A092190 -id:A092190 - cp's OEIS frontend

A323085 Semiprimes that are the sum of the first k terms of A092190 for some k.

4, 14, 8567, 16499, 151211, 344891, 418831, 585197, 1049882, 1186582, 1671029, 2503966, 2989387, 4802311, 8291795, 9769711, 11420129, 13279957, 13677063, 15356513, 16258813, 24318863, 26874293, 39317497, 42862751

Offset: 1

Wilmer Emiro Castrillon Calderon, Jan 03 2019

nonn

If we call the semiprime numbers A001358 level 1, and A092190 level 2, then this sequence is level 3.

			a(2) = 14 = Sum_{i=1..2} A092190(i).
a(3) = 8567 = Sum_{i=1..13} A092190(i).

Wilmer Emiro Castrillon Calderon, Table of n, a(n) for n = 1..100

Cf. A001358, A092190.

f[w_] := Select[Most@ NestWhile[Append[#1, {#2, #2 + #1[[-1, -1]]}] & @@ {#, w[[Length@ # + 1]]} &, {{#, #}} &@ First[w], #[[-1, -1]] <= Max@ w &][[All, -1]], PrimeOmega@ # == 2 &]; Block[{s = Select[Range[10^6], PrimeOmega@ # == 2 &], t}, f@ f@ s] (* Michael De Vlieger, Jan 04 2019 *)

A062198 Sum of first n semiprimes.

Original entry on oeis.org

4, 10, 19, 29, 43, 58, 79, 101, 126, 152, 185, 219, 254, 292, 331, 377, 426, 477, 532, 589, 647, 709, 774, 843, 917, 994, 1076, 1161, 1247, 1334, 1425, 1518, 1612, 1707, 1813, 1924, 2039, 2157, 2276, 2397, 2519, 2642, 2771, 2904, 3038, 3179, 3321, 3464

Offset: 1

Shyam Sunder Gupta, Aug 24 2003

Elements in this sequence can themselves be semiprimes. a(1) = 4 = 2^2. a(2) = 10 = 2 * 5. a(6) = 58 = 2 * 29. a(11) = 185 = 5 * 37. a(12) = 219 = 3 * 73. a(13) = 254 = 2 * 127. a(16) = 377 = 13 * 29. a(20) = 589 = 19 * 31. Etc. Does this happen infinitely often? - Jonathan Vos Post, Dec 11 2004

			a(4) = 29 because the sum of the first 4 semiprimes 4+6+9+10 is 29.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001358, A092189, A092190.

Accumulate[Select[Range[200],PrimeOmega[#]==2&]] (* Harvey P. Dale, Jul 23 2014 *)

is_A062198(N)={ my(n=0); while(N>0, while(bigomega(n++)!=2, ); N-=n); !N}  \\ - M. F. Hasler, Sep 23 2012

A062198(n, list)={my(s=0, N=0); until(!n--, until(bigomega(N++)==2, ); s+=N; list & print1(s", ")); s}  \\ - M. F. Hasler, Sep 23 2012

a(n) = Sum_{i=1..n} A001358(i). - R. J. Mathar, Sep 14 2012

A092189 Numbers k such that the sum of the first k semiprimes is a semiprime.

Original entry on oeis.org

1, 2, 6, 11, 12, 13, 16, 20, 24, 25, 29, 34, 38, 41, 42, 43, 50, 53, 58, 61, 65, 66, 68, 77, 100, 102, 106, 110, 111, 117, 120, 123, 131, 134, 150, 151, 152, 154, 157, 162, 164, 165, 166, 174, 176, 178, 180, 183, 185, 187, 192, 205, 208, 210, 218, 221, 222, 231

Offset: 1

Zak Seidov, Feb 23 2004

			2 is a term because the sum of the first two semiprimes, 4 and 6, is 10, which is a semiprime.

Zak Seidov, Table of n, a(n) for n = 1..50000

Resulting semiprimes: A092190.

A213650 Numbers k such that the sum of the first k primes is semiprime.

Original entry on oeis.org

3, 7, 8, 10, 16, 18, 22, 28, 32, 34, 36, 38, 44, 46, 48, 54, 55, 58, 59, 65, 66, 72, 75, 82, 92, 93, 94, 104, 106, 110, 118, 120, 133, 136, 137, 138, 140, 141, 142, 144, 148, 150, 154, 156, 164, 168, 170, 174, 190, 194, 202, 210, 212, 218, 224, 226, 232, 234

Offset: 1

Michel Lagneau, Jun 17 2012

nonn

Numbers k such that A007504(k) is included in A001358.

			8 is in the sequence because the sum of the first 8 primes is  2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 = 77 = 7*11, which is semiprime.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001358, A007504, A013916, A092189 (numbers n such that sum of first n semiprimes is a semiprime), A092190 (semiprimes that are the sum of first n semiprimes for some n), A180152 (numbers n such that the sum of the first n semiprimes is a prime).

with(numtheory): for n from 1 to 500 do:s:=sum(‘ithprime(k)’, ’k’=1..n):if bigomega(s)=2 then printf(`%d, `, n):else fi:od:

Flatten[Position[Accumulate[Prime[Range[300]]],_?(PrimeOmega[#]==2&)]]

isok(n) = bigomega(vecsum(primes(n))) == 2; \\ Michel Marcus, Sep 18 2017

A217018 Partial sums of 3-almost primes which are again 3-almost primes, i.e., have exactly 3 not necessarily distinct prime factors.

Original entry on oeis.org

8, 20, 964, 1825, 2074, 2637, 3614, 3786, 4503, 5283, 5495, 6414, 6652, 7138, 7383, 9485, 9764, 10330, 10615, 11191, 12427, 12749, 13074, 15475, 16195, 16930, 18446, 19233, 20855, 22108, 22959, 23387, 28273, 28747, 29222, 30676, 32695, 34798, 35871

Offset: 1

M. F. Hasler, Sep 23 2012

nonn

Bigomega=3 analog of the semiprime version A092190. In sequence A086062 it was asked whether there are infinitely many such numbers.

A217018(n,list=0,N=0,S=0)={until(!n--,until(bigomega(S+=N)==3,until(bigomega(N++)==3,));list&print1(S","));S} \\ - M. F. Hasler, Sep 29 2012

A217018 = A086062 intersect A014612.

A265438 Smallest semiprime that is the sum of n consecutive semiprimes.

Original entry on oeis.org

4, 10, 25, 39, 69, 58, 133, 122, 249, 209, 185, 219, 254, 327, 458, 377, 473, 579, 745, 589, 951, 898, 1047, 843, 917, 1382, 1157, 1243, 1247, 1678, 1514, 1895, 1703, 1707, 2138, 2147, 2599, 2157, 2509, 2515, 2519, 2642, 2771, 3566, 4126, 3317, 3599, 3891, 4198, 3755, 4369, 4223, 4227

Offset: 1

Zak Seidov, Dec 09 2015

nonn

The sequence is non-monotonic. But are all the terms distinct?

A092190 is a subsequence. More precisely, a(A092189(k)) = A092190(k). - Altug Alkan, Dec 13 2015

			a(1) = 4 = A001358(1),
a(2) = 10 = A001358(3) = A092192(1) = A001358(1)+A001358(2) = 4+6,
a(3) = 25 = A001358(9) = A131610(1),
a(4) = 39 = A001358(15) = A158339(1),
a(5) = 69 = A001358(24) = A254712(1),
a(6) = 58 = A001358(21) = A266451(1).

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A001358, A092190, A092192, A131610, A158339, A254712.

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

Gerasimov Sergey, Sep 13 2012

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

			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.

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

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

is_A216686(n)={ my(m=bigomega(n),t); while(n>0, while(bigomega(t++)!=m,); n-=t); !n}  \\ - M. F. Hasler, Sep 23 2012

Corrected by R. J. Mathar, Sep 14 2012

cp's OEIS Frontend

A323085 Semiprimes that are the sum of the first k terms of A092190 for some k.

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A062198 Sum of first n semiprimes.

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A092189 Numbers k such that the sum of the first k semiprimes is a semiprime.

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A213650 Numbers k such that the sum of the first k primes is semiprime.

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A217018 Partial sums of 3-almost primes which are again 3-almost primes, i.e., have exactly 3 not necessarily distinct prime factors.

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A265438 Smallest semiprime that is the sum of n consecutive semiprimes.

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

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