A118717 -id:A118717 - cp's OEIS frontend

A181529 Natural numbers equal to the sum of two consecutive semiprimes (A118717), but not an integer multiple of a previous term.

10, 15, 19, 24, 29, 36, 43, 47, 51, 59, 67, 69, 73, 77, 85, 106, 112, 115, 127, 143, 151, 159, 167, 173, 178, 184, 187, 189, 217, 226, 233, 237, 243, 245, 262, 267, 275, 283, 291, 313, 317, 327, 355, 372, 395, 403, 411, 415, 422, 427, 447, 461, 484, 496, 502

Offset: 1

Giovanni Teofilatto, Oct 27 2010

nonn

Essentially A118717 with the multiples removed.

			A118717(16)=95 is not in the sequence since 95=5*A181529(3).
A118717(17)=100 is not in the sequence since 100=10*A181529(1).

Cf. A118717.

Edited and extended by Ray Chandler, Oct 31 2010

A173966 Sums of two consecutive semiprimes.

Original entry on oeis.org

19, 29, 43, 51, 67, 69, 77, 115, 171, 173, 187, 189, 237, 243, 245, 267, 283, 285, 291, 317, 355, 403, 405, 411, 427, 429, 435, 437, 507, 597, 603, 605, 653, 669, 723, 763, 787, 789, 891, 893, 907, 963, 1003, 1029, 1053, 1075, 1085, 1107, 1131, 1245, 1267

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 03 2010

nonn

First 16 terms:19,29,43,51,67,69,77,115,171,173,187,189,237,243,245,267 are the same as in A157483.

These are sums of two consecutive integers which are both semiprimes, whereas A118717 are sums of two semiprimes which are adjacent (consecutive) in A001358. [From R. J. Mathar, Mar 18 2010]

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

f[n_]:=Last/@FactorInteger[n]=={1,1}||Last/@FactorInteger[n]=={2};lst={};Do[If[f[n],If[f[n+1],AppendTo[lst,2*n+1]]],{n,7!}];lst
Total/@Select[Partition[Select[Range[700],PrimeOmega[#]==2&],2,1],#[[2]]- #[[1]] == 1&] (* Harvey P. Dale, Jun 22 2020 *)

A370162 Semiprimes that are the sum of two successive semiprimes and also the sum of three successive semiprimes.

Original entry on oeis.org

134, 597, 614, 898, 982, 998, 1649, 2045, 2078, 2126, 2386, 2705, 2855, 2935, 3394, 3418, 3899, 5533, 5686, 5959, 6982, 7721, 8567, 8986, 9182, 9722, 9998, 10342, 10587, 10862, 10942, 11015, 11363, 11602, 11667, 11962, 13238, 13606, 14054, 14138, 14506, 14614, 15658, 15802, 15898, 16138, 16382

Offset: 1

Zak Seidov and Robert Israel, Feb 26 2024

nonn

			a(3) = 614 is a term because 614 = 2 * 307 is a semiprime, A001358(98) = 305 = 5 * 61 and A001358(99) = 309 = 3 * 103 are two successive semiprimes whose sum is 614, and A001358(65) = 203 = 7 * 29, A001358(66) = 205 = 5 * 41 and A001358(67) = 206 = 2 * 103 are three successive semiprimes whose sum is 614.

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

Cf A001358, A118717. Intersection of A092192 and A131610.

N:= 10^5: # for terms <= N
P:= select(isprime, [2,seq(i,i=3..N/2,2)]):
nP:= nops(P):
SP:= 0:
for i from 1 while P[i]^2 <= N do
  m:= ListTools:-BinaryPlace(P, N/P[i]);
  SP:= SP, op(P[i]*P[i..m]);
od:
SP:= sort([SP]):
SS:= ListTools:-PartialSums(SP):
SS2:= {seq(SS[i]-SS[i-2],i=3..nops(SS))}:
SS3:= {seq(SS[i]-SS[i-3],i=4..nops(SS))}:
A:=SS2 intersect SS3 intersect convert(SP,set):
sort(convert(A,list));

A283873 Smallest number that is the sum of n successive primes and also the sum of n successive semiprimes, n > 1.

Original entry on oeis.org

24, 749, 48, 311, 690, 251, 2706, 2773, 6504, 1081, 2162, 1753, 11356, 6223, 1392, 2303, 9838, 637, 14510, 1995, 3154, 21459, 72960, 5691, 8140, 1475, 2350, 3647, 1593, 7607, 55074, 2719, 9852, 12143, 106562, 12615, 9036, 19883, 15438, 28369, 8560, 8415, 3831

Offset: 2

Zak Seidov, Mar 17 2017

nonn

The sequence is non-monotone.

			a(2) = 24 = A000040(5) + A000040(6) = 11 + 13 = A001358(4) + A001358(5) = 10 + 14,
a(3) = 749 = A000040(53) + A000040(54) + A000040(55) = 241 + 251 + 257 = A001358(79) + A001358(80) + A001358(81) = 247 + 249 + 253.

Alois P. Heinz, Table of n, a(n) for n = 2..1000

Cf. A000040 Primes, A001358 Semiprimes, A118717 Sum of two consecutive semiprimes.

Sum of k consecutive primes: A001043 k=2, A034961 k=3, A034963 k=4, A034964 k=5, A127333 k=6, A127334 k=7, A127335 k=8, A127336 k=9, A127337 k=10, A127338 k=11, A127339 k=12.

issp:= n-> is(not isprime(n) and numtheory[bigomega](n)=2):
ithsp:= proc(n) option remember; local k; for k from 1+
        `if`(n=1, 1, ithsp(n-1)) while not issp(k) do od; k
        end:
ps:= proc(i, j) option remember;
       ithprime(j)+`if`(i=j, 0, ps(i, j-1))
     end:
ss:= proc(i, j) option remember;
       ithsp(j)+`if`(i=j, 0, ss(i, j-1))
     end:
a:= proc(n) option remember; local i, j, k, l, p, s;
      i, j, k, l, p, s:= 1, n, 1, n, ps(1, n), ss(1, n);
      do if p=s then return p
       elif pAlois P. Heinz, Mar 24 2017

Mathematica

sp=Select[Range[4,100000],2==PrimeOmega[#]&];pr=Prime[Range[PrimePi[Max[sp]]]]; Table[Intersection[(Total/@Partition[pr,k,1]),Total/@Partition[sp,k,1]][[1]],{k,2,100}]

Extensions

More terms from Alois P. Heinz, Mar 24 2017

A288949 Numbers that are both the sum of two consecutive primes and the sum of two consecutive semiprimes.

Original entry on oeis.org

24, 36, 100, 112, 120, 240, 288, 320, 372, 472, 532, 576, 600, 810, 828, 864, 882, 924, 990, 1088, 1104, 1164, 1180, 1208, 1236, 1284, 1360, 1392, 1482, 1508, 1560, 1584, 1620, 1632, 1692, 1740, 1818, 1900, 1920, 1938

Offset: 1

Zak Seidov, Jun 20 2017

nonn

Positions of a(n) in A001043 and A118717: {5, 4}, {7, 6}, {15, 17}, {16, 19}, {17, 21}, {30, 39}, {34, 48}, {37, 53}, {42, 60}, {51, 77}.

			24 is a term because 24 = 11+13 and 24 = 10+14.
Alternatively, 24 = A001043(5) = A118717(4), 36 = A001043(7) = A118717(6).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Intersection of A001043 and A118717.

sp=Select[Range[4,1000],2==PrimeOmega[#]&]; Select[Table[sp[[k]]+sp[[k+1]],{k,100}], #==(p=NextPrime[#/2,-1])+NextPrime[p]&]
Module[{nn=2000,sp},sp=Total/@Partition[Select[Range[nn],PrimeOmega[#]==2&],2,1];Intersection[ sp,Total/@Partition[Prime[Range[nn]],2,1]]] (* Harvey P. Dale, Jul 31 2023 *)

issemi(n)=bigomega(n)==2
nextsp(x)=x=ceil(x); while(!issemi(x), x++); x
has(n)=precprime((n-1)/2)+nextprime(n/2)==n
list(lim)=my(v=List(),last=4,t); forfactored(n=6,nextsp(lim\2), if(vecsum(n[2][,2])==2, if(has(t=last+n[1]) && t<=lim, listput(v,t)); last=n[1])); Vec(v) \\ Charles R Greathouse IV, Feb 19 2018

A335076 Numbers that are the sum of two consecutive semiprimes and also the sum of two consecutive 3-almost primes.

Original entry on oeis.org

47, 95, 115, 134, 151, 201, 233, 285, 301, 335, 346, 368, 461, 513, 527, 541, 576, 640, 713, 787, 801, 810, 864, 907, 935, 944, 1104, 1160, 1225, 1245, 1255, 1360, 1397, 1471, 1513, 1521, 1574, 1620, 1692, 1740, 1775, 1782, 1831, 1867, 1873, 1913, 1967, 2009

Offset: 1

Zak Seidov, May 22 2020

nonn

Apparently the sequence is infinite.

			47 = 22 + 25 = A118717(8) = A001358(8) + A001358(9), and
47 = 20 + 27 = A014612(4) + A014612(5).

Zak Seidov, Table of n, a(n) for n = 1..26539 (all terms up to 10^6).

Cf. A001358, A014612, A118717.

p[n_, m_] := Plus @@@ Partition[Select[Range[m], PrimeOmega[#] == n &], 2, 1]; m = 1100; Intersection[p[2, m], p[3, m]] (* Amiram Eldar, May 24 2020 *)

A288955 Integers that are the sum of two consecutive semiprimes, as well as the product of two consecutive semiprimes.

Original entry on oeis.org

24, 1482, 2805, 3596, 7917, 12765, 17157, 17822, 21170, 25122, 29913, 36278, 42230, 45582, 58539, 61503, 62997, 109886, 117986, 145542, 258055, 264710, 268323, 272994, 281957, 294306, 306362, 319790, 324318, 387506, 491397, 599838, 613085, 656091, 679758, 709806, 771762, 793877

Offset: 1

Zak Seidov, Jun 20 2017

nonn

			24 = A118717(4) = A108215(1), 1482 = A118717(223) = A108215(14).

Cf. A001358. Intersection of A118717 and A108215.

s = Partition[Select[Range[10^6], PrimeOmega@ # == 2 &], 2, 1]; Intersection[Map[Total, s], Map[Times @@ # &, s]] (* Michael De Vlieger, Jun 21 2017 *)

More terms from Michel Marcus, Jun 27 2017

cp's OEIS Frontend

A181529 Natural numbers equal to the sum of two consecutive semiprimes (A118717), but not an integer multiple of a previous term.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A173966 Sums of two consecutive semiprimes.

Views

Author

Keywords

Comments

Links

Programs

Mathematica

A370162 Semiprimes that are the sum of two successive semiprimes and also the sum of three successive semiprimes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

A283873 Smallest number that is the sum of n successive primes and also the sum of n successive semiprimes, n > 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A288949 Numbers that are both the sum of two consecutive primes and the sum of two consecutive semiprimes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A335076 Numbers that are the sum of two consecutive semiprimes and also the sum of two consecutive 3-almost primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A288955 Integers that are the sum of two consecutive semiprimes, as well as the product of two consecutive semiprimes.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Extensions