Displaying 1701-1710 of 1778 results found.
One half of the smallest number with prime signature of the multiset defining partition, taken in Abramowitz-Stegun order.
+10
0
1, 2, 3, 4, 6, 8, 12, 16, 15, 18, 24, 32, 30, 36, 48, 64, 60, 72, 96, 128, 90, 120, 108, 144, 192, 256, 105, 180, 240, 216, 288, 384, 512, 210, 360, 480, 432, 576, 768, 1024, 420, 450, 540, 720, 648, 960, 864, 1152, 1536, 2048
COMMENTS
For a list of the multiset repetition class defining partitions in Abramowitz-Stegun (A-St)order see the links under A176725 and A187447. For the A-St ordering of all partitions see A036036. The actual sequence is 2*a(n): 2, 4, 6, 8, 12, 16, 24, 32, 30, 36, 48, 64, 60, 72, 96, 128, 120, 144, 192, 256,... This is similar to A025487 without the leading 1 (products of primorial numbers A002110, ordered increasingly, which is not the case here). The analog sequence for all partitions in A-St order is A185974.
FORMULA
a(n)=((p(1)^e[1])*(p(2)^e^[2])*...*(p(M)^e[M]))/2 with the prime numbers p(j):= A000040(j), and the n-th multiset defining partition with positive integer exponents e[1]>=e[2]>=...>=e[M]>=1; M=M(n)= A176725(n), read as sequence. These partitions are taken in A-St order. See the links to A176725 and A187447 for this partition list.
EXAMPLE
2*a(11)=2*24=48 =2^4*3^1, the smallest number with prime signature e[1]=4, e[2]=1, read as multiset defining partition 1^4,2^1, which is the 11th one in Abramowitz-Stegun order. The corresponding 5-multiset is {1,1,1,1,2}.
a(0) = 1, a(n) = least primorial sharing at least n distinct digits with a(n-1).
+10
0
1, 210, 2310, 223092870, 6469693230, 200560490130, 304250263527210, 614889782588491410, 32589158477190044730, 1922760350154212639070, 117288381359406970983270
COMMENTS
The sequence ends with the smallest pandigital primorial, which is 61# = 117288381359406970983270 (18th primorial).
If you change the criterion from "at least n distinct" to "all" the sequence becomes 1, 210, 2310, 200560490130, 1922760350154212639070, 117288381359406970983270.
EXAMPLE
a(1) = 210 = 7#, which shares the "1" with a(0).
a(2) = 2310 = 11#, which shares more than the needed two digits with 210, namely "0", "1", and "2".
a(3) = 223092870 = 23#.
4084081, 5105101, 8168161, 8678671, 9189181, 10720711, 12762751, 13273261, 13783771, 14804791, 18888871, 21441421, 21951931, 22972951, 25014991, 26546521, 28078051, 31651621, 36246211, 38288251, 40330291
COMMENTS
Infinite by Dirichlet's theorem.
Primes p such that p + 1 or p - 1 is in A066120.
+10
0
2, 3, 23, 8641, 653184001, 1601591599167888308924824752807936000000000000001
COMMENTS
What is the next prime? As of January 2012, there are no known primes ending in 9 with this property.
FORMULA
a(n) are the prime values of p(0)# * (p(0)# * p(1)#) * (p(0)# * p(1)# * p(2)#) * (p(0)# * p(1)# * p(2)# * ... * p(n)#) +/- 1.
MATHEMATICA
Select[Flatten[Table[Product[Product[Product[Prime[i], {i, j}], {j, k}], {k, n}] - 1 + m, {n, 0, 7}, {m, 0, 2, 2}]], PrimeQ]
Quotients of (first) primorial numbers and denominators of Bernoulli numbers B 0, B 1, B 2, B 4, B 6,... .
+10
0
1, 1, 1, 1, 5, 77, 455, 187, 1616615, 437437, 8107385, 607759061, 53773464745, 111446982977, 2180460221945005, 706769865044243, 2275461421392965, 3770118333635711057, 19548063559901161830545, 4094603218587147211, 92990138354449826827902565
COMMENTS
a(2*n+4) is divisible by 5 (because A006954(n+2)=6,30,42,30,... is divisible by A165734(n)=period of length 2: repeat 6,30).
EXAMPLE
a(0) = 1/1, a(1)= 2/2, a(2) = 6/6, a(3) = 30/30, a(4) =210/42=5.
1,
1,
1,
1,
5,
7 * 11,
5 * 7 *13,
11 * 17,
5 * 7 *11 *13 *17 *19,
7 * 11 *13 *19 *23,
5 * 11 *13 *17 *23 *29,
7 * 13 *17 *19 *23 *29 *31,
5 * 7 *11 *13 *17 *19 *29 *31 *37.
MATHEMATICA
a[n_] := Product[ Prime[k], {k, 1, n}] / Denominator[ BernoulliB[2*n-2] ]; a[0] = a[1] = 1; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 15 2013 *)
a(n) = smallest prime p such that p + prime(n)# is next prime after p, where primorial prime(n)# = 2*3*..*prime(n).
+10
0
EXAMPLE
a(2)=23 because 23 is smallest prime such that 23+2*3=29 is next prime.
Primes p such that sum of primorials ( A143293) not including p as a factor is divisible by p.
+10
0
COMMENTS
As in A002110, primorial(0)=1, and primorial(n) = primorial(n-1)*prime(n). The next term, if it exists, is bigger than 10^8.
EXAMPLE
Sum of primorials not including 3 as a factor is 1 + 2 = 3. Because it's divisible by 3, the latter is in the sequence.
Sum of primorials not including 17 as a factor is 1 + 2 + 6 + 6*5 + 30*7 + 210*11 + 2310*13 = 32589. Because 32589 is divisible by 17, the latter is in the sequence.
PROG
(Python)
primes = [2]*2
primes[1] = 3
def addPrime(k):
for p in primes:
if k%p==0: return
if p*p > k: break
primes.append(k)
for n in range(5, 100000000, 6):
addPrime(n)
addPrime(n+2)
sum = 0
primorial = 1
for p in primes:
sum += primorial
primorial *= p
if sum % p == 0: print p,
(PARI) is(p)=if(!isprime(p), return(0)); my(s=Mod(1, p), P=s); forprime(q=2, p-1, s+=P*=q); s==0 \\ Charles R Greathouse IV, Mar 19 2014 (Python)
from itertools import chain, accumulate, count, islice
from operator import mul
from sympy import prime
def A225728_gen(): return (prime(i+1) for i, m in enumerate(accumulate(accumulate(chain((1, ), (prime(n) for n in count(1))), mul))) if m % prime(i+1) == 0)
Numbers n such that the sum of first n primorial numbers is divisible by n.
+10
0
COMMENTS
The k-th primorial number is defined as the product of the first k primes.
The next term, if it exists, is greater than 14000000. - Alex Ratushnyak, Jun 13 2013 If a prime p | a(n) for some n, then p = 2, p = 523, or p > 10^8. Any such prime is itself a member of this sequence. From this (and a small amount of additional calculation) it follows that any other terms below 10^10 are of the form 2^k * p for p > 10^8. - Charles R Greathouse IV, Feb 09 2014
EXAMPLE
2 + 2*3 + 2*3*5 + 2*3*5*7 = 2 + 6 + 30 + 210 = 248, because 248 is divisible by 4, the latter is in the sequence.
MATHEMATICA
With[{nn=2100}, Select[Thread[{Accumulate[FoldList[Times, Prime[ Range[ nn]]]], Range[nn]}], Divisible[ #[[1]], #[[2]]]&]][[All, 2]] (* Harvey P. Dale, Jul 29 2021 *)
PROG
(Python)
primes = []
n = 1
sum = 2
primorial = 6
def addPrime(k):
global n, sum, primorial
for p in primes:
if k%p==0: return
if p*p > k: break
primes.append(k)
sum += primorial
primorial *= k
n += 1
if sum % n == 0: print(n, end=', ')
print(1, end=', ')
for p in range(5, 100000, 6):
addPrime(p)
addPrime(p+2)
(PARI) list(maxx)={n=prime(1); cnt=1; summ=0; scnt=0;
while(n<=maxx, summ=summ+prodeuler(x=1, prime(cnt), x);
if(summ%cnt==0, scnt++; print(scnt, " ", cnt) ); cnt++; n=nextprime(n+1) ); }
\\note MUST increase precision to 10000+ digits \\ Bill McEachen, Feb 04 2014 (PARI) P=1; S=n=0; forprime(p=2, 1e4, S+=P*=p; if(S%n++==0, print1(n", "))) \\ Charles R Greathouse IV, Feb 05 2014 (Python)
from itertools import accumulate, count, islice
from operator import mul
from sympy import prime
def A225841_gen(): return (i+1 for i, m in enumerate(accumulate(accumulate((prime(n) for n in count(1)), mul))) if m % (i+1) == 0)
The Akiyama-Tanigawa algorithm applied to 1/(1,2,3,5,... old prime numbers). Reduced numerators of the second row.
+10
0
1, 1, 2, 8, 20, 12, 28, 16, 36, 60, 22, 72, 52, 28, 60, 96, 102, 36, 114, 80, 42, 132, 92, 144, 200, 104, 54, 112, 58, 120, 434, 128, 198, 68, 350, 72, 222, 228, 156, 240, 246, 84, 430, 88, 180, 92, 564, 576, 196, 100, 204, 312, 106, 540, 330, 336, 342, 116, 354, 240, 122
COMMENTS
1, 1/2, 1/3, 1/5, 1/7,
1/2, 1/3, 2/5, 8/35, = c(n) = a(n)/b(n)
1/6, -2/15, 18/35,
3/10, -136/105,
67/42
a(n) yields to a permutation of A008578 (via 1, 1, 2, 8, 12, 16, 20, 22, ...): 1, 2, 3, 5, 11, 17, 7, 29,... .
FORMULA
a(n) = (n+1)* A001223(n-1), for n>=3.
EXAMPLE
a(n) is the numerators of c(n): c(0) = 1-1/2 = 1/2, c(1) = 2*(1/2-1/3) = 1/3, c(2) = 3*(1/3-1/5) = 2/5, c(3) = 4*(1/5-1/7)=8/35.
a(3) = 4*2 = 8, a(4) = 5*4 = 20.
MATHEMATICA
a[0, 0] = 1; a[0, m_ /; m > 0] := 1/Prime[m]; a[n_, m_] := a[n, m] = (m+1)*(a[n-1, m ] - a[n-1, m+1]); Table[a[1, m] // Numerator, {m, 0, 60}] (* Jean-François Alcover, Jul 04 2013 *)
Primorial expansion of e.
+10
0
2, 1, 1, 1, 3, 9, 3, 0, 1, 1, 16, 25, 8, 3, 32, 32, 37, 24, 53, 17, 28, 67, 52, 2, 21, 81, 56, 88, 9, 3, 80, 42, 15, 37, 107, 52, 32, 120, 49, 46, 84, 3, 129, 29, 159, 103, 90, 172, 128, 98, 202, 138, 207, 150, 249, 131, 132, 66, 9, 86, 137, 191, 236, 141, 222, 285, 8, 205, 310, 250, 63, 173, 288, 93, 294, 84, 66, 104, 28, 154, 93, 229, 96, 254, 333, 89, 126, 393, 388, 396, 418, 424, 356, 299, 482, 64, 114, 60, 513, 471
COMMENTS
The primorial expansion a(n) of a real number x is defined as x = a(0) + sum(i>0, a(i) / prime(i)# ) where a(0) = floor(x) and 0 <= a(i) < prime(i) for all i > 0.
FORMULA
x(0) = e;
a(n) = floor(x(n));
x(n + 1) = prime(n) * (x(n) - a(n));
where prime(n) = A000040(n) is the n-th prime number. a(n) gives the primorial expansion of x(0) = e.
EXAMPLE
e = 2 + 1/prime(1)# + 1/prime(2)# + 1/prime(3)# + 3/prime(4)# + 9/prime(5)# + ...
where prime(n)# = A002110(n) is the n-th primorial.
MATHEMATICA
pe = Block[{x = #, $MaxExtraPrecision = \[Infinity]},
Do[x = Prime[i] (x - Sow[x // Floor]) // Expand, {i, #2 - 1}];
x // Floor // Sow] // Reap // Last // Last // Function;
pe[E, 100]
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