A344324 Let b be the lexicographically earliest weakly increasing sequence of squares such that Sum_{k = 1..m} prime(k) <= Sum_{k = 1..m} b(k) for any m > 0 (where prime(k) denotes the k-th prime number); a(n) = Sum_{k = 1..n} (b(k) - prime(k)).

2, 3, 2, 4, 2, 5, 4, 1, 3, 10, 15, 14, 9, 2, 4, 0, 5, 8, 5, 15, 23, 25, 23, 15, 18, 17, 14, 7, 19, 27, 21, 11, 18, 23, 18, 11, 23, 29, 31, 27, 17, 5, 10, 13, 12, 9, 23, 25, 23, 19, 11, 28, 43, 48, 47, 40, 27, 12, 24, 32, 38, 34, 16, 29, 40, 47, 40, 27, 4, 16

Offset

1,1

Comments

This sequence is a variant of A338699; here we use squares, there powers of 2.

Links

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

Rémy Sigrist, Density plot of the first 1000000 terms

Example

The first terms, alongside the prime numbers (p(n)) and their partial sums (pp(n)), b(n) and their partial sums (bb(n)), are:
  n   a(n)  p(n)  pp(n)  b(n)  bb(n)
  --  ----  ----  -----  ----  -----
   1     2     2      2     4      4
   2     3     3      5     4      8
   3     2     5     10     4     12
   4     4     7     17     9     21
   5     2    11     28     9     30
   6     5    13     41    16     46
   7     4    17     58    16     62
   8     1    19     77    16     78
   9     3    23    100    25    103
  10    10    29    129    36    139
  11    15    31    160    36    175
  12    14    37    197    36    211
  13     9    41    238    36    247
  14     2    43    281    36    283
  15     4    47    328    49    332

Prog

(PARI) pp=0; bb=0; b=0; forprime (p=2, 349, pp+=p; while (bb+b^2<pp, b++); bb+=b^2; print1 (bb-pp", "))

Crossrefs

Cf. A000290, A007504, A338699.

Sequence in context: A162908 A373558 A222817 * A309428 A328368 A052297

Adjacent sequences: A344321 A344322 A344323 * A344325 A344326 A344327

Keyword

nonn

Author

Rémy Sigrist, May 15 2021

Status

approved