A213025 -id:A213025 - cp's OEIS frontend

A086005 Semiprimes sandwiched between semiprimes.

34, 86, 94, 122, 142, 202, 214, 218, 302, 394, 446, 634, 698, 842, 922, 1042, 1138, 1262, 1346, 1402, 1642, 1762, 1838, 1894, 1942, 1982, 2102, 2182, 2218, 2306, 2362, 2434, 2462, 2518, 2642, 2722, 2734, 3098, 3386, 3602, 3694, 3866, 3902, 3958, 4286, 4414

Offset: 1

Reinhard Zumkeller, Jul 07 2003

nonn

These are some of the balanced semiprimes (see A213025). - Alonso del Arte, Jun 04 2012

			94 = 47*2: 94 - 1 = 3*31 and 94 + 1 = 5*19, therefore 94 is in the sequence.

T. D. Noe, Table of n, a(n) for n=1..10000
Eric Weisstein's World of Mathematics, Semiprime

Cf. A001358, A056809, A064911, A100484, A123255.

a086005 n = a086005_list !! (n-1)
a086005_list = filter
   (\x -> a064911 (x - 1) == 1 && a064911 (x + 1) == 1) a100484_list
-- Reinhard Zumkeller, Aug 08 2013, Jun 10 2012

u[n_]:=Plus@@Last/@FactorInteger[n]==2;lst={};Do[If[u[n],sp=n;If[u[sp-1]&&u[sp+1],AppendTo[lst,sp]]],{n,8!}];lst  (* Vladimir Joseph Stephan Orlovsky, Nov 16 2009 *)
(* First run program for A109611 to define semiPrimeQ *) Select[Range[4000], Union[{semiPrimeQ[# - 1], semiPrimeQ[#], semiPrimeQ[# + 1]}] == {True} &] (* Alonso del Arte, Jun 03 2012 *)
Select[Partition[Range@ 4000, 3, 1], Union@ PrimeOmega@ # == {2} &][[All, 2]] (* Michael De Vlieger, Jun 14 2017 *)

from itertools import count, islice
from sympy import factorint, isprime
def agen(): # generator of terms
    nxt = 0
    for k in count(2, 2):
        prv, nxt = nxt, sum(factorint(k+1).values())
        if prv == nxt == 2 and isprime(k//2): yield k
print(list(islice(agen(), 46))) # Michael S. Branicky, Nov 26 2022

a(n) = 2*A086006(n).

a(n) = A056809(n)+1. - Zak Seidov, Sep 30 2012

A258276 Balanced sphenic numbers (of order one): sphenic numbers (A007304) that are the average of the previous sphenic number and the following sphenic number.

Original entry on oeis.org

186, 370, 406, 418, 518, 582, 602, 710, 786, 814, 826, 830, 942, 978, 994, 1010, 1034, 1070, 1162, 1310, 1374, 1394, 1570, 1630, 1686, 1758, 1886, 1978, 2014, 2114, 2158, 2270, 2274, 2278, 2294, 2438, 2510, 2534, 2570, 2630, 2666, 2690, 2774, 2778, 2782, 2806

Offset: 1

Antonio Roldán, May 25 2015

nonn

			406 is in this sequence because 406 = A007304(45) = (402+410)/2 = (A007304(44) + A007304(46))/2.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A006562, A007304, A213025, A245289.

a=30; b=42; c=66; L={}; While[ Length[L]<100, If[b-a == c-b, AppendTo[L, b]]; a=b; b=c++; While[ Last/@ FactorInteger[c] != {1,1,1}, c++]]; L (* Giovanni Resta, Jun 09 2015 *)

issphenic(n)=if(n>0,omega(n)==3&&bigomega(n)==3,0)
nextsph(n)={local(k=n+1);while(!issphenic(k),k+=1);k}
precsph(n)={local(k=n-1);while(!issphenic(k)&&k>0,k-=1);k}
{for(i=1,4*10^3,if(issphenic(i)&&2*i== nextsph(i)+ precsph(i),print1(i, ", ")))}

A213063 Balanced numbers (of order one): k-almost primes that are the average of three successive k-almost primes.

Original entry on oeis.org

5, 34, 53, 68, 86, 94, 102, 122, 142, 157, 171, 173, 185, 188, 194, 202, 204, 211, 214, 218, 245, 257, 258, 262, 263, 285, 289, 302, 314, 321, 338, 342, 358, 366, 371, 373, 394, 404, 407, 413, 415, 422, 429, 435, 446, 471, 489, 490, 493, 497, 507, 513, 517, 524, 535, 562

Offset: 1

Gerasimov Sergey, Jun 03 2012

nonn

Balanced numbers of order one: defined by the union of balanced primes A006562, balanced semiprimes A213025, balanced 3-almost primes (68, 102, 171, 188, 245, 258, 285, 338, 366, 404, 429, 435, 507, 524,..), balanced 4-almost primes (204, 342, 490, 513,..),.., balanced k-almost primes - all of order one.

Balanced numbers of order two are 79, 119, 148, 205, 218, 281, 299, 302, 339, 349, 410, 439, 493,.., defined by the union of balanced primes of order two of A082077, balanced semiprimes of order two (119, 205, 218, 299, 302, 339, 493,..), balanced 3-almost primes of order two (148, 410, 604, 609, 642..),.., balanced k-almost primes of order two.

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

Cf. A001222, A006562, A014612, A014613, A014614, A046306, A046308, A213025.

list(lim)={
lim=lim\1+.5;
my(v=List(),L=log(lim)\log(2),left=vector(L),middle=vector(L),t);
for(n=3,2*lim,
t=bigomega(n);
if(t>L,next);
if(middle[t],
if(2*middle[t] == left[t] + n,
if(middle[t] < lim,
listput(v,middle[t])
,
if(vecmin(middle) > lim, return(vecsort(Vec(v))))
)
);
left[t]=middle[t];
middle[t]=n
,
if(left[t],middle[t]=n,left[t]=n)
)
)
}; \\ Charles R Greathouse IV, Jun 14 2012

A365222 a(n) is the least semiprime such that a(n) - n and a(n) + n are the previous and the next semiprimes.

Original entry on oeis.org

34, 185, 262, 407, 314, 371, 194, 2271, 6218, 4237, 109898, 110645, 53602, 169773, 112298, 163985, 284738, 48529, 1033378, 1781833, 570502, 1963091, 12527458, 6051613, 30377422, 19549343, 66761746, 7926901, 363311378, 5861227, 676386278, 136503631, 72622874, 51204973, 375025874

Offset: 1

Zak Seidov and Robert Israel, Aug 26 2023

nonn

			a(3) = 262 because 262 = 2 * 131 is a semiprime, the previous semiprime is 262 - 3 = 259 = 7 * 37, the next semiprime is 262 + 3 = 265 = 5 * 53, and no semiprime less than 262 works.

Cf. A001358, A213025.

N:= 46: # for a(1) .. a(N)
V:= Vector(N):
b:= 6: d:= 2: count:= 0:
for i from 7 while count < N do
  if numtheory:-bigomega(i) = 2 then
    ib:= i-b;
    if ib = d and ib <= N then
      if V[d] = 0 then V[d]:= b; count:= count+1 fi
    else d:= ib
    fi;
    b:= i;
  fi;
od:
convert(V,list);

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A086005 Semiprimes sandwiched between semiprimes.

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A258276 Balanced sphenic numbers (of order one): sphenic numbers (A007304) that are the average of the previous sphenic number and the following sphenic number.

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A213063 Balanced numbers (of order one): k-almost primes that are the average of three successive k-almost primes.

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A365222 a(n) is the least semiprime such that a(n) - n and a(n) + n are the previous and the next semiprimes.

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