A092192 -id:A092192 - cp's OEIS frontend

A366167 Semiprimes that are the sum of two successive terms of A092192.

25, 146, 201, 221, 249, 302, 365, 529, 662, 681, 849, 949, 1211, 1282, 1318, 1343, 1849, 2517, 3223, 3398, 3466, 3635, 3867, 3949, 4063, 4749, 4819, 4997, 5158, 6049, 6614, 7023, 7041, 7066, 7117, 7921, 8314, 8471, 8709, 8727, 8914, 8981, 9155, 9235, 9299, 9563, 10741, 10895, 10958, 11435, 11962

Offset: 1

Zak Seidov and Robert Israel, Oct 02 2023

nonn

			a(3) = 201 is a term because 201 = 95 + 106 = A092192(7) + A092192(8).

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

Cf. A001358, A092192.

SP:= select(t -> numtheory:-bigomega(t) = 2, [$1..10000]):
A092192:= select(t -> numtheory:-bigomega(t) = 2, SP[2..-1]+SP[1..-2]):
select(t -> numtheory:-bigomega(t) = 2, A092192[2..-1]+A092192[1..-2]);

sim = Select[Range[4, 100000], 2 == PrimeOmega[#];&]; se = Select[Drop[sim, 1]
+ Drop[sim, -1], 2 == PrimeOmega[#] &];    Select[Drop[se, 1] + Drop[se, -1], 2
== PrimeOmega[#] &]

upto(n) = {my(pr = 10, res = List(), semiprimes = List([4,6])); forfactored(i = 9, n, if(bigomega(i[2]) == 2, listpop(semiprimes, 1); listput(semiprimes, i[1]); s = semiprimes[1] + semiprimes[2]; if(bigomega(s) == 2, c = s + pr; if(c > n, return(res)); if(bigomega(c) == 2, listput(res, c)); pr = s))); res} \\ David A. Corneth, Oct 02 2023

A131610 Semiprimes that are the sum of three successive semiprimes.

Original entry on oeis.org

25, 33, 39, 58, 93, 123, 134, 146, 155, 177, 185, 253, 278, 295, 358, 362, 417, 446, 478, 538, 566, 597, 614, 698, 749, 766, 794, 898, 917, 982, 998, 1042, 1059, 1081, 1149, 1159, 1286, 1351, 1393, 1441, 1546, 1589, 1623, 1639, 1649, 1658, 1718, 1766, 1799

Offset: 1

Zak Seidov, Oct 02 2007

nonn

a(n) = A001358(m) = A001358(i)+A001358(i+1)+A001358(i+2), for some m, i. Corresponding values of m and i are: 9, 11, 15, 21, 32, 42, 45, 50, 51, 57, 60, 81, 88, 92, 113, 115, 132, 137, 147, 168, 178, 186, 188; 2, 3, 4, 6, 10, 14, 15, 16, 17, 20, 21, 27, 31, 33, 38, 39, 45, 49, 52, 57, 60, 62, 65.

			25=6+9+10, or A001358(9)=A001358(2)+A001358(3)+A001358(4),
33=9+10+14, or A001358(11)=A001358(3)+A001358(4)+A001358(5).

Zak Seidov, Table of n,a(n) for n=1..10000.

Cf. A001358, A092192.

sp=Select[Range[612],PrimeOmega[#]==2&];L=Length[sp];s={};Do[sm=sp[[n]]+sp[[n+1]]+sp[[n+2]];If[PrimeOmega[sm]==2,AppendTo[s,sm]],{n,L-2}];s (* James C. McMahon, Feb 26 2025 *)

lista(nn) = {vec = vector(nn, i, i); sp = select(i->(bigomega(i) == 2), vec); for (i = 2, #sp-1, sumt = sp[i-1] + sp[i] + sp[i+1]; if (bigomega(sumt) == 2, print1(sumt, ", ")););} \\ Michel Marcus, Oct 13 2013

A092191 Numbers n such that sum of n-th and (n+1)-th semiprimes is a semiprime.

Original entry on oeis.org

1, 2, 9, 12, 14, 15, 16, 18, 20, 23, 24, 26, 30, 32, 34, 35, 36, 38, 43, 44, 49, 50, 54, 55, 56, 57, 58, 61, 62, 63, 66, 67, 68, 69, 73, 75, 80, 84, 88, 91, 93, 98, 99, 100, 103, 107, 108, 110, 112, 114, 118, 119, 124, 125, 128, 133, 137, 138, 146, 147, 150, 151, 153

Offset: 1

Zak Seidov, Feb 23 2004

			9 is a member because sum of 9th and 10th semiprimes, 25 and 26, is 51 which is a semiprime.

Resulting semiprimes: A092192.

A158339 Semiprimes that are the sum of four successive semiprimes.

Original entry on oeis.org

39, 94, 106, 118, 146, 158, 185, 201, 221, 254, 302, 365, 427, 473, 485, 519, 537, 589, 633, 655, 707, 723, 749, 767, 842, 851, 869, 901, 1003, 1145, 1205, 1211, 1219, 1247, 1263, 1337, 1349, 1603, 1646, 1681, 1703, 1731, 1797, 1891, 1903, 1937, 2005, 2019

Offset: 1

Zak Seidov, Mar 16 2009

nonn

			a(1)=39=6+9+10+14, or A001358(15)=A001358(2)+A001358(3)+A001358(4)+A001358(5).

Zak Seidov, Table of n, a(n) for n = 1..100000

Semiprimes that are the sum of k successive semiprimes: A131610 (k=3), A092192 (k=2), A001358 (k=1).

Select[Total/@Partition[Select[Range[600],PrimeOmega[#]==2&],4,1], PrimeOmega[ #]==2&] (* Harvey P. Dale, Aug 14 2014 *)

issemi(n)=bigomega(n)==2
list(lim)=if(lim<39, return([])); my(v=List(), u=v, x=lim\4+log(lim)*4\1+9); forprime(p=2,x\2, forprime(q=2,min(x\p,p), listput(u,p*q))); u=Set(u); while(u[#u-2]+u[#u-1]+u[#u]+x+1<=lim, while(!issemi(x++),); u=concat(u,x)); for(i=1,#u-3, u[i]+=u[i+1]+u[i+2]+u[i+3]); u[#u-2]=u[#u-1]=u[#u]=1; forprime(p=2,lim\2, forprime(q=2,min(lim\p,p), listput(v,p*q))); setintersect(Set(v), u) \\ Charles R Greathouse IV, Mar 05 2017

A254712 Semiprimes that are the sum of five successive semiprimes.

Original entry on oeis.org

69, 82, 166, 417, 451, 545, 614, 679, 717, 731, 763, 779, 799, 813, 851, 1047, 1057, 1077, 1101, 1119, 1138, 1262, 1293, 1346, 1371, 1603, 1639, 1651, 1678, 1739, 1761, 1817, 1897, 1959, 1983, 2049, 2078, 2177, 2263, 2446, 2498, 2533, 2661, 2705, 2734, 2746, 2762

Offset: 1

Zak Seidov, Feb 06 2015

nonn

			69=A001358(24)=A001358(3)+...+A001358(7)= 9 + 10 + 14 + 15 + 21,
82=A001358(27)=A001358(4)+...+A001358(8)= 10 + 14 + 15 + 21 + 22.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A001358, A092192, A131610, A158339.

Module[{sps=Select[Range[750],PrimeOmega[#]==2&]},Select[Total/@ Partition[ sps,5,1],PrimeOmega[ #] ==2&]] (* Harvey P. Dale, Nov 11 2021 *)

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));

A266451 Semiprimes that are the sum of six consecutive semiprimes.

Original entry on oeis.org

58, 91, 123, 142, 161, 205, 278, 473, 566, 614, 706, 718, 802, 838, 851, 889, 1079, 1211, 1226, 1238, 1262, 1286, 1385, 1415, 1633, 1714, 1819, 1891, 1945, 2005, 2123, 2147, 2194, 2217, 2327, 2374, 2427, 2563, 2594, 2653, 2771, 2815, 2854, 2947, 2987, 3118, 3133, 3151, 3199, 3214, 3305, 3379

Offset: 1

Zak Seidov, Dec 29 2015

nonn

			58 = A001358(21) = A001358(1) + ... + A001358(6) = 4+6+9+10+14+15,
91 = A001358(31) = A001358(3) + ... + A001358(8) = 9+10+14+15+21+22.

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

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

N:= 10^4: # to get all terms where the 6 consecutive semiprimes <= N
P:= select(isprime, [2,seq(i,i=3..N/2,2)]): nP:= nops(P):
SP:= NULL:
for i from 1 to nP do
  for j from 1 to nP while P[i]*P[j] <= N do od:
  SP:= SP, op(map(`*`,P[i],P[1..j-1]));
od:
SP:= sort(convert({SP},list)): nSP:= nops(SP):
select(numtheory:-bigomega=2, [seq(convert(SP[i..i+5],`+`),i=1..nSP-5)]): # Robert Israel, Nov 19 2017

Select[(Total[#] & /@ Partition[Select[Range[4, 9999], 2 == PrimeOmega[#] &], 6, 1]), 2 == PrimeOmega[#] &]

A370685 Semiprimes that are also the sums of two, three and four successive semiprimes.

Original entry on oeis.org

2045, 2705, 2855, 14614, 18838, 28437, 31299, 43603, 68807, 76841, 77386, 88041, 108415, 116822, 194605, 213679, 218729, 252094, 255202, 269653, 290449, 294683, 302761, 305362, 310799, 339382, 348242, 361055, 398111, 445066, 445174, 459761, 464567, 489809, 496081, 501386, 515981, 534777, 544405

Offset: 1

Robert Israel, Feb 26 2024

nonn

			a(3) = 2855 is a term because 2855 = 5 * 571 is a semiprime, A001358(423) = 1418 = 2*709 and A001358(424) = 1437 = 3 * 479 are two successive semiprimes whose sum is 2855, A001358(285) = 949 = 13 * 73, A001358(286) = 951 = 3 * 317 and A001358(287) = 955 = 5 * 191 are three successive semiprimes whose sum is 2855, and A001358(216) = 707 = 7 * 101, A001358(217) = 713 = 23 * 31, A001358(218) = 717 = 3 * 239, A001358(219) = 718 = 2 * 359 are four successive semiprimes whose sum is 2855.

Robert Israel, Table of n, a(n) for n = 1..2734

Cf. A001358, A370162. Intersection of A092192, A131610 and A158339.

N:= 10^6: # 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))}:
SS4:= {seq(SS[i]-SS[i-4], i=5..nops(SS))}:
A:=SS2 intersect SS3 intersect SS4 intersect convert(SP, set):
A:= sort(convert(A, list)):

A380316 Sphenic numbers that are the sum of two successive sphenics.

Original entry on oeis.org

385, 555, 759, 897, 935, 957, 1185, 1245, 1265, 1335, 2015, 2037, 2185, 2211, 2261, 2379, 2607, 2821, 2877, 2937, 3059, 3298, 3363, 3434, 3485, 3507, 3538, 3815, 3913, 4029, 4255, 4378, 4433, 4526, 4615, 4738, 4795, 4947, 5181, 5205, 5395, 5405, 5523, 5681, 5829, 5883, 6226

Offset: 1

Massimo Kofler, Jan 20 2025

nonn

The sum of two consecutive sphenic numbers varies as 4n log n/(log log n)^2, so this sequence is not smaller than that. - Charles R Greathouse IV, Jan 21 2025

			385 = 5*7*11 is a member because 385 = 190+195, sum of 18-th and 19-th sphenic number.
555 = 3*5*37 is a member because 555 = 273+292, sum of 28-th and 29-th sphenic number.

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

Cf. A001043, A007304, A092192.

issphenic:= proc(n) local F; F:= ifactors(n)[2]; F[..,2] = [1,1,1] end proc:
S:= select(issphenic, [$1..10000]):
select(issphenic, S[1..-2]+S[2..-1]);
# Robert Israel, Jan 20 2025

sphenicQ[n_] := FactorInteger[n][[;; , 2]] == {1, 1, 1}; Select[MovingMap[Total, Select[Range[3200], sphenicQ], 1], sphenicQ] (* Amiram Eldar, Jan 21 2025 *)

issphenic(n) = my(f=factor(n)); (bigomega(f)==3) && (omega(f)==3);
lista(nn) = my(v=select(issphenic, [1..nn])); select(issphenic, vector(#v-1, k, v[k]+v[k+1])); \\ Michel Marcus, Jan 20 2025

sphen(lim)=my(v=List(), t); forprime(p=2, sqrtnint(lim\=1,3), forprime(q=p+1, sqrtint(lim\p), t=p*q; forprime(r=q+1, lim\t, listput(v, t*r)))); Set(v)
has(n,f=factor(n))=f[,2]==[1,1,1]~
list(lim)=my(v=List(),u=sphen(lim\2)); for(i=2,#u, if(has(u[i]+u[i-1]), listput(v,u[i]+u[i-1]))); forfactored(k=lim\2+1,lim\1-u[#u], if(has(0,k[2]), if(has(k[1]+u[#u]), listput(v,k[1]+u[#u])); break)); Vec(v) \\ Charles R Greathouse IV, Jan 21 2025

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.

cp's OEIS Frontend

A366167 Semiprimes that are the sum of two successive terms of A092192.

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A131610 Semiprimes that are the sum of three successive semiprimes.

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A092191 Numbers n such that sum of n-th and (n+1)-th semiprimes is a semiprime.

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A158339 Semiprimes that are the sum of four successive semiprimes.

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A254712 Semiprimes that are the sum of five successive semiprimes.

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

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A266451 Semiprimes that are the sum of six consecutive semiprimes.

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A370685 Semiprimes that are also the sums of two, three and four successive semiprimes.

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A380316 Sphenic numbers that are the sum of two successive sphenics.

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