A051349 -id:A051349 - cp's OEIS frontend

A351855 Partial sums of nonsquares that are partial sums of nonprimes.

5, 64, 506, 64325, 268723, 480129, 6282620, 64548862, 9657523883, 13480852825, 29766135708, 105223301080, 519861666225, 851245744041, 1378216791896, 581522966976875, 583298551668358, 885441628670251, 1651966084813205, 16868988672306046, 17170433482837259

Offset: 1

J. M. Bergot and Robert Israel, Mar 31 2022

nonn

			a(2) = 64 is a term because 64 = 1+4+6+8+9+10+12+14 = 2+3+5+6+7+8+10+11+12 is the sum of the first 8 nonprimes and the sum of the first 9 nonsquares.

Chai Wah Wu, Table of n, a(n) for n = 1..26

Intersection of A051349 and A086849.

i:= 0: j:= 0: s:= 0: t:= 0:
R:= NULL: count:= 0:
while count < 13 do
  if s <= t then
     i:= i+1;
     if not issqr(i) then
       s:= s+i;
       if s=t then R:= R,s; count:= count+1 fi;
     fi
  else
     j:= j+1;
     if not isprime(j) then
       t:= t+j;
       if s=t then R:= R,t; count:= count+1 fi;
     fi
  fi
od:
R;

from itertools import islice
from sympy import nextprime
def A351855_gen(): # generator of terms
    c, k, ks, m, p, q = 0, 1, 2, 1, 4, 5
    while True:
        for n in range(ks,ks+2*k):
            c += n
            if c == m:
                yield c
            else:
                while c > m:
                    m += p
                    p += 1
                    if p == q:
                        q = nextprime(q)
                        p += 1
        ks += 2*k+1
        k += 1
A351855_list = list(islice(A351855_gen(),20)) # Chai Wah Wu, Apr 04 2022

a(20)-a(21) from Jon E. Schoenfield, Mar 31 2022

A366976 Numbers that cannot be written as sum of two or more consecutive nonprimes.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 15, 16, 20, 21, 24, 25, 30, 32, 35, 40, 42, 44, 47, 48, 52, 56, 61, 66, 70, 72, 73, 80, 88, 93, 96, 98, 100, 107, 110, 119, 120, 140, 141, 144, 167, 174, 179, 186, 190, 196, 204, 205, 234, 236, 252, 253, 260, 275, 290, 292, 299, 303, 310, 312, 313

Offset: 1

Tamas Sandor Nagy, Dec 16 2023

nonn

The complement sequence of sums of two or more consecutive nonprime numbers.

			9 is a term because trying the sums of candidate consecutive nonprimes 1 + 4 = 5 != 9, 1 + 4 + 6 = 11 != 9, 4 + 6 = 10 != 9. All these sums miss the integer 9.
On the other hand, 23 is not a term because 23 = 6 + 8 + 9, which is the sum of three consecutive nonprime numbers.

Cf. A051349, A018252, A367021.

Primes in the sequence: A257393.

A154588 Numbers that can be expressed as the sum of the first j integer numbers or the first k nonprime numbers, with j and k >=1.

Original entry on oeis.org

1, 28, 435, 10296, 415416, 1062153, 3703281, 426626655, 782002378, 102886232631, 1636197988776, 2749764593278, 61972139524851, 813577626225078, 1604393353172781, 3603538956517305, 44000970048906445, 83556903098276790, 208955344344897381

Offset: 1

Paolo P. Lava & Giorgio Balzarotti, Jan 16 2009, Jan 19 2009

nonn

The indices (j,k) where A000217(j) = A051349(k) are (1,1), (7,5), (29,23), (143,123), (911,823), (1457,1327), (2721,2501), (29210,27488), (39547,37295) , (453621,433381) , (1808976,1737137) , (2345107,2253859) , (11133026,10746793), (40338012,39053670), (56646153,54880858) , (84894510,82314170) , (296651209,288273745), (408795555,397457085), (646460121,628975505). - Donovan Johnson, Sep 11 2009

			28 = A000217(7) = A051349(5).
435 = A000217(19) = A051349(23).
10296 = A000217(143) = A051349(123).

Cf. A007504, A051349, A066527, A133784, A154587.

Module[{nn=10^7,np},np=Select[Range[nn],!PrimeQ[#]&];Intersection[Accumulate[Range[ nn]],Accumulate[ np]]] (* The program generates the first 12 terms of the sequence. *) (* Harvey P. Dale, Feb 08 2024 *)

(A000217 INTERSECT A051349) MINUS {0}. - R. J. Mathar, Jan 21 2009

10256 replaced with 10296 and two more terms added by R. J. Mathar, Jan 21 2009

Extended beyond a(9) by Donovan Johnson, Sep 11 2009

A161569 Sum of first n nonprimes minus their indices.

Original entry on oeis.org

0, 2, 5, 9, 13, 17, 22, 28, 34, 40, 47, 55, 63, 71, 80, 89, 98, 107, 116, 126, 137, 148, 159, 170, 181, 193, 205, 217, 230, 244, 258, 272, 287, 302, 317, 332, 347, 363, 379, 395, 411, 427, 444, 462, 480, 498, 516, 534, 553, 572, 591, 611, 632, 653, 674, 695, 716

Offset: 1

Juri-Stepan Gerasimov, Jun 14 2009

The sum of first n nonprimes is in A051349.

Partial sums of A073425. - Jaroslav Krizek, Jun 27 2009

			a(1) = 1-1 = 0; a(2) = 0+4-2 = 2, a(3) = 2+6-3 = 5; a(4) = 5+8-4 = 9.

Cf. A073425, A051349, A018252, A112997.

a(n) = A051349(n)-n*(n+1)/2.

a(n) = a(n-1)+A018252(n)-n, a(1) = 0. - Klaus Brockhaus, Dec 16 2010

a(54)-a(57) from Stefano Spezia, May 26 2025

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A351855 Partial sums of nonsquares that are partial sums of nonprimes.

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A366976 Numbers that cannot be written as sum of two or more consecutive nonprimes.

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A154588 Numbers that can be expressed as the sum of the first j integer numbers or the first k nonprime numbers, with j and k >=1.

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A161569 Sum of first n nonprimes minus their indices.

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