A014614 -id:A014614 - cp's OEIS frontend

A114966 Prime(n) + Semiprime(n) + 3AlmostPrime(n) + 4AlmostPrime(n) + 5AlmostPrime(n).

62, 93, 140, 157, 214, 224, 248, 326, 344, 364, 384, 423, 451, 516, 538, 568, 589, 600, 630, 672, 689, 736, 807, 837, 871, 892, 916, 937, 964, 993, 1030, 1052, 1090, 1100, 1164, 1192, 1250, 1294, 1320, 1359, 1373, 1387, 1435, 1454, 1487, 1526, 1547, 1584

Offset: 1

Jonathan Vos Post, Feb 21 2006

Primes in this sequence include a(4) = 157, a(28) = 937, a(41) = 1373, a(45) = 1487, a(49) = 1609.

			a(1) = Prime(1) + Semiprime(1) + 3AlmostPrime(1) +
4AlmostPrime(1) + 5AlmostPrime(1) = 2 + 4 + 8 + 16 + 32 = 62.
a(6) = A114944(6) + A014614(6) = 112 + 112 = 224.

Harvey P. Dale, Table of n, a(n) for n = 1..3000

Cf. A000040, A001358, A014612, A014613, A014613, A114382, A114944.

Module[{nn=1000,p1,p2,p3,p4,p5,len},p1=Prime[Range[nn]];p2= Select[ Range[ nn], PrimeOmega[ #] ==2&];p3=Select[ Range[nn], PrimeOmega[ #]==3&];p4=Select[ Range[ nn],PrimeOmega[#]==4&];p5=Select[ Range[ nn], PrimeOmega[ #]==5&];len=Min[Length/@{p1,p2,p3,p4,p5}]; Total/@Thread[ {Take[ p1,len], Take[p2,len],Take[p3,len], Take[p4,len],Take[p5,len]}]] (* Harvey P. Dale, Apr 16 2015 *)

a(n) = A000040(n) + A001358(n) + A014612(n) + A014613(n) + A014614(n). a(n) = A114944(n) + A014614(n).

Corrected by Harvey P. Dale, Apr 16 2015

A131175 Table, read by rows, of coefficients of characteristic polynomials of almost prime matrices.

Original entry on oeis.org

1, -2, 1, -8, 1, -26, -4, 1, -66, -36, 1, -174, -196, 1, -398, -676, 1, -878, -3044, 1, -2174, -6852, -192, 1, -4862, -18628, -704, 1, -10494, -45508, -1216, 1, -22014, -141252, -6336, 1, -47614, -315332, -10432, 1, -100862, -858052, -55488, 1, -225278, -1878980, -245952

Offset: 1

Jonathan Vos Post, Sep 24 2007

Because the first column of A is a column vector of powers of 2, the determinant (for n>1) is always 0. Hence the rank is always (for n>1) less than n. A[n.n] = n-th n-almost prime A101695. The second column of the table is the negative of the trace of the matrices.

			A_1 = [2], with determinant = 2 and characteristic polynomial = x-2, with coefficients (1, -2) so a(a) = 1 and a(2) = -2.
A_2 =
[2.3]
[4.6]
with determinant = 0, polynomial x^2 - 8x, so the coefficients are (1, -8), hence a(3) = 1 and a(4) = -8.
A_3 =
[2..3..5]
[4..6..9]
[8.12.18]
with determinant = 0, polynomial = x^3 - 26x^2, -4x, so coefficients are (1, -26, -4), hence a(5) = 1, a(6) = -26, a(7) = -4.

Cf. A000040, A001358, A014612, A014613, A014614, A101695.

A078840 := proc(n,m) local p,k ; k := 1 ; p := 2^n ; while k < m do p := p+1 ; while numtheory[bigomega](p) <> n do p := p+1 ; od; k := k+1 ; od: RETURN(p) ; end: A131175 := proc(nrow,showall) local A,row,col,pol,T,a ; A := linalg[matrix](nrow,nrow) ; for row from 1 to nrow do for col from 1 to nrow do if row = col then A[row,col] := x-A078840(row,col) ; else A[row,col] := -A078840(row,col) ; fi ; od: od: pol := linalg[det](A) ; T := [] ; for col from nrow to 0 by -1 do a := coeftayl(pol,x=0,col) ; if a <> 0 or showall then T := [op(T),a] ; fi ; od; RETURN(T) ; end: for n from 1 to 15 do print(op(A131175(n,false))) ; od: # R. J. Mathar, Oct 26 2007

Row n of the table consists of the coefficients of x^n, x^n-1, ... of the characteristic polynomial of the n X n matrix A whose first row is the first n primes (1-almost primes) (A000040), 2nd row is the first n semiprimes (2-almost primes) A001358, 3rd row is the first n 3-almost primes A014612.

Corrected and extended by R. J. Mathar, Oct 26 2007

A156620 Primes p such that p^2 - 2 is a 5-almost prime.

Original entry on oeis.org

1201, 2999, 4001, 4273, 5009, 7151, 8467, 9769, 10427, 10937, 11701, 11897, 12011, 12113, 12323, 13339, 13681, 14087, 14563, 15187, 15277, 15809, 16139, 16699, 17209, 17383, 17483, 17623, 18757, 19051, 19267, 19697, 20107, 20129, 20297

Offset: 1

Rick L. Shepherd, Feb 11 2009

nonn

Corresponding 5-almost primes are A156621.

This sequence is infinite: Ribenboim states that Rieger proved in 1969 that "there exist infinitely many primes p such that p^2 - 2 [is an element of] P_5", this being a particular case of a general theorem proved (also in 1969) by Richert: (again quoting Ribenboim) "Let f(X) be a polynomial with integral coefficients, positive leading coefficient, degree d >= 1 (and different from X). Assume that for every prime p, the number [rho](p) of solutions of f(X) = 0 (mod p) is less than p; moreover if p <= d+1 and p does not divide f(0) assume also that [rho](p) < p-1. Then, there exist infinitely many primes p such that f(p) is a (2d+1)-almost prime."

H. Halberstam and H. E. Richert, Sieve Methods, Academic Press, NY, 1974.
P. Ribenboim, The Little Book of Big Primes, Springer-Verlag, NY, 1991, p. 184.
G. J. Rieger, On polynomials and almost-primes, Bull. Amer. Math. Soc., 75 (1969), 100-103.

Cf. A156621, A014614.

Select[Prime[Range[5000]],PrimeOmega[#^2-2]==5&] (* Harvey P. Dale, Jul 11 2014 *)

forprime(p=2, prime(2500), if(bigomega(p^2-2)==5, print1(p,", ")))

A156621 5-almost primes of the form p^2 - 2, where p is prime.

Original entry on oeis.org

1442399, 8993999, 16007999, 18258527, 25090079, 51136799, 71690087, 95433359, 108722327, 119617967, 136913399, 141538607, 144264119, 146724767, 151856327, 177928919, 187169759, 198443567, 212080967, 230644967, 233386727, 249924479

Offset: 1

Rick L. Shepherd, Feb 11 2009

nonn

Corresponding primes are A156620.

This sequence is infinite: See A156620 for comments and references.

Cf. A156620, A014614.

forprime(p=2, prime(2500), ap=p^2-2; if(bigomega(ap)==5, print1(ap,", ")))

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

A288517 Least integer k such that A001358(k) + A001358(k+1) is the product of exactly n prime factors (counting multiplicity).

Original entry on oeis.org

3, 1, 28, 4, 19, 39, 48, 89, 120, 551, 447, 589, 3707, 10137, 21644, 28456, 22998, 44494, 86132, 166930, 703448, 628371, 1220814, 1608668, 11153853, 6091437, 56676014, 268389220, 146153797, 193010987, 916382785, 738246947, 4702317172, 2830095027, 12627951809

Offset: 1

Zak Seidov, Jun 10 2017

nonn

			n=1: k=3, A001358(3) + A001358(4) = 9 + 10 = 19 = A000040(8) (8th prime),
n=2: k=1, A001358(1)+A001358(2) = 4+6 = 10 = 2*5 = A001358(4) (4th semiprime),
n=11: k=447, A001358(447)+A001358(448) = 1535+1537 = 3072 = 2^10*3 = A069272(2) (2nd 11-almost prime).

Charles R Greathouse IV, Table of n, a(n) for n = 1..50

Cf. A000040, A001358, A014612, A014614, A046306, A046308, A046310, A046312, A046314, A069272.

a(21)-a(35) from Charles R Greathouse IV, Jun 10 2017

A321169 a(n) is the smallest prime p such that p + 2 is a product of n primes (counted with multiplicity).

Original entry on oeis.org

3, 2, 43, 79, 241, 727, 3643, 15307, 19681, 164023, 1673053, 885733, 2657203, 18600433, 23914843, 100442347, 358722673, 645700813, 4519905703, 18983603959, 48427561123, 31381059607, 261508830073, 1307544150373, 3295011258943, 24006510600883, 12709329141643, 53379182394907, 190639937124673, 2615579937350539

Offset: 1

Amiram Eldar and Zak Seidov, Jan 10 2019

nonn

a(n) ~ c * 3^n. - David A. Corneth, Jan 11 2019

			a(1) = 3 as 3 + 2 = 5 (prime),
a(2) = 2 as 2 + 2 = 4 = 2*2 (semiprime),
a(3) = 43 as 43 + 2 = 45 = 3*3*5  (3-almost prime),
a(4) = 79 as 79 + 2 = 81 = 3*3*3*3 (4-almost prime).

David A. Corneth, Table of n, a(n) for n = 1..801

Cf. A001222, A001358, A014612, A014613, A014614, A046306, A046308, A046310, A046312, A046314, A073919, A118883.

ptns[n_, 0] := If[n == 0, {{}}, {}]; ptns[n_, k_] := Module[{r}, If[n < k, Return[{}]]; ptns[n, k] = 1 + Union @@ Table[PadRight[#, k] & /@ ptns[n - k, r], {r, 0, k}]]; a[n_] := Module[{i, l, v}, v = Infinity; For[i = n, True, i++, l = (Times @@ Prime /@ # &) /@ ptns[i, n]; If[Min @@ l > v, Return[v]]; minp = Min @@ Select[l - 2, PrimeQ]; If[minp < v, v = minp]]] ; Array[a, 10] (* after Amarnath Murthy at A073919 *)

a(n) = forprime(p=2, , if (bigomega(p+2) == n, return (p))); \\ Michel Marcus, Jan 10 2019

a(n) = {my(p3 = 3^n, u, c); if(n <= 2, return(4 - n)); if(isprime(p3 - 2), return(p3 - 2)); forprime(p = 5, oo, if(isprime(p3 / 3 * p - 2), u = p3 / 3 * p - 2; break ) ); for(i = 2, n, if(p3 * (5/3)^i > u, return(u)); for(j = 1, oo, if(p3 * j \ 3^i > u, next(2)); if(bigomega(j) == i, if(isprime(p3 / 3^(i) * j - 2), u = p3 / 3^(i) * j - 2; next(2) ) ) ) ); return(u) } \\ David A. Corneth, Jan 11 2019

A342246 Numbers k such that k-1, k and k+1 are all composite with four, five and six (not necessarily distinct) prime factors respectively.

Original entry on oeis.org

11151, 13455, 23375, 26271, 31311, 33776, 36125, 40375, 45495, 46375, 48411, 49049, 49167, 61335, 63125, 74151, 77895, 78111, 78351, 80271, 82575, 83511, 84591, 86031, 87375, 88749, 90207

Offset: 1

Sean Lestrange, Mar 07 2021

nonn

			For k=11151 we have 11150 = 2 * 5^2 * 223 which is composite with four prime factors, 11151 = 3^3 * 7 * 59 which is composite with five prime factors and 11152 = 2^4 * 17 * 41 which is composite with six prime factors.

Dumitru Damian, Table of n, a(n) for n = 1..10456

Subsequence of A342258.

Cf. A001222, A014613, A014614, A046306.

SequencePosition[PrimeOmega[Range[100000]],{4,5,6}][[;;,1]]+1 (* Harvey P. Dale, Jul 30 2024 *)

for(n=3,100000,if(bigomega(n-1)==4&&bigomega(n)==5&&bigomega(n+1)==6,print1(n,", "))) \\ Hugo Pfoertner, Mar 07 2021

# The following SageMath algorithm will generate all terms up to 100000
L=[]
for n in [1..100000]:
    sum1, sum2, sum3 = 0,0,0
    for f in list(factor(n)):
        sum1+=f[1]
    for f in list(factor(n+1)):
        sum2+=f[1]
    for f in list(factor(n+2)):
        sum3+=f[1]
    if sum1==4 and sum2==5:
        if sum3==6:
            L.append(n+1)
print(L)

A363391 Numbers k such that k, k+1, k+2, k+3 have 2, 3, 4, 5 prime factors respectively, counted with multiplicity.

Original entry on oeis.org

493, 2413, 3013, 3427, 3873, 4333, 4885, 5029, 5893, 6697, 7373, 8373, 10113, 10533, 13011, 14005, 14677, 15122, 16373, 17173, 17869, 18613, 19693, 20053, 20613, 22417, 23073, 23077, 23137, 23573, 24493, 24613, 24937, 25141, 26101, 26193, 26917, 27637, 27973, 28357, 29713, 29941, 31861, 32393

Offset: 1

Zak Seidov and Robert Israel, Jun 23 2023

nonn

Numbers k such that A001222(k+j) = 2+j for j = 0,1,2,3.
The first k in the sequence such that A001222(k+4) = 6 is a(232) = 153221.
The first k in the sequence such that A001222(k+4) = 6 and A001222(k+5) = 7 is a(4716) = 2940571.

			a(3) = 3013 is a term because 3013 = 23 * 131 has 2 prime factors counted by multiplicity, 3014 = 2 * 11 * 137 has 3, 3015 = 3^2 * 5 * 67 has 4, and 3016 = 2^3 * 13 * 29 has 5.

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

Cf. A001222, A001358, A014612, A014613, A014614, A112998.

R:= NULL: state:= 0: count:= 0:
for x from 1 while count < 50 do
  v:= numtheory:-bigomega(x);
  if v = 2 then state:= 2
  elif v = state+1 and state >= 2 then state:=state+1
  else state:= 0
  fi;
  if state = 5 then count:= count+1; R:= R,x-3;
  fi;
od:
R;

A248165 a(1)=252; for n>=1, a(n+1) is the smallest palindromic 5-almost prime with a(n) as a central substring.

Original entry on oeis.org

252, 92529, 189252981, 1218925298121, 212189252981212, 12121892529812121, 8121218925298121218, 781212189252981212187, 1678121218925298121218761, 216781212189252981212187612, 22167812121892529812121876122, 2221678121218925298121218761222

Offset: 1

Michel Lagneau, Dec 01 2014

The 5-almost primes are the numbers that are the product of exactly five (not necessarily distinct).

			a(1) = 252 = 2*2*3*3*7;
a(2) = 92529 = 3*3*3*23*149.

Cf. A014614, A247483, A247484, A248047.

d[n_]:= IntegerDigits[n]; t = {x = 252}; Do[i = 1; While[!PrimeOmega[y = FromDigits[Flatten[{z = d[i], d[x], Reverse[z]}]]]==5, i++]; AppendTo[t, x = y], {n, 13}]; t

cp's OEIS Frontend

A114966 Prime(n) + Semiprime(n) + 3AlmostPrime(n) + 4AlmostPrime(n) + 5AlmostPrime(n).

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A131175 Table, read by rows, of coefficients of characteristic polynomials of almost prime matrices.

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A156620 Primes p such that p^2 - 2 is a 5-almost prime.

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PARI

A156621 5-almost primes of the form p^2 - 2, where p is prime.

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PARI

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

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PARI

A288517 Least integer k such that A001358(k) + A001358(k+1) is the product of exactly n prime factors (counting multiplicity).

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Extensions

A321169 a(n) is the smallest prime p such that p + 2 is a product of n primes (counted with multiplicity).

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PARI

PARI

A342246 Numbers k such that k-1, k and k+1 are all composite with four, five and six (not necessarily distinct) prime factors respectively.

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