A051935 -id:A051935 - cp's OEIS frontend

A085626 Partial sums of A051935.

2, 5, 11, 19, 29, 41, 59, 79, 101, 127, 157, 191, 227, 269, 313, 359, 409, 461, 521, 587, 659, 733, 809, 887, 967, 1049, 1151, 1259, 1373, 1489, 1607, 1733, 1861, 1993, 2129, 2267, 2411, 2557, 2707, 2861, 3019, 3181, 3347, 3517, 3691, 3877, 4073, 4271, 4481

Offset: 1

Patrick Capelle, Jul 10 2003

nonn

Same as A070865 after first term. - David Wasserman, Jun 27 2005

			a(3) = 11 because it is the sum of the first 3 terms of A051935: 2+3+6 = 11.

Cf. A051935, A070865.

A051956 a(n) = smallest number > a(n-1) such that a(1)a(2)...a(n) + 1 and a(1)a(2)...a(n) - 1 are primes.

Original entry on oeis.org

4, 15, 17, 19, 29, 57, 77, 285, 318, 354, 379, 405, 520, 521, 595, 625, 938, 1706, 1738, 2085, 2345, 2817, 4319, 4529, 7005, 8207, 8232, 9451, 9839, 11044, 11170, 12386, 12421, 12722, 14153, 15220, 15371, 17052, 17965, 18279, 18811, 19890, 21628, 22667, 22746

Offset: 1

Felice Russo, Dec 21 1999

Amiram Eldar, Table of n, a(n) for n = 1..184

Cf. A051935, A051957.

a[1] = 4; a[n_] := a[n] = For[k = a[n-1]+1, True, k++, p = Times @@ Array[a, n-1]; If[PrimeQ[k*p+1] && PrimeQ[k*p-1], Print[k]; Return[k]]]; Array[a, 45] (* Jean-François Alcover, Oct 23 2016 *)

lista(nn) = {my (v = vector(nn)); for (n = 1, nn, if (n == 1, p = 1; k = 0; , p = prod(j=1, n-1, v[j]); k = v[n-1]+1); while (! isprime(p*k+1) || ! isprime(p*k-1), k++); v[n] = k; print1(k, ", "););} \\ Michel Marcus, Sep 28 2013

More terms from Michel Marcus, Sep 28 2013

A078706 a(n) = smallest integer >=a(n-1) such that sum of first n terms is prime.

Original entry on oeis.org

2, 3, 6, 6, 6, 6, 8, 10, 12, 12, 12, 14, 16, 18, 18, 18, 24, 32, 34, 36, 38, 42, 46, 48, 54, 56, 64, 68, 78, 90, 90, 94, 102, 114, 122, 124, 134, 144, 148, 150, 152, 160, 168, 170, 178, 182, 190, 192, 200, 216, 220, 222, 234, 234, 234, 246, 260, 264, 268, 270, 278, 280

Offset: 1

Neil Fernandez, Dec 18 2002

			a(1)=2, the first prime. The smallest integer >=2 that yields a prime when added to 2 is 3, so a(2)=3. The smallest integer >=3 that yields a prime when added to 2+3 is 6 so a(3)=6.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A051935.

s = 0; NestList[(s += #; NextPrime[s + # - 1] - s) &, 2, 61] (* Ivan Neretin, May 14 2015 *)
nxt[{t_,a_}]:=Module[{k=a},While[CompositeQ[t+k],k++];{t+k,k}]; NestList[nxt,{2,2},70][[;;,2]] (* Harvey P. Dale, Sep 01 2024 *)

A219634 a(n) is the smallest number > a(n-1) such that 1 + a(1)^2 + a(2)^2 + ... + a(n)^2 is a prime.

Original entry on oeis.org

1, 3, 6, 12, 30, 54, 60, 72, 120, 126, 144, 174, 198, 210, 294, 300, 318, 354, 408, 420, 426, 432, 480, 498, 522, 564, 588, 594, 600, 624, 630, 648, 666, 714, 720, 852, 864, 978, 1002, 1050, 1056, 1080, 1098, 1122, 1146, 1152, 1170, 1176, 1200, 1206, 1458

Offset: 1

Michel Lagneau, Nov 24 2012

nonn

The corresponding primes are 2, 11, 47, 191, 1091, 4007, 7607, 12791, 27191, 43067, ...

			a(1) = 1 because 1 + 1^2 = 2 is prime;
a(2) = 3 because 1 + 1^2 + 2^2 = 6 is not prime, but 1 + 1^2 + 3^2 = 11 is prime;
a(3) = 6 because neither 1 + 1^2 + 3^2 + 4^2 = 27 nor 1 + 1^2 + 3^2 + 5^2 = 36 is prime, but 1 + 1^2 + 3^2 + 6^2 = 47 is prime.

Cf. A051935.

p=1;lst={p};Do[If[PrimeQ[p+n^2],AppendTo[lst,n];p=p+n^2],{n,1,1500}];lst

A233334 a(1) = 1; for n > 1, a(n) is the smallest number > a(n-1) such that a(1) + a(2) + ... + a(n) is a composite number.

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Michel Lagneau, Dec 18 2013

{a(n)} = {1, 3, 4, 6, 7} union {9, 10, 11, 12, ...} and the sum s(n) = a(1) + a(2) + ... + a(n) is always composite because s(1) = 1, s(2) = 4, s(3) = 8, s(4) = 14 and for n = 5,6,7,... s(n) = (n-2)*(n+9)/2 = 21, 30, 40, 51, ... = A056115(n) for n >= 3.

			The third term is 4 because 1+3+4=8 is composite.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A051935, A056115.

p=1; lst={p}; Do[If[!PrimeQ[p+n], AppendTo[lst, n]; p=p+n], {n, 3, 70}]; lst
nxt[{c_,a_}]:=Module[{k=a+1},While[!CompositeQ[c+k],k++];{c+k,k}]; NestList[nxt,{1,1},70][[;;,2]] (* Harvey P. Dale, Dec 05 2023 *)

From Chai Wah Wu, Jan 28 2024: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 7.
G.f.: x*(-x^6 + x^5 - x^4 + x^3 - x^2 + x + 1)/(x - 1)^2. (End)

A360061 Lexicographically earliest increasing sequence such that a(1) = 2 and for n >= 2, a(1)^2 + a(2)^2 + ... + a(n)^2 is a prime.

Original entry on oeis.org

2, 3, 4, 12, 48, 54, 66, 138, 144, 162, 168, 180, 198, 234, 252, 264, 330, 360, 366, 372, 402, 420, 444, 462, 480, 534, 546, 552, 564, 576, 600, 630, 642, 678, 702, 744, 756, 846, 852, 858, 882, 966, 1008, 1206, 1242, 1254, 1266, 1272, 1296, 1302, 1338, 1650

Offset: 1

Win Wang, Jan 23 2023

			For n >= 2, partial sums of squares are (showing primality): 2^2 + 3^2 = 13; 13 + 4^2 = 29; 29 + 12^2 = 173; 173 + 48^2 = 2477; ...

Cf. A051935, A137326.

import Math.NumberTheory.Primes.Testing (isPrime)
a360061_list = 2 : 3 : recurse 4 13 where
  recurse n p
    | isPrime(n^2 + p) = n : recurse (n+1) (n^2 + p)
    | otherwise        = recurse (n+1) p
-- Peter Kagey, Jan 25 2023

s:= proc(n) option remember; `if`(n<1, 0, a(n)^2+s(n-1)) end:
a:= proc(n) option remember; local k, m;
      k:= s(n-1); for m from 1+a(n-1)
      while not isprime(k+m^2) do od; m
    end: a(1):=2:
seq(a(n), n=1..52);  # Alois P. Heinz, Jan 26 2023

s[n_] := s[n] = If[n < 1, 0, a[n]^2 + s[n-1]];
a[n_] := a[n] = Module[{k, m},
   k = s[n-1]; For[m = 1 + a[n-1],
   !PrimeQ[k + m^2], m++]; m];
a[1] = 2;
Table[a[n], {n, 1, 52}] (* Jean-François Alcover, Feb 03 2025, after Alois P. Heinz *)

cp's OEIS Frontend

A085626 Partial sums of A051935.

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A051956 a(n) = smallest number > a(n-1) such that a(1)*a(2)*...*a(n) + 1 and a(1)*a(2)*...*a(n) - 1 are primes.

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A078706 a(n) = smallest integer >=a(n-1) such that sum of first n terms is prime.

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A219634 a(n) is the smallest number > a(n-1) such that 1 + a(1)^2 + a(2)^2 + ... + a(n)^2 is a prime.

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A233334 a(1) = 1; for n > 1, a(n) is the smallest number > a(n-1) such that a(1) + a(2) + ... + a(n) is a composite number.

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A360061 Lexicographically earliest increasing sequence such that a(1) = 2 and for n >= 2, a(1)^2 + a(2)^2 + ... + a(n)^2 is a prime.

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A051956 a(n) = smallest number > a(n-1) such that a(1)a(2)...a(n) + 1 and a(1)a(2)...a(n) - 1 are primes.