A386369 -id:A386369 - cp's OEIS frontend

A386687 Partial sums of A386369.

0, 1, 3, 5, 7, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 99, 117, 135, 153, 171, 189, 207, 225, 251, 277, 303, 329, 355, 381, 407, 433, 459, 485, 511, 537, 563, 589, 615, 641, 667, 693, 719, 745, 771, 797, 823, 849, 875, 901, 927, 953, 979, 1005, 1031, 1057, 1083, 1109

Offset: 1

Paolo Xausa, Jul 29 2025

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A386369, A386370, A386688.

Module[{s = 0, a = 0},Table[If[IntegerQ[Sqrt[s += a]], a = n]; s, {n, 100}]]

a(n) = Sum_{k=1..n} A386369(k).

A386370 Square roots of partial sums of A386369 that are square numbers.

Original entry on oeis.org

0, 1, 3, 9, 15, 37, 47, 117, 327, 957, 2847, 8517, 25527, 33129, 33291, 46653, 66855, 147661, 161689, 218691, 503481, 629673, 919347, 2076921, 3007455, 7100533, 7931517, 9022563, 9100641, 11272057, 11437383, 15080379, 32539617, 37443597, 37821775, 53419727

Offset: 1

Rémy Sigrist, Jul 19 2025

nonn

			A386369(1) + ... + A386369(6) = 3^2, so 3 belongs to this sequence.

Cf. A386369, A386687, A386688.

Module[{s = 0, a = 0}, Table[If[IntegerQ[#], a = k; #, Nothing] & [Sqrt[s += a]], {k, 10^5}]] (* Paolo Xausa, Jul 29 2025, after Rémy Sigrist  *)

{ t = 0; v = 0; for (n = 1, 86903746, t += v; if (issquare(t), print1 (sqrtint(t) ", "); v = n;);); }

a(n) = sqrt(A386688(n)). - Paolo Xausa, Jul 29 2025

Offset changed to 1 by Paolo Xausa, Jul 29 2025

A385254 Distinct terms in A386369.

Original entry on oeis.org

0, 1, 2, 6, 18, 26, 70, 82, 222, 642, 1902, 5682, 17022, 51042, 59778, 59958, 77774, 107258, 268870, 285010, 361086, 930666, 1084314, 1498134, 3813282, 5053994, 13240150, 14183598, 15487758, 15579122, 18418666, 18622506, 23809998, 58728474, 64572254, 65013058

Offset: 1

David A. Corneth and Paolo Xausa, Jul 29 2025

nonn

a(1) = 0. a(n) is the smallest positive integer > a(n-1) such that Sum_{m = 1..n-1} (a(m+1)-a(m))*a(m) is a perfect square.

			The first 6 terms of A386369 are 0, 1, 2, 2, 2, 2 which has partial sum 9. We have A386369(7) = 6. To find a(4) we look for the next term in A386369 that is larger than 6 i.e. solve 6*(k-6) + 9 = s^2 for some k. Rewrite gives 6*(k-6) = s^2 - 9 = (s-3)(s + 3). So we have 4 cases:
   1 | s - 3, 6 | s + 3
   2 | s - 3, 3 | s + 3
   3 | s - 3, 2 | s + 3
   6 | s - 3, 1 | s + 3
Solving for smallest t > 6 gives s = 9. So 6*(k-6) = 9^2 - 9 = 72 and so k = 18.

David A. Corneth, Table of n, a(n) for n = 1..1018
David A. Corneth, PARI program

Cf. A386369.

Module[{s = 0, a = 0}, Table[If[IntegerQ[Sqrt[s += a]], a = k-1, Nothing], {k, 10^5}]]

PARI
```
\\ See Corneth link
```

More terms from Michael De Vlieger, Jul 29 2025

A385986 a(1) = 2, and for any n > 1, a(n) is the largest k < n such that a(1) + ... + a(k) is prime.

Original entry on oeis.org

2, 1, 2, 3, 3, 5, 5, 5, 5, 9, 9, 9, 9, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 65, 65, 65

Offset: 1

Rémy Sigrist, Jul 14 2025

In other words: a(1) = 2, and for any n > 0, if a(1) + ... + a(n) is prime then a(n+1) = n, otherwise a(n+1) = a(n).

This sequence is unbounded: for any n > 1, let P = a(1) + ... + a(a(n)); P is prime and a(a(n)+1) = a(n); as P > a(n), P and a(n) are coprime, hence, by Dirichlet's theorem on arithmetic progressions, P + k*a(n) is prime for some minimal k > 0, and a(a(n)+k+1) = a(n)+k > a(n).

			Sequence begins:
  n   a(n)  a(1)+...+a(n)  Prime?
  --  ----  -------------  ------
   1     2              2  Yes
   2     1              3  Yes
   3     2              5  Yes
   4     3              8  No
   5     3             11  Yes
   6     5             16  No
   7     5             21  No
   8     5             26  No
   9     5             31  Yes
  10     9             40  No
  11     9             49  No
  12     9             58  No
  13     9             67  Yes
  14    13             80  No

Rémy Sigrist, Table of n, a(n) for n = 1..10000

See A385988 and A386369 for similar sequences.

Cf. A385987 (corresponding prime numbers).

v = 2;t = 0;values={};Do[AppendTo[values,v];t+=v;If[PrimeQ[t],v=n],{n, 1, 68}];values (* James C. McMahon, Jul 22 2025 *)

{ v = 2; t = 0; for (n = 1, 68, print1 (v", "); if (isprime(t += v), v = n);); }

A386688 Squares in A386687.

Original entry on oeis.org

0, 1, 9, 81, 225, 1369, 2209, 13689, 106929, 915849, 8105409, 72539289, 651627729, 1097530641, 1108290681, 2176502409, 4469591025, 21803770921, 26143332721, 47825753481, 253493117361, 396488086929, 845198906409, 4313600840241, 9044785577025, 50417568884089, 62908961921289

Offset: 1

Paolo Xausa, Jul 29 2025

nonn

Cf. A386369, A386370.

Intersection of A000290 and A386687.

Module[{s = 0, a = 0}, Table[If[IntegerQ[Sqrt[s += a]], a = k; s, Nothing], {k, 10^5}]]

a(n) = A386370(n)^2.

cp's OEIS Frontend

A386687 Partial sums of A386369.

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A386370 Square roots of partial sums of A386369 that are square numbers.

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A385254 Distinct terms in A386369.

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A385986 a(1) = 2, and for any n > 1, a(n) is the largest k < n such that a(1) + ... + a(k) is prime.

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A386688 Squares in A386687.

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