A056171 -id:A056171 - cp's OEIS frontend

A235623 Numbers n for which in the prime power factorization of n!, the numbers of exponents 1 and >1 are equal.

0, 1, 4, 7, 8, 9, 13, 19, 20, 21

Offset: 1

Vladimir Shevelev, Apr 20 2014

Number n is in the sequence, if and only if pi(n) = 2*pi(n/2), where pi(x) is the number of primes<=x. Indeed, all primes from interval (n/2, n] appear in prime power factorization of n! with exponent 1, while all primes from interval (0, n/2] appear in n! with exponents >1. However, it follows from Ehrhart's link that, for n>=22, pi(n) < 2*pi(n/2). Therefore, a(9)=21 is the last term of the sequence.

m is in this sequence if and only if the number of prime divisors of [m/2]! equals the number of unitary prime divisors of m! - Peter Luschny, Apr 29 2014

			21! = 2^20*3^9*5^4*7^3*11*13*17*19. Here 4 primes with exponent 1 and 4 primes with exponents >1, so 21 is in the sequence.

Eugene Ehrhart, On prime numbers, Fibonacci Quarterly 26:3 (1988), pp. 271-274.

Cf. A056171, A177329, A177333, A177334, A240537, A240588, A240606, A240619, A240620, A240668, A240669, A240670, A240672, A240695, A240751, A240755, A240764, A240905, A240906, A241123, A241124, A241139, A241148, A241289.

with(numtheory): a := proc(n) factorset(n!); factorset(iquo(n,2)!);
`if`(nops(%% minus %) = nops(%), n, NULL) end: seq(a(n), n=0..30); # Peter Luschny, Apr 28 2014

isok(n) = {f = factor(n!); sum(i=1, #f~, f[i,2] == 1) == sum(i=1, #f~, f[i,2] > 1);} \\ Michel Marcus, Apr 20 2014

A298071 Number of primes between floor(3n/2) and 2n (inclusive).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 3, 2, 3, 3, 3, 4, 4, 3, 4, 3, 3, 3, 4, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 5, 5, 4, 5, 4, 5, 6, 6, 6, 7, 6, 7, 7, 7, 6, 6, 6, 6, 7, 7, 7, 7, 6, 7, 7, 7, 6, 6, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 9, 9, 8, 9, 8, 8, 9, 10, 9

Offset: 1

Bruno Berselli, Jan 11 2018

nonn

Cf. A000040, A035250, A056171.

a[n_] := PrimePi[2*n] - PrimePi[Floor[3*n/2]] + If[PrimeQ[Floor[ 3*n/2]], 1, 0]; Array[a, 100] (* Jean-François Alcover, Jan 11 2018 *)

A298071 = lambda n: len([p for p in (3*n//2..2*n) if is_prime(p)])
print([A298071(n) for n in (1..97)]) # Peter Luschny, Jan 11 2018

A076225 Counts of the maximum value in n-th row of A076221.

Original entry on oeis.org

1, 2, 3, 2, 3, 2, 3, 3, 3, 2, 3, 3, 4, 3, 3, 3, 4, 4, 5, 5, 5, 4, 5, 5, 5, 4, 4, 4, 5, 5, 6, 6, 6, 5, 5, 5, 6, 5, 5, 5, 6, 6, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 7, 8, 8, 9, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 11, 10, 10, 10, 10, 10, 11, 11, 11, 10, 11, 11, 11, 10, 10, 10, 11, 11, 11, 11, 11

Offset: 1

Olivier Gérard

nonn

Does a(n) always equal 1 + pi(n) - pi(n/2), where pi(x) is number of primes <= x? If so a(n) = A056171(n)+1. - Leroy Quet, Feb 06 2003.

This is true, because A076221(n,k) = n-1 iff either k = 1 or k is a prime with n/2 < k <= n. - Robert Israel, Aug 29 2016

Ethan Berkove and Michael Brilleslyper, Subgraphs of Coprime Graphs on Sets of Consecutive Integers, Integers (2022) Vol. 22, #A47, see p. 8.

Cf. A056171, A076221.

seq(1+numtheory:-pi(n) - numtheory:-pi(floor(n/2)),n=1..100); # Robert Israel, Aug 29 2016

A280380 First occurrence of A280379(k) = n.

Original entry on oeis.org

1, 2, 30, 46, 374, 2146, 5945, 14855, 24702

Offset: 0

John W. Nicholson, Jan 09 2017

nonn

n < log(a(n)) for a(n) < 1.1*10^6.

			For prime(30)=113, A056171(113) = 14, A104272(12)=107 and A104272(13) = 127, so 14 - 12 = 2 (First occurrence).

Cf. A056171, A104272.

\\RR[x] is a list of Ramanujan primes, A104272.
{plimit=1.1*10^6;i=n=s=0;
forprime(p=2,plimit,
s++;
if(p==RR[n+1],n++);
if(i==s-primepi(floor(p/2))-n,print(i," ",s);i++)
)
}

A360285 Triangle read by rows: T(n,k) is the number of subsets of {1,...,n} of cardinality k in which no two elements are coprime; n >= 0, 0 <= k <= floor(n/2) + [n=1].

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 4, 1, 1, 5, 1, 1, 6, 4, 1, 1, 7, 4, 1, 1, 8, 7, 4, 1, 1, 9, 9, 5, 1, 1, 10, 14, 11, 5, 1, 1, 11, 14, 11, 5, 1, 1, 12, 21, 24, 16, 6, 1, 1, 13, 21, 24, 16, 6, 1, 1, 14, 28, 39, 36, 21, 7, 1, 1, 15, 34, 48, 41, 22, 7, 1, 1, 16, 41, 69, 76, 57, 28, 8, 1

Offset: 0

Marcel K. Goh, Feb 01 2023

			Triangle T(n,k) begins:
   n/k 0  1  2  3  4 5 6
   0   1
   1   1  1
   2   1  2
   3   1  3
   4   1  4  1
   5   1  5  1
   6   1  6  4  1
   7   1  7  4  1
   8   1  8  7  4  1
   9   1  9  9  5  1
  10   1 10 14 11  5 1
  11   1 11 14 11  5 1
  12   1 12 21 24 16 6 1
  ...
For n=8 and k=3 the T(8,3)=4 sets are {2,4,6}, {2,4,8}, {2,6,8}, and {4,6,8}.

Marcel K. Goh and Jonah Saks, Alternating-sum statistics for certain sets of integers, arXiv:2206.12535 [math.CO], 2022.

Cf. A056171, A355146.

cp's OEIS Frontend

A235623 Numbers n for which in the prime power factorization of n!, the numbers of exponents 1 and >1 are equal.

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PARI

A298071 Number of primes between floor(3*n/2) and 2*n (inclusive).

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Mathematica

Sage

A076225 Counts of the maximum value in n-th row of A076221.

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Maple

A280380 First occurrence of A280379(k) = n.

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Programs

PARI

A360285 Triangle read by rows: T(n,k) is the number of subsets of {1,...,n} of cardinality k in which no two elements are coprime; n >= 0, 0 <= k <= floor(n/2) + [n=1].

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Author

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A298071 Number of primes between floor(3n/2) and 2n (inclusive).