cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A112600 Smallest prime factor of A111392(n).

Original entry on oeis.org

2, 5, 11, 37, 13, 23, 19, 23, 37, 127, 47, 61, 61, 47, 67, 61, 277, 83, 79, 97, 127, 83, 101, 131, 269, 109, 131, 109, 113, 157, 137, 181, 157, 181, 151, 173, 173, 179, 173, 211, 223, 251, 193, 197, 223, 233, 223, 251, 271, 241, 239, 269, 293, 281, 313, 347, 293
Offset: 1

Views

Author

Yasutoshi Kohmoto, Dec 15 2005

Keywords

Comments

For all i, if iA111392(n))=1, where p_i is i-th prime.
A111392: a(n) = Product_{i=1..n-1} (Product_{k=1..i} p_k + Product_{k=i+1..n} p_k). - Robert G. Wilson v, Dec 22 2005

Crossrefs

Cf. A111392.

Programs

  • Mathematica
    f[n_] := Product[(Product[Prime[k], {k, i}] + Product[Prime[k], {k, i + 1, n}]), {i, n - 1}]; f[1] = 2; g[n_] := Block[{k = 1}, While[Mod[f[n], Prime[k]] != 0, k++ ]; Prime@k]; Array[g, 20] (* Robert G. Wilson v *)

Extensions

More terms from Robert G. Wilson v, Dec 22 2005

A112601 a(n) = prime(b(n)) where b(n) = b(n-2) + a(n-1) (with b(1)=1, b(2)=2).

Original entry on oeis.org

2, 3, 7, 23, 103, 613, 4751, 47137, 582511, 8758339, 156819893, 3283370969, 79174605361, 2171048919947, 66970610115763, 2302616062156639, 87542957597514007, 3654858165039471959
Offset: 1

Views

Author

Yasutoshi Kohmoto, Dec 15 2005

Keywords

Crossrefs

Cf. A111392, A112237.

Programs

  • Mathematica
    a[n_] := a[n] = Prime[b[n]]; b[1] = 1; b[2] = 2; b[n_] := b[n] = b[n - 2] + a[n - 1]; Array[a, 15] (* Robert G. Wilson v, Dec 22 2005 *)

Extensions

Better definition from Dean Hickerson, more terms from Emeric Deutsch, Dec 17 2005
a(12)-a(15) from Robert G. Wilson v, Dec 22 2005
a(16)-a(18) from Amiram Eldar, Sep 12 2022

A112404 a(n) = (Product_{e_i=0..1} (Product_{i=1..n} p_i^e_i + Product_{i=1..n} p_i^(1-e_i)))^(1/2) where p_i is the i-th prime.

Original entry on oeis.org

3, 35, 75361, 105640931881921, 368107372881122974005026861194791580321, 1068920105772796102633531337368359482127315843763564268088796774223747755119986736765386063992951681
Offset: 1

Views

Author

Yasutoshi Kohmoto

Keywords

Comments

This is a "Proof of existence of infinitely many primes" sequence. Proof. Let N = (Product_{e_i=0..1} (Product_{i=1..n} p_i^e_i + Product_{i=1..n} p_i^(1-e_i)))^(1/2). Suppose there are only a finite number of primes p_i, 1 <= i <= n. If N is prime, then for all i, N != p_i because, for all i, p_i < N. If N is composite, then it must have a prime divisor p which is different from primes p_i because, for all i, N !== 0 (mod p_i).
The numbers of decimal digits of a(n) are 1, 2, 5, 15, 39, 100, 246, 590, 1387, 3215, 7321, 16507, 36823, 81305, 178212, 388495, 842638, 1816984, ..., . - Robert G. Wilson v
The numbers of prime factors of a(n) are 1, 2, 4, 8, 16, 33, 69, 136, 280, 566, 1107, ..., . - Robert G. Wilson v

Examples

			a(3) = ((1 + p_1*p_2*p_3)*(p_3 + p_1*p_2)*(p_2 + p_1*p_3)*(p_2*p_3 + p_1)*(p_1 + p_2*p_3)*(p_1*p_3 + p_2)*(p_1*p_2 + p_3)*(p_1*p_2*p_3 + 1))^(1/2)
= (1 + p_1*p_2*p_3)*(p_3 + p_1*p_2)*(p_2 + p_1*p_3)*(p_2*p_3 + p_1) = 31*11*13*17.

Crossrefs

Cf. A111392.

Programs

  • Mathematica
    f[n_] := Block[{a = 1, p = Prime@Range@n, k = 0, lmt = 2^(n - 1)}, While[k < lmt, e = IntegerDigits[k, 2, n]; a = a*(Times @@ (p^e) + Times @@ (p^(1 - e))); k++ ]; a]; Array[f, 7] (* Robert G. Wilson v *)

Extensions

Edited by Robert G. Wilson v, Dec 10 2005

A113270 a(n) = sqrt(Product_{k=1..2^n} (Product_{i=1..n} p_i^e_{k,i} + Product_{i=1..n} p_i^(1-e_{k,i}))) * Sum_{i=1..n} ((1/p_i)*Product_{k=1..n} p_k) where p_i is the i-th prime and e_{k,i} is a vector of length n that runs through all combinations of {0,1}.

Original entry on oeis.org

3, 175, 2336191, 26093310174834487, 1077450280423046944912713622717154955599567
Offset: 1

Views

Author

Yasutoshi Kohmoto, Jan 07 2006

Keywords

Comments

This is a "Proof of existence of infinitely many primes" sequence. Proof. Let N = (Product_{e_i=0..1} (Product_{i=1..n} p_i^e_i + Product_{i=1..n} p_i^(1-e_i)))^(1/2) * (Sum_{i=1..n} (1/p_i*Product_{k=1..n} p_k)). Suppose there are only a finite number of primes p_i, 1 <= i <= n. If N is prime, then for all i, N != p_i because, for all i, p_i < N. If N is composite, then it must have a prime divisor p which is different from primes p_i because, for all i, N !== 0 (mod p_i).

Examples

			a(3) = ((1 + p_1*p_2*p_3)*(p_3 + p_1*p_2)*(p_2 + p_1*p_3)*(p_2*p_3 + p_1)*(p_1 + p_2*p_3)*(p_1*p_3 + p_2)*(p_1*p_2 + p_3)*(p_1*p_2*p_3 + 1))^(1/2) * (p_2*p_3 + p_1*p_3 + p_1*p_2)
= (1 + p_1*p_2*p_3)*(p_3 + p_1*p_2)*(p_2 + p_1*p_3)*(p_2*p_3 + p_1) * (p_2*p_3 + p_1*p_3 + p_1*p_2)
= 31*11*13*17*31.

Links

Crossrefs

Cf. A111392.

Programs

  • Mathematica
    a[n_] := Module[{e = Tuples[{0, 1}, n]}, (Product[Product[Prime[i]^e[[j]][[i]], {i, 1, n}] + Product[Prime[i]^(1 - e[[j]][[i]]), {i, 1, n}], {j, 1, 2^n}])^(1/2) ]*Sum[1/Prime[i], {i, 1, n}]*  Product[Prime[i], {i, 1, n}]; Array[a, 6] (* Amiram Eldar, Nov 23 2018 *)

Extensions

Name clarified by and a(5) from Amiram Eldar, Nov 23 2018
Showing 1-4 of 4 results.

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