cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A173663 Numbers k that divide the k-th partial sum of all semiprimes.

Original entry on oeis.org

1, 2, 9, 19, 29, 44, 632, 11829, 19262, 25286, 26606, 29824, 247273, 310556, 491240, 1419166, 1601984, 9509238, 113333959, 220531559, 1034662494, 8323088842, 13102043650, 14053673678, 23505911647
Offset: 1

Views

Author

Jonathan Vos Post, Nov 24 2010

Keywords

Comments

a(26) > 3*10^10. - Donovan Johnson, Nov 26 2010

Examples

			a(1) = 1 because 1 divides the first semiprime 4, trivially also the first partial sum of all semiprimes.
a(2) = 2 because A062198(2) = A001358(1) + A001358(2) = 4 + 6 = 10 is divisible by 2.
a(3) = 9 because A062198(9) = 126 = 2 * 3^2 * 7 is divisible by 9.
a(4) = 19 because A062198(19) = 532 = 2^2 * 7 * 19 is divisible by 19.
a(5) = 29 because A062198(29) = 1247 = 29 * 43 is divisible by 29.
a(6) = 44 because A062198(44) = 2904 = 44 * 66.

Crossrefs

Cf. A001358, A007506.

Programs

  • Mathematica
    SemiprimeQ[n_Integer] := If[Abs[n]<2, False, (2==Plus@@Transpose[FactorInteger[Abs[n]]][[2]])]; nn=10^6; sm=0; cnt=0; Reap[Do[If[SemiprimeQ[n], cnt++; sm=sm+n; If[Divisible[sm, cnt], Sow[cnt]]], {n, nn}]][[2, 1]]
  • PARI
    s=0; p=0; for(n=1, 1e9, until(bigomega(p++)==2,); (s+=p)%n || print1(n", ")) \\ M. F. Hasler, Nov 24 2010

Formula

{k: k | Sum_{i=1..k} A001358(i)}.

Extensions

Extended by T. D. Noe, Nov 24 2010
a(1)-a(17) double-checked and a(18) from M. F. Hasler, Nov 25 2010
a(19) from Ray Chandler, Nov 25 2010
a(20)-a(25) from Donovan Johnson, Nov 26 2010

A216686 Numbers n such that n appears in the partial sums of the m-almost primes, where m=bigomega(n).

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 16, 17, 20, 32, 40, 41, 58, 64, 80, 128, 160, 185, 197, 219, 254, 256, 281, 320, 377, 512, 589, 640, 843, 917, 964, 1024, 1247, 1280, 1652, 1707, 1804, 1825, 2048, 2074, 2157, 2519, 2560, 2637, 2642, 2727, 2771, 3614, 3755, 3786, 4046, 4096, 4227
Offset: 1

Views

Author

Gerasimov Sergey, Sep 13 2012

Keywords

Comments

A013918 is a subsequence. - Zak Seidov, Sep 17 2012
Or: Numbers n equal to the sum of the first k numbers x having bigomega(x)=bigomega(n), for some k. - M. F. Hasler, Sep 23 2012

Examples

			2 is in the sequence because 2 appears in A007504.
4 is in the sequence because 4 appears in A062198.
5 is in the sequence because 5 appears in A007504.
6 is not in the sequence because 6 is not in A062198.
8 is in the sequence because 8 appears in A086062,
10 is in the sequence because 10 appears in A062198.

Crossrefs

Cf. A001222, A007504, A013918, A062198, A092190, A086052, A086062.

Programs

  • Maple
    alm := proc(n,m) # n-th m-almost prime
    option remember;
    if n =1 then
        2^m ;
    else
        for a from procname(n-1,m)+1 do
            if numtheory[bigomega](a) = m then
                return a;
            end if;
        end do:
    end if;
end proc:
almP := proc(n,m) #n-th partial sum of the m-almost primes
    add(alm(i,m),i=1..n) ;
end proc:
isA216686 := proc(n) # is n in the sequence?
    local m ,k,ps;
    m := numtheory[bigomega](n) ;
    for k from 1 do
        ps := almP(k,m) ;
        if ps = n then
            return true;
        elif ps > n then
            return false;
        end  if;
    end do:
end proc:
for n from 1 to 4300 do
    if isA216686(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Sep 14 2012
  • PARI
    is_A216686(n)={ my(m=bigomega(n),t); while(n>0, while(bigomega(t++)!=m,); n-=t); !n}  \\ - M. F. Hasler, Sep 23 2012

Extensions

Corrected by R. J. Mathar, Sep 14 2012

A378764 Sum of the semiprimes which are less than or equal to n minus the sum of the primes which are less than or equal to n.

Original entry on oeis.org

0, -2, -5, -1, -6, 0, -7, -7, 2, 12, 1, 1, -12, 2, 17, 17, 0, 0, -19, -19, 2, 24, 1, 1, 26, 52, 52, 52, 23, 23, -8, -8, 25, 59, 94, 94, 57, 95, 134, 134, 93, 93, 50, 50, 50, 96, 49, 49, 98, 98, 149, 149, 96, 96, 151, 151, 208, 266, 207, 207, 146, 208, 208, 208, 273, 273, 206, 206, 275, 275
Offset: 1

Views

Author

Robert G. Wilson v and Luca Bencini-Tibo, Dec 20 2024

Keywords

Comments

After a(32), a(n) always exceeds 0. See A243906(32).

Examples

			a(6) = 0, because (4+6) - (2+3+5) = 0.

Crossrefs

Cf. A001358, A062198, A000040, A007504, A243906.

Programs

  • Mathematica
    a[n_] := Plus @@ Select[ Range@ n, PrimeOmega@ # == 2 &] - Plus @@ Select[ Range@ n, PrimeOmega@ # == 1 &]; Array[a, 70]
  • PARI
    a(n) = my(vf=apply(factor, [1..n])); vecsum(Vec(select(x->(bigomega(x)==2), vf, 1))) - vecsum(Vec(select(x->(bigomega(x)==1), vf, 1))); \\ Michel Marcus, Dec 28 2024
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