Displaying 1771-1778 of 1778 results found.
a(n) = n for n <= 2. Thereafter a(n) is the least novel multiple of the greatest prior term which is coprime to a(n-1).
+10
0
1, 2, 3, 4, 6, 5, 12, 10, 9, 20, 18, 15, 8, 30, 7, 60, 14, 45, 28, 90, 21, 40, 42, 25, 84, 50, 63, 100, 126, 75, 56, 150, 35, 36, 70, 27, 200, 189, 400, 378, 125, 756, 250, 567, 800, 1134, 375, 112, 750, 49, 1600, 1701, 3200, 3402, 500, 5103, 6400, 10206, 625, 20412
COMMENTS
The only way a prime can enter the sequence is consequent to a primorial term, thus prime(k) follows A002110(k). However it seems that only the first 4 primorial numbers (1,2,6,30) appear in the sequence, and that consequently primes > 7 do not become terms. Conjecture: All terms are 7-smooth numbers. Is this a permutation of A002473? Sequence enters a multiplicative recurrence for n > 110 such that a(73(m+1)+j)/a(73m+j) is a prime power p(k)^e(k) in S = {2, 2^6, 2^12, 3^10, 5^4, 7}.
Since a(1..110) are all 7-smooth and so is S, the sequence is in A002473 but is not a permutation, since we are missing 105, 210, and 2^k or 3^k for k > 3, and there is no way to produce these given S. There are 4 primes, 9 squarefree composites in the sequence; together with a(1) = 1, these are 14 of the 16 divisors of 210. {2^k : k=0..3}, {3^k : k=0..3}, A000351 and A000420 are each contained in this sequence. a(73m+50) = 7^(m+2), m >= 0. Aside from these, the sequence is in A126706 (numbers neither prime powers nor squarefree). (End)
MATHEMATICA
nn = 2^12; c[_] := False; m[_] := 1;
Array[Set[{a[#], c[#]}, {#, True}] &, 2]; w = {1}; j = 2;
Do[k = SelectFirst[w, CoprimeQ[j, #] &];
While[c[k*m[k]], m[k]++]; k *= m[k];
w = Insert[w, j, LengthWhile[w, # > j &] + 1];
Set[{a[n], c[k], j}, {k, True, k}], {n, 3, nn}];
PROG
(PARI) findgcd(v, k) = v = vecsort(v, , 4); for (i=1, #v, if (gcd(v[i], k) == 1, return(v[i])); );
findmult(k, v) = v = vecsort(v); for (i=1, oo, if (!vecsearch(v, i*k), return (i*k)));
lista(nn) = my(va = vector(nn)); va[1] = 1; va[2] = 2; for (n=3, nn, my(v = Vec(va, n-1)); my(x = findgcd(v, va[n-1])); my(y = findmult(x, v)); va[n] = y; ); va; \\ Michel Marcus, Sep 08 2023
a(n) = binomial(primorial(n), n).
+10
0
1, 2, 15, 4060, 78738660, 545754554499462, 1018081517447240182211275, 1793004475784081302284255717158418120, 1943305407393725342965469143054357602760779899437185, 3772316402417100592416011698371929155605067111502494326520988270728160
PROG
(Python)
from sympy import binomial, primorial
a = lambda n: binomial(primorial(n), n)
print([a(n) for n in range(1, 10)])
(PARI) a(n) = binomial(vecprod(primes(n)), n); \\ Michel Marcus, Sep 14 2023
Primes of the form (k-th primorial) - (k+1)st prime.
+10
0
23, 199, 2297, 30013, 9699667, 3217644767340672907899084554047, 267064515689275851355624017992701, 23768741896345550770650537601358213, 1492182350939279320058875736615841068547583863326864530259
COMMENTS
Conjecture: sequence is infinite.
EXAMPLE
primorial(4) - prime(4+1) = 2*3*5*7 - prime(5) = 210 - 11 = 199, which is prime, so 199 is a term.
CROSSREFS
A038708 with subtraction instead of addition.
Smallest nonprime that is the n-th prime plus a multiple of the (n-1)-st primorial.
+10
0
4, 9, 35, 187, 221, 2323, 120137, 1021039, 19399403, 223092899, 6469693261, 200560490167, 7420738134851, 304250263527253, 39248283995010137, 614889782588491463, 65178316954380089519, 3845520700308425278201, 117288381359406970983337, 7858321551080267055879161
EXAMPLE
For n = 1, the n-th prime (2) plus a multiple m of the (n-1) primorial (1) is 2+m, giving the smallest nonprime, a(1)=4, when m=2.
For n = 4, the n-th prime (7) plus a multiple m of the (n-1) primorial (30) is 7+30m, giving the smallest nonprime, a(4)=187, when m=6.
MATHEMATICA
a[n_] := Module[{p = Prime[n], r = Product[Prime[i], {i, 1, n - 1}]}, While[p += r; PrimeQ[p]]; p]; Array[a, 20] (* Amiram Eldar, Dec 07 2023 *)
3, 11, 61, 457, 5237, 1226677, 34543329507310391, 1636619248175258407, 5186576044693944076609, 742779051038516950393163206833793, 1506853388294906471801157206440769816406928024502711, 651879075122842895567706351814676957742356330143458665568047
COMMENTS
Primes which are the sum of the numerator and the denominator of partial sums of the reciprocals of primes.
Each term of this sequence can be expressed as the sum of an expression with exactly one odd term and n even terms, where the odd term is A002110(n)/ A002110(1), and n > 0 (see Alexander Adamchuk comment in A024528). a(2) = 11 = 3 + 2 + 6 contains the only prime odd term 3.
EXAMPLE
3 is a term because 1/2 = 1/2 and 1 + 2 = 3 which is prime.
11 is a term because 1/2 + 1/3 = 5/6 and 5 + 6 = 11 which is prime.
61 is a term because 5/6 + 1/5 = 31/30 and 31 + 30 = 61 which is prime.
457 is a term because 31/30 + 1/7 = 247/210 and 247 + 210 = 457 which is prime.
a(n) is the least integer m such that p#*m - 1 is prime for all primes p <= prime(n).
+10
0
2, 2, 2, 2, 9, 9, 9, 224719, 384427, 16114470, 259959472, 13543584514, 100318016379, 100318016379
PROG
(PARI) isok(m, n) = for(i=1, n, my(P = vecprod(primes(i))); if (! isprime(P*m-1), return(0)); ); return(1);
a(n) = my(m=1); while (!isok(m, n), m++); m;
a(n) is the largest k such that tau(k)^n >= k.
+10
0
2, 1260, 27935107200, 29564884570506808579056000
COMMENTS
Let prime(j)# denote the product of the first j primes, A002110(j); then a(1) = prime(1)# = 2,
a(2) = 6*prime(4)# = 1260,
a(3) = 2880*prime(8)# = 2.7935...*10^10,
a(4) = 907200*prime(16)# = 2.9564...*10^25,
a(5) >= 259459200*prime(30)# = 8.2015...*10^54,
a(6) >= 3238237626624000*prime(52)# = 3.4403...*10^111,
a(7) >= 248818180782850398720000*prime(91)# = 5.4351...*10^218.
EXAMPLE
27935107200 = 2^7 * 3^3 * 5^2 * 7^1 * 11^1 * 13^1 * 17^1 * 19^1,
so tau(27935107200) = (7+1)*(3+1)*(2+1)*(1+1)*(1+1)*(1+1)*(1+1)*(1+1) = 8*4*3*2*2*2*2*2 = 3072; 3072^3 = 28991029248 > 27935107200, and there is no larger number k such that tau(k)^3 >= k, so a(3) = 27935107200.
Square array T(n, k), n > 1 and k >= 1, read by antidiagonals in ascending order, give the smallest number that starts a sequence of exactly k consecutive numbers, each having exactly n distinct prime factors (counted without multiplicity), or -1 if no such number exists.
+10
0
6, 30, 14, 210, 230, 20, 2310, 7314, 644, 33, 30030, 254540, 37960, 1308, 54, 510510, 11243154, 1042404, 134043, 2664, 91, 9699690, 965009045, 323567034, 21871365, 357642, 6850, 142
COMMENTS
All positive terms are composite.
EXAMPLE
T(2,3) = 20 = 2^2 * 5, because both 21 and 22 have the same omega. Thus, 20 is the starting number of a run of 3 numbers that each have same omega, i.e. 2. No lesser number has this property, so T(2,3) = 20.
Table begins (upper left corner = T(2,1)):
6 14 20 33 ...
30 230 644 1308 ...
210 7314 37960 134043 ...
2310 254540 1042404 21871365 ...
30030 11243154 323567034 7933641735 ...
... ... ... ... ...
Search completed in 0.953 seconds
|