A372209 -id:A372209 - cp's OEIS frontend

A375008 Products m of k = 3 consecutive primes p_1..p_k, where only p_1 < m^(1/k).

105, 1001, 4199, 20677, 47027, 65231, 146969, 190747, 290177, 347261, 478661, 871933, 1009091, 1201289, 1879981, 2494633, 3127361, 3864241, 4273697, 5171489, 5605027, 6672203, 7566179, 9363547, 10681031, 11592209, 13420567, 15546187, 16965341, 18181979, 19172437

Offset: 1

Michael De Vlieger, Sep 11 2024

In other words, products m of k = 3 consecutive primes p_1..p_k, where floor(log_p_1 m) >= k but floor(log_p_j m) = k-1, j > 1.
For m = 105, floor(log_3 105) > k but floor(log_p_j 105) = k-1 for j > 1.
For m > 105, floor(log_p_1 m) = k but floor(log_p_j m) = k-1 for j > 1.
Superset of A372419.
Does not intersect A372319.

			105 is in the sequence since m = 3*5*7 = 105 is such that 3 is less than the cube root of 105, but both 5 and 7 exceed it.
385 is not in the sequence because m = 5*7*11 = 385 is such that both 5 and 7 are less than the cube root.
1001 is in the sequence since m = 7*11*13 = 1001 is such that 7 < 1001^(1/3), but both 11 and 13 are larger than 1001^(1/3), etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A372209, A372319, A372419.

k = 3; s = {1}~Join~Prime[Range[k - 1]]; Reap[Do[s = Append[Rest[s], Prime[i + k - 1]]; r = Surd[Times @@ s, k]; If[Count[s, _?(# < r &)] == 1, Sow[Times @@ s] ], {i, 120}] ][[-1, 1]]

A375974 Primes p_1 where products m of k = 4 consecutive primes p_1..p_k are such that only p_1 < m^(1/k).

Original entry on oeis.org

113, 139, 181, 211, 293, 337, 421, 631, 811, 839, 863, 887, 953, 1021, 1051, 1069, 1129, 1259, 1307, 1327, 1409, 1471, 1583, 1637, 1669, 1759, 1951, 2069, 2113, 2179, 2221, 2251, 2311, 2423, 2647, 2777, 2819, 2939, 2971, 3137, 3229, 3271, 3517, 3659, 3739, 3779

Offset: 1

Michael De Vlieger, Sep 12 2024

nonn

Let gap g(j) = p_j - p_(j+1), j < k. Primes p_1 such that g(1) is at least as large as g(2) + g(3).
Proper subset of A372209.
Does not intersect A007530.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007530, A372209.

k = 4; s = {1}~Join~Prime[Range[k - 1]]; Reap[Do[s = Append[Rest[s], Prime[i + k - 1]]; r = Surd[Times @@ s, k]; If[Count[s, _?(# < r &)] == 1, Sow[Prime[i]] ], {i, 4000}] ][[-1, 1]]

A374873 Smallest primes p_1 where products m of n consecutive primes p_1..p_n are such that only p_1 < m^(1/n).

Original entry on oeis.org

2, 3, 113, 3229, 15683, 736279, 8332427, 37305713, 4948884397, 6193302809, 316781230427

Offset: 2

Michael De Vlieger, Sep 19 2024

			a(2) = 2 since m = 2*3 = 6 and 3 > sqrt(6).
a(3) = 3 since m = 3*5*7 = 105 and 5 > 105^(1/3).
a(4) = 113 since m = 113 * 127 * 131 * 137 = 257557397 and 127 > 257557397^(1/4), etc.

Cf. A372209, A375974, A376136.

k = 1; Table[r = Range[0, n - 1]; While[(Set[{p, q, m}, {#[[1]], #[[2]], Times @@ #}]; q < Surd[m, n]) &[Prime[k + r]], k++]; p, {n, 2, 6}]

a(n) = {my(ps = vector(n, k, prime(k))); forprime(p = prime(n+1), , if(ps[2]^n > vecprod(ps), return(ps[1])); ps = concat(vecextract(ps, "^1"), p));} \\ Amiram Eldar, Sep 23 2024

a(10)-a(12) from Amiram Eldar, Sep 23 2024

cp's OEIS Frontend

A375008 Products m of k = 3 consecutive primes p_1..p_k, where only p_1 < m^(1/k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A375974 Primes p_1 where products m of k = 4 consecutive primes p_1..p_k are such that only p_1 < m^(1/k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A374873 Smallest primes p_1 where products m of n consecutive primes p_1..p_n are such that only p_1 < m^(1/n).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions