A062198 -id:A062198 - cp's OEIS frontend

A110208 1 + sum of first n semiprimes.

1, 5, 11, 20, 30, 44, 59, 80, 102, 127, 153, 186, 220, 255, 293, 332, 378, 427, 478, 533, 590, 648, 710, 775, 844, 918, 995, 1077, 1162, 1248, 1335, 1426, 1519, 1613, 1708, 1814, 1925, 2040, 2158, 2277, 2398, 2520, 2643, 2772, 2905, 3039, 3180, 3322, 3465

Offset: 0

Jonathan Vos Post, Sep 06 2005

First differences are the sequence of semiprimes (A001358). Hence a(n) is the least positive sequence whose first differences are the sequence of semiprimes. Primes in this sequence include: a(1) = 5, a(2) = 11, a(6) = 59, a(9) = 127, a(14) = 293, a(33) = 1613. Semiprimes in this sequence include: a(17) = 427 = 7 * 61, a(18) = 478 = 2 * 239, a(19) = 533 = 13 * 41, a(26) = 995 = 5 * 199, a(27) = 1077 = 3 * 359, a(35) = 1814 = 2 * 907, a(42) = 2643 = 3 * 881, a(45) = 3039 = 3 * 1013.

Cf. A001358, A062198, A110209, A110226.

Accumulate[Join[{1},Select[Range[200],PrimeOmega[#]==2&]]] (* Harvey P. Dale, Apr 29 2018 *)

a(0) = 1; for n>0, a(n) = 1 + A062198(n).

A180152 Numbers k such that the sum of the first k semiprimes is a prime.

Original entry on oeis.org

3, 4, 5, 7, 8, 15, 21, 22, 37, 56, 59, 62, 82, 85, 89, 91, 114, 119, 121, 129, 139, 146, 168, 169, 189, 195, 197, 214, 227, 258, 286, 295, 312, 333, 341, 352, 360, 361, 387, 400, 419, 426, 434, 437, 440, 466, 470, 483, 495, 497, 504, 556, 595, 610, 619, 629, 636

Offset: 1

Jason G. Wurtzel, Aug 13 2010

nonn

			21 is a term because the sum of the first 21 semiprimes is 647, which is prime.

K. Brockhaus, Table of n, a(n) for n = 1..10000

Cf. A062198, A000040.

SP:=[ n: n in [2..3000] | &+[ k[2]: k in Factorization(n) ] eq 2 ]; V:=[]; s:=0; for j in [1..640] do s+:=SP[j]; if IsPrime(s) then Append(~V, j); end if; end for; V; // Klaus Brockhaus, Aug 14 2010

A062198 := proc(n) add( A001358(i),i=1..n) ; end proc:
isA180152 := proc(n) isprime( A062198(n)) ; end proc:
for n from 1 to 1000 do if isA180152(n) then printf("%d,",n) ; end if; end do: # R. J. Mathar, Aug 14 2010

Position[Accumulate[Select[Range[10000],PrimeOmega[#]==2&]],?PrimeQ] // Flatten (* _Harvey P. Dale, Feb 17 2021 *)

More terms from Klaus Brockhaus and R. J. Mathar, Aug 14 2010

A347413 a(n) = (product of first n semiprimes) mod (sum of first n semiprimes).

Original entry on oeis.org

0, 4, 7, 14, 11, 40, 17, 17, 0, 8, 85, 147, 62, 16, 292, 26, 138, 18, 0, 570, 167, 257, 360, 156, 525, 882, 372, 918, 0, 0, 0, 0, 0, 150, 0, 0, 1070, 2136, 1172, 0, 1265, 1502, 663, 0, 0, 0, 0, 1208, 306, 2995, 0, 1404, 1389, 636, 0, 272, 0, 1944, 5216, 2268, 1548, 1160, 3300, 0, 924, 84, 0, 3723

Offset: 1

J. M. Bergot and Robert Israel, Aug 31 2021

nonn

A surprising number of terms are 0: 3124 of the first 10000 terms.

			The first 3 semiprimes are 4, 6, 9, so a(3) = (4*6*9) mod (4+6+9) = 216 mod 19 = 7.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001358, A062198, A112141, A347421.

R:= NULL:
s:= 0: p:= 1: count:= 0:
for n from 4 while count < 100 do
  if numtheory:-bigomega(n) = 2 then
    count:= count+1: s:= s+n; p:= p*n;
    R:= R, p mod s;
  fi
od:
R;

from sympy import factorint
def aupton(terms):
    alst, i, n, s, p = [], 1, 0, 0, 1
    while n < terms:
        if sum(factorint(i).values()) == 2:
            n += 1; s += i; p *= i
            alst.append(p%s)
        i += 1
    return alst
print(aupton(68)) # Michael S. Branicky, Sep 01 2021

a(n) = A112141(n) mod A062198(n).

A347421 Numbers k such that the product of the first k semiprimes is divisible by the sum of the first k semiprimes.

Original entry on oeis.org

1, 9, 19, 29, 30, 31, 32, 33, 35, 36, 40, 44, 45, 46, 47, 51, 55, 57, 64, 67, 70, 71, 72, 74, 81, 83, 84, 92, 94, 95, 96, 97, 103, 104, 105, 107, 108, 109, 113, 116, 118, 124, 125, 127, 130, 131, 132, 133, 136, 138, 140, 142, 144, 158, 159, 160, 167, 177, 182, 184, 188, 191, 196, 202, 203, 206

Offset: 1

J. M. Bergot and Robert Israel, Aug 31 2021

nonn

What are the asymptotics of a(n)/n as n -> infinity?

			a(2) = 9 is a term because the first 9 semiprimes are 4, 6, 9, 10, 14, 15, 21, 22, 25, and 4*6*9*10*14*15*21*22*25 = 5239080000 is divisible by 4+6+9+10+14+15+21+22+25 = 126.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001358, A062198, A112141, A347413.

using Nemo
function A347421List(upto)
    c, s, p = 0, ZZ(0), ZZ(1)
    list = Int32[]
    for n in 4:typemax(Int32)
        if 2 == sum([e for (p, e) in factor(n)])
            s += n; p *= n; c += 1
            if divisible(p, s)
                c > upto && return list
                push!(list, c)
            end
        end
    end
end
A347421List(206) |> println # Peter Luschny, Aug 31 2021

R:= NULL:
s:= 0: p:= 1: zcount:= 0: scount:= 0:
for n from 4 while zcount < 100 do
  if numtheory:-bigomega(n) = 2 then
    s:= s+n; p:= p*n;
    scount:= scount+1;
    if p mod s = 0 then zcount:= zcount+1; R:= R, scount fi
  fi
od:
R;

sp = Select[Range[700], PrimeOmega[#] == 2 &]; Position[Divisible[Rest @ FoldList[Times, 1, sp], Accumulate @ sp], True] // Flatten (* Amiram Eldar, Aug 31 2021 *)

from sympy import factorint
def aupto(limit):
    alst, i, k, s, p = [], 1, 0, 0, 1
    while k < limit:
        if sum(factorint(i).values()) == 2:
            k += 1; s += i; p *= i
            if p%s == 0: alst.append(k)
        i += 1
    return alst
print(aupto(206)) # Michael S. Branicky, Aug 31 2021

A140234 Sum of the semiprimes <= n.

Original entry on oeis.org

0, 0, 0, 0, 4, 4, 10, 10, 10, 19, 29, 29, 29, 29, 43, 58, 58, 58, 58, 58, 58, 79, 101, 101, 101, 126, 152, 152, 152, 152, 152, 152, 152, 185, 219, 254, 254, 254, 292, 331, 331, 331, 331, 331, 331, 331, 377, 377, 377, 426, 426, 477, 477, 477, 477, 532, 532, 589

Offset: 0

Jonathan Vos Post, May 13 2008

This is to semiprimes A001358 as A034387 is to primes A000040. From the prime number theorem A034387(n) has the asymptotic expression: a(n) ~ n^2 / (2 log n), so what is the asymptotic expression for a(n)?

Cf. A001358, A034387, A051352, A062198, A072000, A100959, A140235.

a[n_]:=Total[Select[Range[n],PrimeOmega[#]==2&]];Array[a,58,0] (* James C. McMahon, Jul 06 2025 *)

a(n) = Sum_{j such that j is in A001358 and j<=n} = A062198(A072000(n)).

A140235 Partial sum of non-semiprimes A100959.

Original entry on oeis.org

1, 3, 6, 11, 18, 26, 37, 49, 62, 78, 95, 113, 132, 152, 175, 199, 226, 254, 283, 313, 344, 376, 412, 449, 489, 530, 572, 615, 659, 704, 751, 799, 849, 901, 954, 1008, 1064, 1123, 1183, 1244, 1307, 1371, 1437, 1504, 1572, 1642, 1713, 1785, 1858, 1933, 2009

Offset: 1

Jonathan Vos Post, May 13 2008

This is to semiprimes A001358 as A051352 is to primes A000040. Equivalently, this is to non-semiprimes A100959 as A051349 is to nonprimes A018252.

			a(5) = 18 = 1 + 2 + 3 + 5 + 7.

Cf. A000217, A001358, A018252, A034387, A051349, A051352, A062198, A072000, A100959, A140234.

Accumulate[Select[Range[100],PrimeOmega[#]!=2&]] (* Harvey P. Dale, Aug 22 2021 *)

a(n) = Sum{k=1..n} A100959(k).

Corrected and edited by Giovanni Resta, Jun 20 2016

A182081 Next semiprime after the partial sum of the first n semiprimes.

Original entry on oeis.org

6, 14, 21, 33, 46, 62, 82, 106, 129, 155, 187, 221, 259, 295, 334, 381, 427, 478, 533, 591, 649, 713, 778, 849, 921, 995, 1077, 1165, 1253, 1337, 1437, 1522, 1618, 1711, 1814, 1927, 2041, 2159, 2279, 2402, 2533, 2643, 2773, 2906, 3039, 3183, 3326, 3466, 3611

Offset: 1

Jonathan Vos Post, Apr 10 2012

This is to A202301 next prime after the partial sum of the first n primes as A001358 semiprimes is to A000040 primes.

			a(10) = 155 because 4 + 6 + 9 + 10 + 14 + 15 + 21 + 22 + 25 + 26 = 152, and the next semiprime after 152 is 155.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A001358, A062198, A202301.

h:= proc(m) local k;
      for k from m+1 while isprime(k) or
        add (i[2], i=ifactors(k)[2])<>2 do od; k
    end:
b:= proc(n)
      b(n):= h(b(n-1))
    end: b(0):=0:
s:= proc(n)
      s(n):= b(n) +s(n-1)
    end: s(0):=0:
a:= n-> h(s(n)):
seq (a(n), n=1..60); # Alois P. Heinz, Apr 10 2012

a(n) = min { k > A062198(n) and k in A001358 }.

More terms from Alois P. Heinz, Apr 10 2012

A217736 Sum of first n squares of semiprimes.

Original entry on oeis.org

16, 52, 133, 233, 429, 654, 1095, 1579, 2204, 2880, 3969, 5125, 6350, 7794, 9315, 11431, 13832, 16433, 19458, 22707, 26071, 29915, 34140, 38901, 44377, 50306, 57030, 64255, 71651, 79220, 87501, 96150, 104986, 114011, 125247, 137568, 150793, 164717, 178878

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, Mar 22 2013

nonn

There are 68 such numbers less than one million; a(68) = 988849.

			a(3) = 4^2 + 6^2 + 9^2 = 133.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Cf. A062198. Equals sum(A001358^2), and sum(A074985).

Accumulate[Select[Range[200],PrimeOmega[#]==2&]^2] (* Harvey P. Dale, Mar 13 2018 *)

A347366 Primes that are partial sums of the semiprimes.

Original entry on oeis.org

19, 29, 43, 79, 101, 331, 647, 709, 2039, 4723, 5261, 5827, 10271, 11057, 12163, 12743, 20183, 22039, 22807, 25999, 30319, 33563, 44777, 45319, 56843, 60623, 61927, 73583, 83077, 108013, 133447, 142183, 159541, 182659, 191833, 204803, 214463, 215689, 248789, 266239, 292573, 302593, 314339, 318823

Offset: 1

J. M. Bergot and Robert Israel, Aug 29 2021

nonn

			a(3) = 43 is a term because 43 = A062198(5) is prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Primes in A062198. Cf. A001358.

SP:= select(t -> numtheory:-bigomega(t) = 2, [$2..10000]):
PSSP:= ListTools:-PartialSums(SP):
select(isprime,PSSP);

Select[Accumulate @ Select[Range[1500], PrimeOmega[#] == 2 &], PrimeQ] (* Amiram Eldar, Aug 29 2021 *)

from sympy import factorint, isprime
def aupto(limit):
    alst, k, s = [], 1, 0
    for k in range(1, limit+1):
        if sum(factorint(k).values()) == 2:
            s += k
            if s > limit: break
            if isprime(s): alst.append(s)
    return alst
print(aupto(320000)) # Michael S. Branicky, Aug 29 2021

A382831 a(n) is the n-th n-almost-prime that is a partial sum of the sequence of n-almost-primes.

Original entry on oeis.org

2, 10, 964, 1804, 7820, 48120, 830817, 4895208, 11308160, 162802560, 394129476, 3763612800, 19823090472, 1018716103620, 9744542956800, 3989325082624, 329306801920000, 2978224618328064, 11804664377696256, 128906665137012736

Offset: 1

Robert Israel, Apr 28 2025

nonn

			The first three members of A086062 that are 3-almost-primes are 8 = 2^3, 20 = 2^2 * 5 = 8 + 12, and 964 = 2^2 * 241 = 8 + 12 + 18 + ... + 92, so a(3) = 964.

Cf. A007504, A062198, A086062, A086046, A086047, A086052, A086059, A086061.

f:= proc(n) uses priqueue;
  local pq,t,s,x,p,i,count;
  initialize(pq);
  insert([-2^n,2$n],pq);
  s:= 0; count:= 0:
  do
    t:= extract(pq);
    x:= -t[1];
    s:= s + x;
    if numtheory:-bigomega(s) = n  then count:= count+1; if count = n then return s fi fi;
    p:= nextprime(t[-1]);
    for i from n+1 to 2 by -1 while t[i] = t[-1] do
          insert([t[1]*(p/t[-1])^(n+2-i), op(t[2..i-1]), p$(n+2-i)], pq)
    od;
  od
end proc:
map(f, [$1..20]);

cp's OEIS Frontend

A110208 1 + sum of first n semiprimes.

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A180152 Numbers k such that the sum of the first k semiprimes is a prime.

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A347413 a(n) = (product of first n semiprimes) mod (sum of first n semiprimes).

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A347421 Numbers k such that the product of the first k semiprimes is divisible by the sum of the first k semiprimes.

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Julia

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A140234 Sum of the semiprimes <= n.

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A140235 Partial sum of non-semiprimes A100959.

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A182081 Next semiprime after the partial sum of the first n semiprimes.

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Maple

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A217736 Sum of first n squares of semiprimes.