A112998 -id:A112998 - cp's OEIS frontend

A186696 a(n)=(A112998(n)-1)/12.

5, 6, 16, 23, 33, 35, 51, 55, 63, 91, 103, 121, 138, 178, 195, 216, 233, 243, 268, 348, 385, 436, 510, 511, 518, 588, 628, 675, 693, 696, 710, 723, 741, 751, 773, 810, 818, 863, 913, 930, 973, 975, 1025, 1035, 1036, 1045, 1070, 1071, 1115, 1161, 1260, 1273, 1311, 1336, 1365, 1368, 1386, 1438, 1441, 1448, 1490, 1501, 1508, 1521, 1540, 1546, 1623, 1651, 1655, 1720

Offset: 1

Zak Seidov, Feb 25 2011

nonn

All terms in A112998 are congruent to 1 modulo 12, hence this sequence.

Zak Seidov, Table of n, a(n) for n = 1..1500

Cf. A112998.

A328160 Terms k of A112998 such that k+2 is nonsquarefree.

Original entry on oeis.org

61, 73, 277, 421, 2797, 6217, 8521, 9277, 9817, 10357, 11161, 12301, 12841, 13381, 15121, 17377, 17881, 18097, 19861, 25657, 30517, 30661, 33037, 35521, 36241, 36457, 48121, 50821, 51481, 54421, 56437, 58417, 60217, 66601, 66697, 67057, 71341, 74077, 77641, 79801, 88117, 94777, 96181, 98017

Offset: 1

Robert Israel, Oct 05 2019

nonn

Complement of A328137 in A112998.

Each term is either 3*x^2-2 where x, 3*x^2-2 and (3*x^2-1)/2 are prime or it is 9*x-2 where x, 9*x-2 and (9*x-1)/2 are prime.

			a(3)=277 is a term because 277 is prime, 277+1=2*139 where 139 is prime, and 279=3^2*31 is a 3-almost prime that is nonsquarefree.

Robert Israel, Table of n, a(n) for n = 1..7487

Cf. A112998, A328137.

N:= 100000:
A1:= map(x -> 3*x^2-2, select(x -> isprime(x) and isprime(3*x^2-2) and isprime((3*x^2-1)/2), {seq(i,i=3..floor(sqrt((N+2)/3)),2)})):
A2:= map(x -> 9*x-2, select(x -> isprime(x) and isprime(9*x-2) and isprime((9*x-1)/2), {seq(i,i=3..(N+2)/9,2)})):
sort(convert(A1 union A2,list));

Select[Prime@ Range[10^4], And[PrimeOmega /@ {# + 1, # + 2} == {2, 3}, ! SquareFreeQ[# + 2]] &] (* Michael De Vlieger, Oct 06 2019 *)

A072875 Smallest start for a run of n consecutive numbers of which the i-th has exactly i prime factors.

Original entry on oeis.org

2, 3, 61, 193, 15121, 838561, 807905281, 19896463921, 3059220303001, 3931520917431241

Offset: 1

Rick L. Shepherd, Jun 30 2002 and Jens Kruse Andersen, Jul 28 2002

By definition, each term of this sequence is prime.
a(11) <= 1452591346605212407096281241 (Frederick Schneider), see primepuzzles link. - sent by amd64(AT)vipmail.hu, Dec 21 2007
Prime factors are counted with multiplicity. - Harvey P. Dale, Mar 09 2021

			a(3)=61 because 61 (prime), 62 (=2*31), 63 (=3*3*7) have exactly 1, 2, 3 prime factors respectively, and this is the smallest solution;
a(6)=807905281: 807905281 is prime; 807905281+1=2*403952641;
807905281+2=3*15733*17117; 807905281+3=2*2*1871*107951;
807905281+4=5*11*43*211*1619; 807905281+5=2*3*3*3*37*404357;
807905281+6=7*7*7*7*29*41*283; 807905281 is the smallest number m such that m+k is product of k+1 primes for k=0,1,2,3,4,5,6.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 61, p. 22, Ellipses, Paris 2008.

alt.math.recreational thread, Consecutive numbers with counting prime factors.
Carlos Rivera, Puzzle 425. Consecutive numbers, increasing quantity of prime factors, The Prime Puzzles & Problems Connection.

Cf. A001222, A093552, A093550, A086560, A124592.

a(1) = A000040(1), a(2) = A005383(1), a(3) = A112998(1), a(4) = A113000(1), a(5) = A113008(1), a(6) = A113150(1).

(* This program is not suitable to compute a large number of terms. *) nmax = 6; kmax = 10^6; a[1] = 2; a[n_] := a[n] = For[k = a[n-1]+n-1, k <= kmax, k++, If[AllTrue[Range[0, n-1], PrimeOmega[k+#] == #+1&], Return[k] ] ]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, nmax}] (* Jean-François Alcover, Sep 06 2017 *)

a(7) found by Mark W. Lewis
a(8) and a(9) found by Jens Kruse Andersen
a(10) found by Jens Kruse Andersen; probably a(11) > 10^20. - Aug 24 2002
Entry revised by N. J. A. Sloane, Jan 26 2007
Cross-references and editing by Charles R Greathouse IV, Apr 20 2010

A113000 Numbers k such that k, k+1, k+2 and k+3 are 1,2,3,4-almost primes.

Original entry on oeis.org

193, 421, 661, 1093, 1657, 2137, 2341, 2593, 6217, 7057, 8101, 9817, 12421, 12853, 15121, 16033, 16417, 17257, 17881, 19813, 20641, 21817, 25033, 25657, 27337, 28921, 30661, 31081, 31321, 31333, 32377, 35521, 36457, 38281, 40693, 45553

Offset: 1

Zak Seidov, Jan 03 2006

Subsequence of A112998: a(1) = 193 = A112998(3), a(2) = 421= A112998(6), a(3) = 661 = A112998(8). - Zak Seidov, Oct 14 2012

			193 is prime, 194=2*97 is semiprime, 195=3*5*13 is 3-almost prime, 196=2*2*7*7 is 4-almost prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Cf. A112998.

[n: n in PrimesUpTo(5*10^4) | forall{k: k in [1..3] | &+[f[j, 2]: j in [1..#f]] eq k+1 where f is Factorization(n+k)}]; // Vincenzo Librandi, Sep 24 2012

Do[p=Prime[n];If[Table[Total[FactorInteger[p+k]][[2]], {k, 3}]=={2, 3, 4}, Print[p]], {n, 1, 10000}]

list(lim)=my(v=List(), L=(lim+2)\3, t); forprime(p=3, L\3, forprime(q=3, min(L\p, p), t=3*p*q-2; if(t%12==1 && isprime(t) && isprime((t+1)/2) && bigomega(t+3)==4, listput(v, t)))); Set(v) \\ Charles R Greathouse IV, Feb 01 2017

Edited by Charles R Greathouse IV, Apr 20 2010

A113008 Numbers k such that k, k+1, k+2, k+3 and k+4 are respectively 1,2,3,4,5-almost primes.

Original entry on oeis.org

15121, 35521, 52321, 117841, 235441, 313561, 398821, 516421, 520021, 531121, 570601, 623641, 761113, 838561, 941041, 1117321, 1190821, 1317361, 1333621, 1336177, 1372081, 1413793, 1424041, 1431361, 1488901, 1513921, 1560121

Offset: 1

Zak Seidov, Jan 03 2006

All listed terms are congruent to 1 modulo 12.

			15121 is prime (or 1-almost prime), 15122=2*7561 is semiprime (or 2-almost prime), 15123=3*71*71 is 3-almost prime, 15124=2*2*29*199 is 4-almost prime, 15125=5*5*5*11*11 is 5-almost prime.

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

Cf. A112998, A113000.

[n: n in PrimesUpTo(2*10^6) | forall{k: k in [1..4] | &+[f[j, 2]: j in [1..#f]] eq k+1 where f is Factorization(n+k)}]; // Vincenzo Librandi, Sep 24 2012

f[n_] := Plus @@ Last /@ FactorInteger@n; t = {}; Do[p = Prime[n]; If[Array[ f[p + # ] &, 4] == {2, 3, 4, 5}, AppendTo[t, p]], {n, 126483}]; t (* Robert G. Wilson v *)
aprQ[p_]:=Total[FactorInteger[#][[All,2]]]&/@Range[p+1,p+4]=={2,3,4,5}; Select[ Prime[ Range[120000]],aprQ] (* Harvey P. Dale, Dec 17 2022 *)

list(lim)=my(v=List(), L=(lim+2)\3, t); forprime(p=3, L\3, forprime(q=3, min(L\p, p), t=3*p*q-2; if(t%12==1 && isprime(t) && isprime((t+1)/2) && bigomega(t+3)==4 && bigomega(t+4)==5, listput(v, t)))); Set(v) \\ Charles R Greathouse IV, Feb 05 2017

More terms from Robert G. Wilson v, Jan 05 2006

A113150 Primes p such that p+1, p+2, p+3, p+4, p+5 are resp. 2-, 3-, 4-, 5-, 6-almost primes.

Original entry on oeis.org

838561, 1190821, 2116921, 3318421, 3456721, 3720361, 3776881, 4185121, 5712241, 5811241, 6455521, 6457621, 6793321, 7450501, 7981801, 8321881, 8391001, 9903721, 11420041, 11980921, 12806041, 13311301, 13748521, 14326021, 14566261

Offset: 1

Zak Seidov, Jan 04 2006

nonn

All terms == 1 (mod 12).

Zak Seidov, Table of n, a(n) for n = 1..2000
Eric Weisstein's World of Mathematics, Almost Prime.

Cf. A005383, A112998, A113000, A113008.

[ n: n in PrimesUpTo(15000000) | forall{ k: k in [1..5] | &+[ f[j, 2]: j in [1..#f] ] eq k+1 where f is Factorization(n+k) } ]; // Klaus Brockhaus, Jan 24 2011

Select[Prime[Range[10^6]], PrimeOmega[#+1]==2 && PrimeOmega[#+2]==3 && PrimeOmega[#+3]==4 && PrimeOmega[#+4]==5 && PrimeOmega[#+5]==6&] (* James C. McMahon, Jun 16 2024 *)

isok(n) = bigomega(n)==1 && bigomega(n+1)==2 && bigomega(n+2)==3 && bigomega(n+3)==4 && bigomega(n+4)==5 && bigomega(n+5)==6; \\ Michel Marcus, Oct 23 2014

Edited by Charles R Greathouse IV, Apr 20 2010

A201147 Numbers m such that m, m-1 and m-2 are 1,2,3-almost primes respectively.

Original entry on oeis.org

47, 107, 167, 263, 347, 359, 467, 479, 563, 863, 887, 983, 1019, 1187, 1283, 1907, 2039, 2063, 2099, 2447, 2819, 2879, 3023, 3167, 3203, 3623, 3803, 3947, 4139, 4919, 5387, 5399, 5507, 5879, 6599, 6659, 6983, 7079, 7187, 7523, 7559, 7703, 8423, 8699, 8963

Offset: 1

Antonio Roldán, Nov 27 2011

nonn

m-2 is multiple of 3.
m is  of the form 12k-1.
This sequence is subset of A005385.
Following a suggestion of Claudio Meller.

			2099 is prime, 2098=2*1049 is semiprime, 2097=3*3*233 is 3-almost prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..8400

Cf. A005385, A005383, A112998, A113000, A113008, A072875, A093552.

primeCount[n_] := Plus @@ Transpose[FactorInteger[n]][[2]]; Select[Range[10000], primeCount[#] == 1 && primeCount[#-1] == 2 && primeCount[#-2] == 3 &] (* T. D. Noe, Nov 28 2011 *)
Select[Range[10000],PrimeOmega[Range[#,#+2]]=={3,2,1}&]+2 (* Harvey P. Dale, Dec 10 2011 *)

list(lim)=my(v=List(),L=(lim-2)\3,t); forprime(p=3,L\3, forprime(q=3,min(p,L\p), t=3*p*q+2; if(isprime(t) && isprime((t-1)/2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 01 2017

A328137 Primes p such that p+1 is the product of two distinct primes and p+2 is the product of three distinct primes.

Original entry on oeis.org

193, 397, 613, 661, 757, 1093, 1237, 1453, 1657, 2137, 2341, 2593, 2917, 3217, 4177, 4621, 5233, 6121, 6133, 7057, 7537, 8101, 8317, 8353, 8677, 8893, 9013, 9721, 10957, 11677, 11701, 12421, 12433, 12541, 12853, 13933, 15277, 15733, 16033, 16381, 16417, 16633, 17257, 17293, 18013, 18253, 18481

Offset: 1

J. M. Bergot and Robert Israel, Oct 04 2019

nonn

All terms == 1 (mod 12).

Members k of A112998 such that k+2 is squarefree.

			a(3)=613 is in the sequence because 613 is prime, 614=2*307 is the product of two distinct primes, and 615=3*5*41 is the product of three distinct primes.

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

Contained in A005383, A100363 and A112998.

[p:p in PrimesUpTo(20000)| IsPrime((p+1) div 2) and IsSquarefree(p+2) and #PrimeDivisors(p+2) eq 3]; // Marius A. Burtea, Oct 04 2019

select(t -> isprime(t) and isprime((t+1)/2) and numtheory:-issqrfree(t+2) and numtheory:-bigomega(t+2)=3, [seq(i,i=1..10^5,12)]);

A255092 Least prime p such that p+n is product of (n+1) primes (with multiplicity).

Original entry on oeis.org

2, 3, 43, 13, 239, 59, 171869, 569, 32797, 2551, 649529, 6133, 1708984363, 57331, 103630981, 65521, 301327031, 262127, 82244873046857, 11943917, 38354628391, 26214379, 679922958173, 37748713, 584125518798828101, 553648103, 7625597484961, 2281701349, 882592301503097, 8153726947

Offset: 0

Zak Seidov, Feb 14 2015

For n>0, terms with odd indices 3, 13, 59, 569... are much smaller than neighbor terms with even indices.

For n > 0, a(n) >= A053669(n)^(n+1) - n. - Robert Israel, Sep 25 2024

			2+0=2(prime), 3+1=4=2*2, 43+2=45=3*3*5, 13+3=16=2^4, 239+4=243=3^5,59+5=64=2^6,171869+6=171875=5^6*11,569+7=574=2^6*3^2,
32797+8=32805=3^5*5, 2551+9=2590=2^9*5, 649529+10=649539=3^10*11, 6133+11=6143=2^11*3.

Robert Israel, Table of n, a(n) for n = 0..1000

Cf. A053669, A072875, A112998, A113000, A113008.

f:= proc(n)
    uses priqueue;
      local pq, t, v, p,w,i;
      initialize(pq);
      p:= 2;
      while n mod p = 0 do p:= nextprime(p) od;
      insert([-p^(n+1),[p$(n+1)]],pq);
      do
        t:= extract(pq);
        v:= -t[1]; w:= t[2];
        if isprime(v-n) then return v-n fi;
        p:= nextprime(w[-1]);
      while n mod p = 0 do p:= nextprime(p) od:
       for i from n+1 to 1 by -1 while w[i] = w[n+1] do
        insert([t[1]*(p/w[n+1])^(n+2-i),[op(w[1..i-1]),p$(n+2-i)]],pq);
     od od
end proc:
f(0):= 2:
map(f, [$0..40]); # Robert Israel, Sep 25 2024

More terms from Robert Israel, Sep 25 2024

A201220 Numbers m such that m, m-1, m-2 and m-3 are 1,2,3,4-almost primes respectively.

Original entry on oeis.org

107, 263, 347, 479, 863, 887, 1019, 2063, 2447, 3023, 3167, 3623, 5387, 5399, 5879, 6599, 6983, 7079, 8423, 8699, 9743, 9887, 10463, 11807, 12263, 12347, 14207, 15383, 15767, 18959, 20663, 22343, 23039, 23567, 24239, 27239, 32183, 33647, 33767, 37799

Offset: 1

Antonio Roldán, Nov 28 2011

nonn

Following a suggestion of Claudio Meller.

m is of the form 12k-1, so m-2 is a multiple of 3 and m-3 is a multiple of 4.

			6599 is prime, 6598=2*3299 is semiprime, 6597=3*3*733 is 3-almost prime, 6596=2*2*17*97 is 4-almost prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1700

Subsequence of A005385 and of A201147.

Cf. A005383, A112998, A113000, A113008, A072875, A093552.

primeCount[n_] := Plus @@ Transpose[FactorInteger[n]][[2]]; Select[Range[40000], primeCount[#] == 1 && primeCount[#-1] == 2 && primeCount[#-2] == 3 && primeCount[#-3] == 4 &] (* T. D. Noe, Nov 28 2011 *)
Select[Range[40000],PrimeOmega[Range[#,#+3]]=={4,3,2,1}&]+3 (* Harvey P. Dale, Dec 10 2011 *)
SequencePosition[PrimeOmega[Range[40000]],{4,3,2,1}][[;;,2]] (* Harvey P. Dale, Oct 08 2023 *)

list(lim)=my(v=List(), L=(lim-2)\3, t); forprime(p=3, L\3, forprime(q=3, min(p, L\p), t=3*p*q+2; if(isprime(t) && isprime((t-1)/2) && bigomega(t-3)==4, listput(v, t)))); Set(v) \\ Charles R Greathouse IV, Feb 02 2017

cp's OEIS Frontend

A186696 a(n)=(A112998(n)-1)/12.

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A328160 Terms k of A112998 such that k+2 is nonsquarefree.

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Maple

Mathematica

A072875 Smallest start for a run of n consecutive numbers of which the i-th has exactly i prime factors.

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Mathematica

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A113000 Numbers k such that k, k+1, k+2 and k+3 are 1,2,3,4-almost primes.

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Magma

Mathematica

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A113008 Numbers k such that k, k+1, k+2, k+3 and k+4 are respectively 1,2,3,4,5-almost primes.

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Magma

Mathematica

PARI

Extensions

A113150 Primes p such that p+1, p+2, p+3, p+4, p+5 are resp. 2-, 3-, 4-, 5-, 6-almost primes.

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Author

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Programs

Magma

Mathematica

PARI

Extensions

A201147 Numbers m such that m, m-1 and m-2 are 1,2,3-almost primes respectively.

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Mathematica

PARI