A061214 -id:A061214 - cp's OEIS frontend

A361761 a(n) = Product_{i=prime(n)..prime(n+1)} i.

6, 60, 210, 55440, 1716, 742560, 5814, 4037880, 7866331200, 26970, 51889178880, 89927760, 74046, 184072680, 776881123200, 1719393207840, 215940, 4383026968320, 1562389080, 373176, 14609718723600, 3484127520, 34726953602880, 518607878946393600, 9505049400, 1061106

Offset: 1

Karl-Heinz Hofmann, Mar 23 2023

nonn

Winston de Greef, Table of n, a(n) for n = 1..10000

Cf. A006094, A061214, A072472, A112231, A361760.

a(n) = my(x=1); for(i=prime(n), prime(n+1), x*=i); x; \\ Michel Marcus, Mar 28 2023

from sympy import prod, sieve
def A361761(n): return prod(range(sieve[n],sieve[n+1]+1))

a(n) = A006094(n)*A061214(n).

A109919 a(1) = 1, then product of consecutive composite numbers sandwiched between primes.

Original entry on oeis.org

1, 2, 1, 3, 4, 5, 6, 7, 720, 11, 12, 13, 3360, 17, 18, 19, 9240, 23, 11793600, 29, 30, 31, 45239040, 37, 59280, 41, 42, 43, 91080, 47, 311875200, 53, 549853920, 59, 60, 61, 1072431360, 67, 328440, 71, 72, 73, 2533330800, 79, 531360, 83, 4701090240, 89

Offset: 1

Amarnath Murthy, Jul 16 2005

a(1) = a(3) = 1 as empty product is defined to be 1.

The odd numbered terms are in A061214. - T. D. Noe, Oct 02 2012

Cf. A109920.
Cf. A072472.
Cf. A061214 (product of composite numbers between primes).

A109919 := proc(n) local p; if n mod 2 = 0 then ithprime(n/2) ; elif n = 1 then 1 ; else p := ithprime((n-1)/2) ; mul(i,i=p+1..nextprime(p)-1) ; fi ; end: for n from 1 to 80 do printf("%d, ",A109919(n)) ; od ; # R. J. Mathar, May 02 2007

a(2n) = prime(n) and a(2n+1)= product of composite numbers between prime(n) and prime(n+1).

a(2n) = A000040(n). a(2n+1) = A072472(n)/A000040(n+1). - R. J. Mathar, May 02 2007

More terms from R. J. Mathar, May 02 2007

A276133 Exponent of highest power of 2 dividing the product of the composite numbers between the n-th prime and the (n+1)-st prime.

Original entry on oeis.org

0, 2, 1, 4, 2, 5, 1, 3, 6, 1, 8, 4, 1, 3, 7, 5, 2, 8, 3, 3, 4, 5, 6, 9, 3, 1, 4, 2, 5, 11, 8, 6, 1, 10, 1, 6, 7, 3, 6, 6, 2, 8, 6, 3, 1, 12, 10, 6, 2, 4, 4, 4, 8, 11, 4, 6, 1, 7, 4, 1, 11, 13, 3, 3, 3, 15, 7, 8, 2, 6, 4, 7, 7, 5, 3, 10, 7, 5, 7

Offset: 1

Juri-Stepan Gerasimov, Aug 29 2016

Robert Israel, Table of n, a(n) for n = 1..10000

Supersequence of A205649 (Hamming distance between twin primes).
Cf. A000040, A007814, A061214.
First differences of A080085.

A:= Vector(100): q:= 2:
for n from 1 to 100 do
  p:= q; q:= nextprime(q);
  t:= 0;
  for i from p+1 to q-1 do t:= t + padic:-ordp(i,2) od;
  A[n]:= t
od:
convert(A,list); # Robert Israel, Apr 11 2021

IntegerExponent[#,2]&/@(Times@@Range[#[[1]]+1,#[[2]]-1]&/@Partition[ Prime[ Range[ 80]],2,1]) (* Harvey P. Dale, Aug 12 2024 *)

a(n) = valuation(prod(k=prime(n)+1, prime(n+1)-1, k), 2); \\ Michel Marcus, Aug 31 2016

a(n) = my(p=prime(n+1),q=prime(n)); p-hammingweight(p) - (q-hammingweight(q)); \\ Kevin Ryde, Apr 11 2021

from sympy import prime
def A276133(n): return (p:=prime(n+1)-1)-p.bit_count()-(q:=prime(n))+q.bit_count() # Chai Wah Wu, Jul 10 2022

a(n) = A007814(A061214(n)).

a(n+1) = Sum_{k = A000040(n+1)..A000040(n+2)} A007814(k).

a(16) corrected by Robert Israel, Apr 11 2021

A061216 a(n) = product of all even numbers between n-th prime and (n+1)-st prime.

Original entry on oeis.org

1, 4, 6, 80, 12, 224, 18, 440, 17472, 30, 39168, 1520, 42, 2024, 124800, 175392, 60, 261888, 4760, 72, 438672, 6560, 635712, 74718720, 9800, 102, 11024, 108, 12320, 356925975275520, 16640, 2405568, 138, 61857653760, 150, 3651648, 4095360

Offset: 1

Amarnath Murthy, Apr 22 2001

nonn

Previous name used "even composite numbers", but if an even number is strictly between two primes, it is composite. So the word 'composite' isn't needed in the title. - David A. Corneth, Aug 21 2016

			a(4) = 80 = 8 * 10, as 7 is the 4th prime and 11 is the 5th prime.
a(9) = 17472. Let p_(n) = prime(n). p_(9) = 23, p_(10) = 29. The number of even numbers between 23 and 29 is floor((29 - 23) / 2) = 3. So a(9) is 2^3 * (23 + 1)/2 * ... * (29 - 1)/2 = 17472. - _David A. Corneth_, Aug 21 2016

Harry J. Smith, Table of n, a(n) for n = 1..2000

Cf. A061214, A061215.

f:= proc(n) local p,q;
  p:= ithprime(n); q:= ithprime(n+1);
  2^((q-p)/2)*floor(q/2)!/floor(p/2)!
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Aug 28 2016

f[n_]:=Module[{pn=Prime[n],pn1=Prime[n+1]},Times@@Range[pn+1,pn1,2]]; Table[f[i], {i, 45}] (* Harvey P. Dale, Jan 16 2011 *)

for(n=1,50,p=1;for(k=prime(n)+1, prime(n+1)-1,if(k%2==0,p=p*k));print1(p","))

n=0; q=2; forprime (p=3, prime(2001), a=1; for (i=q + 1, p - 1, if (i%2==0, a*=i)); q=p; write("b061216.txt", n++, " ", a) ) \\ Harry J. Smith, Jul 19 2009

a(n) = {my(p1 = prime(n), p2 = nextprime(p1 + 1));
2^((p2-p1)\2) * prod(i=(p1+1)\2,(p2-1)\2,i)} \\ David A. Corneth, Aug 21 2016

a(n) = 2^((prime(n+1)-prime(n))/2) * ((prime(n+1)-1)/2)!/((prime(n)-1)/2)! for n >= 2. - Robert Israel, Aug 28 2016

Corrected and extended by Ralf Stephan, Mar 22 2003

Name simplified by David A. Corneth, Aug 21 2016

A077217 Prime(k) such that the prime power with largest exponent that divides the product P(k) of composite numbers between prime(k) and prime(k+1) is an odd number, i.e., if p^r and 2^s divide P(k) then r >= s, p is an odd prime.

Original entry on oeis.org

2, 5, 17, 29, 41, 101, 107, 137, 149, 179, 197, 269, 281, 457, 461, 499, 521, 569, 593, 617, 641, 673, 727, 809, 821, 827, 857, 881, 1049, 1061, 1229, 1277, 1289, 1301, 1321, 1451, 1453, 1481, 1483, 1619, 1697, 1721, 1753, 1777, 1861, 1873, 1877, 1949, 1997, 2027

Offset: 1

Amarnath Murthy, Nov 02 2002

nonn

In most cases a power of 2 has a larger exponent than any odd prime power.

Primes p = prime(k) such that A051903(A000265(A061214(k))) >= A007814(A061214(k)). - Amiram Eldar, Apr 01 2021

			5 is a member as 6 is divisible by 3^1 as well as by 2^1.
17 is a member as 18 is divisible by 3^2 but not by 2^2.

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

Cf. A000265, A007814, A051903, A061214.

q[p_] := Module[{prod = Product[k, {k, p + 1, NextPrime[p] - 1}], e2}, e2 = IntegerExponent[prod, 2]; Max[FactorInteger[prod/2^e2][[;; , 2]]] >= e2]; Select[Range[2000], PrimeQ[#] && q[#] &] (* Amiram Eldar, Apr 01 2021 *)

f(p) = prod(k=p+1, nextprime(p+1)-1, k);\\ A061214
isok(p) = {my(prd = f(p), e = valuation(prd, 2), ofprd = prd/2^e); if (prd > 1, (ofprd == 1) || (e <= vecmax(factor(ofprd)[,2])));} \\ Michel Marcus, Apr 01 2021

Wrong term removed and more terms added by Amiram Eldar, Apr 01 2021

A361806 Sum of distinct prime factors of all composite numbers between n-th and (n+1)st primes.

Original entry on oeis.org

0, 2, 5, 10, 5, 17, 5, 28, 30, 10, 45, 42, 12, 44, 47, 76, 10, 72, 57, 5, 97, 51, 117, 150, 28, 22, 83, 5, 65, 321, 66, 131, 28, 298, 10, 108, 172, 145, 109, 205, 10, 276, 5, 127, 16, 441, 582, 130, 24, 80, 232, 10, 276, 195, 270, 256, 10, 218, 187, 52, 388, 701, 162

Offset: 1

Karl-Heinz Hofmann, Mar 26 2023

			a(6): 6th prime = 13 and the (6+1)th prime = 17; the composites between are {14,15,16} and the distinct prime factors of this set are {2,7,3,5} (no duplicates allowed); so a(6) = 2 + 7 + 3 + 5 = 17.

Winston de Greef, Table of n, a(n) for n = 1..10000

Cf. A052297, A077218, A008472, A061214.

a[n_] := Plus @@ Union@ (Join @@ (FactorInteger[#][[;; , 1]] & /@ Range[Prime[n] + 1, Prime[n + 1] - 1])); Array[a, 65] (* Amiram Eldar, Mar 27 2023 *)

a(n) = my(list=List()); for(i=prime(n)+1, prime(n+1)-1, my(f=factor(i)[,1]); for (k=1, #f, listput(list, f[k]))); vecsum(Set(list)); \\ Michel Marcus, Mar 27 2023

from sympy import primefactors, sieve
def A361806(n):
    primeset = []
    for composites in range (sieve[n]+1, sieve[n+1]):
        for p in primefactors(composites): primeset.append(p)
    return(sum(set(primeset)))

a(n) = A008472(A061214(n)).

A362296 Greatest common divisor of composite numbers between the n-th and (n+1)st primes.

Original entry on oeis.org

4, 6, 1, 12, 1, 18, 1, 1, 30, 1, 1, 42, 1, 1, 1, 60, 1, 1, 72, 1, 1, 1, 1, 1, 102, 1, 108, 1, 1, 1, 1, 138, 1, 150, 1, 1, 1, 1, 1, 180, 1, 192, 1, 198, 1, 1, 1, 228, 1, 1, 240, 1, 1, 1, 1, 270, 1, 1, 282, 1, 1, 1, 312, 1, 1, 1, 1, 348, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 2

Chai Wah Wu, Apr 15 2023

nonn

Winston de Greef, Table of n, a(n) for n = 2..10000

Cf. A056831 (LCM), A061214 (product).

a(n) = gcd([prime(n)+1..prime(n+1)-1]); \\ Michel Marcus, Apr 16 2023

from sympy import prime, isprime
def A362296(n): return m-1 if isprime(m:=prime(n)+2) else 1

For n > 1, a(n)=prime(n)+1 if and only if prime(n+1)=prime(n)+2 and a(n)=1 otherwise.

A276376 Exponent of highest power of 3 dividing product of composite numbers between n-th prime and (n+1)-st prime.

Original entry on oeis.org

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

Offset: 1

Juri-Stepan Gerasimov, Sep 01 2016

Cf. A007849, A061214, A276133.

Table[IntegerExponent[Times@@Range[Prime[n]+1,Prime[n+1]-1],3],{n,100}] (* Harvey P. Dale, Mar 23 2021 *)

a(n) = A007949(A061214(n)).

a(n) = Sum_{k = A000040(n)..A000040(n + 1)} A007949(k) for n > 2.

cp's OEIS Frontend

A361761 a(n) = Product_{i=prime(n)..prime(n+1)} i.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Python

Formula

A109919 a(1) = 1, then product of consecutive composite numbers sandwiched between primes.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Formula

Extensions

A276133 Exponent of highest power of 2 dividing the product of the composite numbers between the n-th prime and the (n+1)-st prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Python

Formula

Extensions

A061216 a(n) = product of all even numbers between n-th prime and (n+1)-st prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

Extensions

A077217 Prime(k) such that the prime power with largest exponent that divides the product P(k) of composite numbers between prime(k) and prime(k+1) is an odd number, i.e., if p^r and 2^s divide P(k) then r >= s, p is an odd prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A361806 Sum of distinct prime factors of all composite numbers between n-th and (n+1)st primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A362296 Greatest common divisor of composite numbers between the n-th and (n+1)st primes.

Views

Author

Keywords

Links