Lubomir Alexandrov - cp's OEIS frontend

A065505 Let p(k) denote k-th prime; consider solutions (p(n),p(m)) of Diophantine equation p(p(n)+1)-6.p(p(m))=1 (*), where p(p(n)) belongs to A060213 and p(p(m))=(p(p(n))+1)/6; sequence gives values of p(n).

7, 2309, 2753, 2789, 26183, 46933, 53597, 58411, 61357, 69481, 87691, 111487, 124991, 134327, 140659, 144651, 147551, 236519, 247711, 164643, 270223, 291359

Offset: 0

Lubomir Alexandrov, Nov 25 2001

nonn

			p(n)=395581 and p(m)=75277 satisfy equation (*) at the primes p(p(n))=5730617 and p(p(m))=955103.

A060213

A065511 Let p(k) denote k-th prime; consider solutions (n,m) of the Diophantine system {p(p(n)+1)-p(p(n))=2, p(p(n))-6.p(p(m))=-1} (*); sequence gives values of m.

Original entry on oeis.org

1, 92, 105, 106, 689, 1138, 1280, 1373, 1432, 1600, 1960, 2416, 2683, 2846, 2968, 3042, 3091, 4694, 4884, 5191, 5284, 5642, 6905, 6949, 7074, 7095, 7213, 7274, 7418

Offset: 0

Lubomir Alexandrov, Nov 26 2001

nonn

( p(p(n)), p(p(n)+1) ) is twin prime pair with average 6.p(p(m)) (A060213).

			n = 402 and m = 105 satisfy system (*) at the primes p(p(n)) = 24917 and p(p(m)) = 4153; n = 33521 and m = 7418 satisfy system (*) at the primes p(p(n)) = 5730617 and p(p(m)) = 955103.

A060213

A063502 a(n+1) = p, where p is the a(n)-th twin prime (p,p+2), with a(0) = 1.

Original entry on oeis.org

1, 3, 11, 137, 5639, 641129, 152921807, 65818751039, 46091763604421

Offset: 0

Lubomir Alexandrov, Jul 30 2001

Instead of starting with a(0) = 1 for the first twin prime (3,5) other sequences can be formed for a(0) = 2, i.e. 2nd twin prime: 2, 5, 29, 641, 44381, 7212059, etc., a(0) = 4: 4, 17, 239, 12161, 1583927, etc., a(0) = 6: 6, 41, 1151, 93251, 16989317, etc., a(0) = 7: 7, 59 1931,176021, 35263691, etc., a(0) = 8: 8, 71, 2339,221201, 45749309 and so on.

			a(3) = 137 because a(2) = 11 and the 11th twin prime is (137,139).

Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x) [From _M. F. Hasler_, Mar 02 2009]

Cf. A007097.

(* Computes up to a(6) only *) tp[n_] := (* = A001359 *) tp[n] = (p = NextPrime[tp[n-1]]; While[ !PrimeQ[p+2], p = NextPrime[p]]; p); tp[1] = 3; Do[tp[n], {n, 2, 10^6}]; a[n_] := a[n] = tp[a[n-1]]; a[0]=1; Table[ Print[ a[n]]; a[n], {n, 0, 6}] (* Jean-François Alcover, Dec 13 2011 *)

a(n+1) = A001359(a(n)); a(0)=1. [M. F. Hasler, Mar 02 2009]

Edited by Frank Ellermann, Jan 25 2002
Offset and example corrected by Farideh Firoozbakht, Dec 07 2008
a(7) = 65818751039 from Zak Seidov and Farideh Firoozbakht, Dec 13 2008
Computed a(8)=46091763604421 using data from T. Oliveira e Silva. - M. F. Hasler, Mar 02 2009

A064110 Let s(n) = n-th single prime (cf. A007510). Sequence is defined by recurrence a(n+1) = s(a(n)), n = 0,1,2,..., a(0)=1.

Original entry on oeis.org

1, 2, 23, 263, 2917, 38639, 603311, 11093633, 236524303, 5782539281

Offset: 0

Lubomir Alexandrov, Sep 07 2001

This is the "isolated prime Eratosthenes progression at base 1 (ipep(1))". The next ipep are: ipep(3) = 3, 37, 397, 4751, 64403, 1038629, 19661749,...; ipep(4) = 4, 47, 491, 5897, 81131, 1328167, 25467419,...; ipep(5) = 5, 53, 557, 6709, 93287, 1541191, 29778547,...; ...; ipep(22)= 22, 257, 2861, 37799, 589181, 10821757, 230452837,... ipep(24)= 24, 277, 3079, 40823, 640121, 11807167, 252480587,... and so on.

In the terminology of A007097 the name is "isolated_prime-th recurrence ..."

"Isolated Primes", by Richard L. Francis, J. Rec. Math., 11 (1978), 17-22.

Cf. A007097, A063502.

a(9) from Sean A. Irvine, Jun 12 2023

A065503 Indices k of primes p(k) such that p(k) is in A060213.

Original entry on oeis.org

5, 7, 10, 13, 26, 33, 60, 113, 116, 142, 265, 288, 313, 332, 353, 384, 408, 484, 498, 542, 625, 636, 719, 805, 864, 1064, 1070, 1194, 1328, 1456, 1477, 1528, 1538, 1571, 1623, 1627, 1651, 1660, 1867, 2003, 2216, 2244, 2309, 2311, 2418, 2438, 2469, 2616, 2753

Offset: 1

Lubomir Alexandrov, Nov 25 2001

nonn

Original name: Let p(k) denote k-th prime; consider solutions (x,y) of Diophantine equation p(x+1)-6p(y)=1 (*), where p(x) belongs to A060213 and p(m)=(p(n)+1)/6; sequence gives values of x.

			x=13084 and y=2612 satisfy equation (*) at the primes p(x)=140837 and p(y)=23473.

Cf. A060213, A065505.

Offset corrected, more terms, and title clarified by Sean A. Irvine, Sep 03 2023

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User: Lubomir Alexandrov

A065505 Let p(k) denote k-th prime; consider solutions (p(n),p(m)) of Diophantine equation p(p(n)+1)-6.p(p(m))=1 (*), where p(p(n)) belongs to A060213 and p(p(m))=(p(p(n))+1)/6; sequence gives values of p(n).

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A065511 Let p(k) denote k-th prime; consider solutions (n,m) of the Diophantine system {p(p(n)+1)-p(p(n))=2, p(p(n))-6.p(p(m))=-1} (*); sequence gives values of m.

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A063502 a(n+1) = p, where p is the a(n)-th twin prime (p,p+2), with a(0) = 1.

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A064110 Let s(n) = n-th single prime (cf. A007510). Sequence is defined by recurrence a(n+1) = s(a(n)), n = 0,1,2,..., a(0)=1.

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A065503 Indices k of primes p(k) such that p(k) is in A060213.

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