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.

A111096 Partial sums of A137701.

Original entry on oeis.org

16, 232, 59281, 10059281, 4049575228945, 1950244643588320, 30041944445326335483061, 32095019157463691981298869, 142108579247039194637916834814494, 108199957883829576141601541930838816381470, 118558455387984539329682688832638841343258239487
Offset: 1

Views

Author

Jonathan Vos Post, Oct 13 2005

Keywords

Comments

a(n) is prime for n = 3, 4, ..., a(n) is semiprime for n = 7, 8, 11, ...

Examples

			a(1) = 16 because semiprime(1)^prime(1) = 4^2 = 16.
a(2) = 232 because 4^2 + 6^3 = 232.
a(3) = 59281 = 4^2 + 6^3 + 9^5, which is a prime.
a(4) = 10059281 = 4^2 + 6^3 + 9^5 + 10^7, which is a prime.
a(7) = 4^2 + 6^3 + 9^5 + 10^7 + 14^11 + 15^13 + 21^17 = 428081461 * 70178102025601, which is semiprime.
a(8) = 4^2 + 6^3 + 9^5 + 10^7 + 14^11 + 15^13 + 21^17 + 22^19 = 47 * 682872748031142382580827, which is semiprime.
a(11) = 4^2 + 6^3 + 9^5 + 10^7 + 14^11 + 15^13 + 21^17 + 22^19 + 25^23 + 26^29 + 33^31 = 17 * 6974026787528502313510746401919931843721072911 which is semiprime.

Crossrefs

Cf. A000040, A001358, A137701, A125580.

Formula

a(n) = Sum_{i=1..n} A001358(i)^A000040(i).

A125850 Sums of two or more distinct terms from A137701.

Original entry on oeis.org

232, 59065, 59265, 59281, 10000016, 10000216, 10000232, 10059049, 10059065, 10059265, 10059281, 4049565169680, 4049565169880, 4049565169896, 4049565228713, 4049565228729, 4049565228929, 4049565228945, 4049575169664, 4049575169680, 4049575169880, 4049575169896
Offset: 1

Views

Author

Jonathan Vos Post, May 03 2008

Keywords

Crossrefs

Cf. A111096, A137701.

Extensions

More terms from Jinyuan Wang, Jul 29 2022

A140052 Indices m such that A114850(m)+A114850(k) is prime for some k

Original entry on oeis.org

6, 9, 9, 19, 20, 25, 33, 38, 40, 59, 69, 76, 99, 111, 126, 141, 147, 167, 188, 202, 211, 211, 220, 238, 263, 264, 279, 284, 297, 329, 336, 354, 407, 407, 407, 410, 426, 540, 568, 652, 683, 696, 769, 780, 948, 951
Offset: 1

Views

Author

Jonathan Vos Post, May 03 2008

Keywords

Comments

The associated primes b(n), which grow too quickly for many to be given as a sequence themselves, are {primes of the form A114850(a) + A114850(b)} = {primes of the form A114850(a) + A114850(b)} and begin as follows. b(1) = 437893890380859631 = 256 + 437893890380859375 = 4^4 + 15^15 = semiprime(1)^semiprime(1) + semiprime(6)^semiprime(6).
b(2) = 88817841970012523233890533447265881 = 256 + 88817841970012523233890533447265625 = 4^4 + 25^25 = semiprime(1)^semiprime(1) + semiprime(9)^semiprime(9).
b(3) = 46656 + 88817841970012523233890533447265625 = 6^6 + 24^25 = semiprime(2)^semiprime(2) + semiprime(9)^semiprime(9). This is to A068145 "Primes of the form a^a + b^b" as A001358 semiprimes is to A000040 primes; and as A114850 "(n-th semiprime)^(n-th semiprime)" is to A051674 "(n-th prime)^(n-th prime)."
M. F. Hasler gave the present definition which allows us to list merely the indices, which in the 3 examples above, are [6, 1],[9, 1],[9, 2]. The first 13 [m,k] value pairs are (as found by M. F. Hasler as an extension) are [6, 1], [9, 1], [9, 2], [19, 5], [20, 8], [25, 7], [33, 11], [38, 6], [40, 33], [59, 14], [69, 62], [76, 57], [99, 22]. Hence our sequence begins a(1) = 6, a(2) = 9, a(3) = 9. For the sequence of corresponding k values {1, 1, 2, 5, 8, ...}, see A140053.

Examples

			a(1) = 6 because semiprime(6)^semiprime(6) + semiprime(1)^semiprime(1) = 15^15 + 4^4 = 437893890380859375 + 256 = 437893890380859631 is prime.

Crossrefs

Cf. A000040, A001358, A051674, A068145, A114850, A137701, A140053.

Programs

Formula

A001358(a(n))^A001358(a(n)) + A001358(A140053(n))^A001358(A140053(n)) is prime.

Extensions

a(14)-a(46) from Donovan Johnson, Nov 11 2008

A140053 Indices k such that A114850(m)+A114850(k) is prime for some m>k.

Original entry on oeis.org

1, 1, 2, 5, 8, 7, 11, 6, 33, 14, 62, 57, 22, 7, 86, 61, 28, 70, 66, 134, 77, 131, 107, 58, 161, 252, 240, 52, 155, 32, 152, 322, 167, 200, 284, 258, 28, 173, 95, 563, 369, 57, 58, 126, 113, 369
Offset: 1

Views

Author

Jonathan Vos Post, May 03 2008

Keywords

Comments

The associated primes b(n), which grow too quickly for many to be given as a sequence themselves, are {primes of the form A114850(a) + A114850(b)} = {primes of the form A114850(a) + A114850(b)} and begin as follows. b(1) = 437893890380859631 = 256 + 437893890380859375 = 4^4 + 15^15 = semiprime(1)^semiprime(1) + semiprime(6)^semiprime(6).
b(2) = 88817841970012523233890533447265881 = 256 + 88817841970012523233890533447265625 = 4^4 + 25^25 = semiprime(1)^semiprime(1) + semiprime(9)^semiprime(9).
b(3) = 46656 + 88817841970012523233890533447265625 = 6^6 + 24^25 = semiprime(2)^semiprime(2) + semiprime(9)^semiprime(9). This is to A068145 "Primes of the form a^a + b^b" as A001358 semiprimes is to A000040 primes; and as A114850 "(n-th semiprime)^(n-th semiprime)" is to A051674 "(n-th prime)^(n-th prime)."
M. F. Hasler gave the present definition which allows us to list merely the indices, which in the 3 examples above, are [6, 1],[9, 1],[9, 2]. The first 13 [m,k] value pairs are (as found by M. F. Hasler as an extension) are [6, 1], [9, 1], [9, 2], [19, 5], [20, 8], [25, 7], [33, 11], [38, 6], [40, 33], [59, 14], [69, 62], [76, 57], [99, 22]. Hence our sequence begins a(1) = 6, a(2) = 9, a(3) = 9. For the sequence of corresponding k values {6, 9, 9, 19, 20, ...}, see A140052.

Examples

			a(1) = 1 because semiprime(6)^semiprime(6) + semiprime(1)^semiprime(1) = 15^15 + 4^4 = 437893890380859375 + 256 = 437893890380859631 is prime.

Crossrefs

Cf. A000040, A001358, A051674, A068145, A114850, A137701, A140052.

Programs

Formula

A001358(a(n))^A001358(a(n)) + A001358(A140052(n))^A001358(A140052(n)) is prime.

Extensions

a(14)-a(46) from Donovan Johnson, Nov 11 2008
Showing 1-4 of 4 results.

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