k) = A051903(k) is the maximal exponent in prime factorization of k.

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A329338 a(n) = {1{0}^(A268336(n)-1)}^(n-1){1}{0}^A051903(n): upper bound for A329126(n).