A065311 -id:A065311 - cp's OEIS frontend

A065312 Primes which occur exactly once in A065308 (prime(n - pi(n))).

7, 11, 19, 23, 37, 41, 47, 53, 59, 61, 73, 79, 83, 89, 101, 103, 113, 127, 137, 139, 149, 151, 163, 167, 173, 179, 193, 197, 199, 211, 227, 229, 241, 251, 257, 263, 271, 277, 283, 293, 307, 311, 317, 331, 337, 347, 349, 353, 367, 373, 389, 397, 419, 421, 433

Offset: 1

Labos Elemer, Oct 29 2001

nonn

In A065308 each odd prime seems to appear once or twice. Prime 2 arises there 3 times.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000040, A000720, A054546, A065308, A065309, A065310, A065311.

With[{s = Array[Prime[# - PrimePi[#]] &, 120]}, Most@ Select[Split[s], Length@ # == 1 &][[All, 1]] ] (* Michael De Vlieger, Jun 19 2018 *)

{ n=0; p=1; f=2; m=1; for (i=1, 10^9, a=0; p=nextprime(p + 1); while (p==f, a++; m++; f=prime(m - primepi(m))); if (a==1, write("b065312.txt", n++, " ", p); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 16 2009

A065610 Smallest number m so that n^2 + A000330(m) is also a square, i.e., n^2 + (1 + 4 + 9 + 16 + ... + m^2) = w^2 for some w.

Original entry on oeis.org

1, 47, 2, 5, 767, 16, 1727, 22, 17, 13, 18, 112, 10, 70, 8, 10799, 12287, 21, 82, 17327, 31, 15, 255, 16, 10, 13, 9, 5, 49, 40367, 43199, 117, 17, 1630, 7, 58799, 10, 65711, 34, 73007, 49, 13, 64, 29, 17, 6, 9, 30, 42, 309, 8, 124847, 17, 31, 139967, 13, 150527, 15

Offset: 0

Labos Elemer, Nov 07 2001

nonn

I.e., a(n) is the least solution to n^2 + (x(x+1)(2x+1)/6) = w^2; a(n) is the length of shortest sum of consecutive squares from 1 to a(n) which when added to n^2 gives a new square.

			n = 3: a(3) = 5 because n^2 + 1 + 4 + 9 + 16 + 25 = 9 + (1 + 4 + 9 + 16 + 25) = 64 = 8*8; n = 4: a(4) = 767 because n^2 + (1 + 4 + ... + 767^2) = 150700176 = 12276*12726, where 767 is the length of the shortest such consecutive-square sequence which provides (when summed) a new square, namely 12276^2. Often the least solution is rather large. E.g., at n = 93, a(n) = 415151, which means that 93^2 + A000330(415151) = 8649 + (long square sum) = 154436265^2 = 23850559947150225 is the smallest such square number, sum odd distinct consecutive squares except one repetition(8649).

Cf. A000330, A065311, A065312, A065313, A065314, A065315.

s=n^2 Do[s=s+m^2; If[IntegerQ[Sqrt[s]], Print[m]], {m, 1, 500000}] (* gives solutions of which the smallest is entered into the sequence *)

n^2 + (1 + 4 + 9 + ... + a(n)^2) = w^2, where w depends also on n; i.e., sum of consecutive squares from 1, 4, ... to a(n)^2 + n^2 is also a square.

A065611 Let k be the least integer such that n^2 + Sum_{m=1..k} m^2 is a perfect square, then a(n) is the resulting square.

Original entry on oeis.org

1, 35721, 9, 64, 150700176, 1521, 1718434116, 3844, 1849, 900, 2209, 474721, 529, 116964, 400, 419845682025, 618399795456, 3600, 187489, 1734149230641, 10816, 1681, 5560164, 2025, 961, 1444, 961, 784, 41209, 21926752125201

Offset: 0

Labos Elemer, Nov 07 2001

nonn

I.e., n^2 + {1 + 4 + 9 + 16 + ... + m^2} = a(n) = A065612(n)^2 = A065311(n). a(n) is the smallest square obtained as n^2 + x*(x+1)*(2x+1)/6 where x = A065610(n).

			n = 3: a(3) = 64 because n^2 + 1 + 4 + 9 + 16 + 25 = 9 + (1 + 4 + 9 + 16 + 25) = 64 = 8^2;
n = 4: a(4) = 150700176 because n^2 + (1 + 4 + ... + 767^2) = 150700176 = 12276^2, where 767 is the length of the shortest such consecutive-square sequence which provides(when summed) a new square, namely 12276^2. Often the least solution is rather large. E.g., a(93) = 23850559947150225 which means that 93^2 + A000330(415151) = 8649 + [a long square sum] = 154436265^2 = 23850559947150225.

Harry J. Smith, Table of n, a(n) for n = 0..500

Cf. A000330, A065610, A065612.

Do[s = n^2; k = 1; While[s = s + k^2; !IntegerQ[ Sqrt[s]], k++ ]; Print[s], {n, 0, 30} ]

{ for (n = 0, 500, s=n^2 + 1; k=1; while (!issquare(s), k++; s+=k^2); write("b065611.txt", n, " ", s) ) } \\ Harry J. Smith, Oct 23 2009

a(n) = my(s=n^2+1, k=1); while (!issquare(s), k++; s+=k^2); s; \\ Michel Marcus, Mar 24 2020

Edited by Jon E. Schoenfield, Jun 14 2018

Name clarified by Michel Marcus, Mar 24 2020

A065994 a(n) = prime(prime(n) - n).

Original entry on oeis.org

2, 2, 3, 5, 13, 17, 29, 31, 43, 67, 71, 97, 107, 109, 131, 157, 181, 191, 223, 233, 239, 269, 281, 313, 359, 379, 383, 401, 409, 431, 503, 523, 569, 571, 619, 631, 659, 691, 719, 751, 787, 797, 857, 859, 881, 883, 971, 1039, 1061, 1063, 1091, 1117, 1123

Offset: 1

Reinhard Zumkeller, Dec 10 2001

a(n) = A065311(n-2) for 3 < n <= 10000. - Georg Fischer, Oct 19 2018

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000040, A014689, A065311.

A065994:=n->ithprime(ithprime(n)-n): seq(A065994(n), n=1..100); # Wesley Ivan Hurt, Jan 21 2017

Table[Prime[Prime[n]-n],{n,60}] (* Harvey P. Dale, Sep 04 2011 *)

{ for (n=1, 1000, a=prime(prime(n) - n); write("b065994.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 06 2009

a(n) = A000040(A014689(n)). - Reinhard Zumkeller, Aug 06 2003

cp's OEIS Frontend

A065312 Primes which occur exactly once in A065308 (prime(n - pi(n))).

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A065610 Smallest number m so that n^2 + A000330(m) is also a square, i.e., n^2 + (1 + 4 + 9 + 16 + ... + m^2) = w^2 for some w.

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A065611 Let k be the least integer such that n^2 + Sum_{m=1..k} m^2 is a perfect square, then a(n) is the resulting square.

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A065994 a(n) = prime(prime(n) - n).

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