A014613 -id:A014613 - cp's OEIS frontend

A110226 1 + sum of first n 4-almost primes.

1, 17, 41, 77, 117, 171, 227, 287, 368, 452, 540, 630, 730, 834, 960, 1092, 1227, 1363, 1503, 1653, 1805, 1961, 2145, 2334, 2530, 2728, 2932, 3142, 3362, 3587, 3815, 4047, 4281, 4529, 4779, 5039, 5315, 5609, 5905, 6202, 6508, 6816, 7131, 7459, 7789, 8129

Offset: 0

Jonathan Vos Post, Sep 06 2005

First differences are the sequence of 4-almost primes (A014613). Hence a(n) is the least positive sequence whose first differences are the sequence of 4-almost primes. Primes in this sequence include: a(1) = 17, a(2) = 41, a(6) = 227, a(35) = 5039, a(43) = 7459, a(44) = 7789. Semiprimes in this sequence include: a(3) = 77 = 7 * 11, a(7) = 287 = 7 * 41, a(16) = 1227 = 3 * 409, a(17) = 1363 = 29 * 47, a(21) = 1961 = 37 * 53, a(27) = 3142 = 2 * 1571, a(29) = 3587 = 17 * 211, a(32) = 4281 = 3 * 1427, a(33) = 4529 = 7 * 647, a(36) = 5315 = 5 * 1063, a(37) = 5609 = 71 * 79, a(38) = 5905 = 5 * 1181, a(42) = 7131 = 3 * 2733a(45) = 8129 = 11 * 739. 3-almost primes in this sequence include: a(4) = 117 = 3^2 * 13, a(5) = 171 = 3^2 * 19, a(9) = 452 = 2^2 * 113, a(12) = 730 = 2 * 5 * 73, a(13) = 833 = 2 * 3 * 139, a(18) = 3^2 * 167, a(19) = 1653 = 3 * 19 * 29, a(20) = 1805 = 5 * 19^2, a(23) = 2534 = 2 * 3 * 389, a(26) = 2932 = 2^2 * 733, a(28) = 3362 = 2 * 41^2, a(30) = 3815 = 5 * 7 * 109, a(31) = 4047 = 3 * 19 * 71, a(39) = 2 * 7 * 443, a(40) = 6508 = 2^2 * 1627. 4-almost primes in this sequence include: a(22) = 2145 = 3 * 5 * 11 * 13, a(24) = 2530 = 2 * 5 * 11 * 23.

Cf. A014613, A086046, A110208, A110209.

Accumulate[Join[{1},Select[Range[500],PrimeOmega[#]==4&]]] (* Harvey P. Dale, Dec 13 2018 *)

a(0) = 1; for n>0, a(n) = 1 + A086046(n).

A114618 Numbers k such that the k-th octagonal number is 4-almost prime.

Original entry on oeis.org

4, 9, 27, 39, 49, 57, 59, 69, 75, 85, 87, 105, 109, 117, 119, 121, 125, 143, 147, 153, 161, 169, 175, 177, 185, 187, 199, 207, 217, 219, 231, 235, 239, 245, 249, 265, 267, 269, 275, 283, 285, 289, 291, 299, 301, 305, 311, 319, 321, 327, 329, 333, 335, 345, 349, 357, 359, 361, 363, 371, 381, 385

Offset: 1

Jonathan Vos Post, Feb 17 2006

It is necessary but not sufficient that k must be prime (A000040), semiprime (A001358), or 3-almost prime (A014612).

			a(1) = 4 because OctagonalNumber(4) = Oct(4) = 4*(3*4-2) = 40 = 2^3 * 5 has exactly 4 prime factors (3 are all equally 2; factors need not be distinct).
a(2) = 9 because Oct(9) = 9*(3*9-2) = 225 = 3^2 * 5^2, a 4-almost prime [225 is also a square, hence a square octagonal number A036428, as is Oct(121)].
a(3) = 27 because Oct(27) = 27*(3*27-2) = 2133 = 3^3 * 79.
a(4) = 39 because Oct(39) = 39*(3*39-2) = 4485 = 3 * 5 * 13 * 23 has exactly 4 prime factors, in this case distinct.
a(26) = 187 because Oct(187) = 187*(3*187-2) = 104533 = 11 * 13 * 17 * 43 [a 4-brilliant number, that is with 4 prime factors that are each the same number of digits in length].

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Octagonal Number.

Cf. A000040, A000567, A001222, A001358, A014612, A014613, A036428.

Select[Range[400],PrimeOmega[#(3#-2)]==4&] (* Harvey P. Dale, Sep 07 2011 *)

Numbers k such that k*(3*k-2) has exactly four prime factors (with multiplicity).
Numbers k such that A000567(k) is a term of A014613.
Numbers k such that A001222(A000567(k)) = 4.
Numbers k such that A001222(k) + A001222(3*k-2) = 4.
Numbers k such that [(3*k-2)*(3*k-1)*(3*k)]/[(3*k-2)+(3*k-1)+(3*k)] is a term of A014613.

265 inserted by R. J. Mathar, Dec 22 2010

A114621 Numbers k such that the k-th octagonal number is 5-almost prime.

Original entry on oeis.org

8, 10, 12, 20, 26, 28, 45, 58, 63, 68, 76, 81, 82, 92, 99, 106, 115, 116, 129, 146, 159, 165, 171, 172, 188, 195, 202, 212, 213, 218, 225, 236, 255, 259, 261, 268, 273, 279, 298, 309, 325, 339, 343, 351, 362, 375, 387, 395, 399

Offset: 1

Jonathan Vos Post, Feb 17 2006

It is necessary but not sufficient that k must be prime (A000040), semiprime (A001358), 3-almost prime (A014612), or 4-almost prime (A014613).

			a(1) = 8 because OctagonalNumber(8) = Oct(8) = 8*(3*8-2) = 176 = 2^4 * 11 has exactly 5 prime factors (four are all equally 2; factors need not be distinct). Also, 176 = Oct(8) = Oct(Oct(2)), an iterated octagonal number. Also, 176 is a pentagonal number, hence a term of A046189 octagonal pentagonal numbers.
a(2) = 10 because Oct(10) = 10*(3*10-2) = 280 = 2^3 * 5 * 7 is 5-almost prime.
a(4) = 20 because Oct(20) = 20*(3*20-2) = 1160 = 2^3 * 5 * 29.
a(5) = 26 because Oct(26) = 26*(3*26-2) = 1976 = 2^3 * 13 * 19.
a(19) = 129 because Oct(129) = 129*(3*129-2) = 49665 = 3 * 5 * 7 * 11 * 43 is 5-almost prime (in this case, the 5 prime factors are distinct).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Octagonal Number.

Cf. A000040, A000567, A001222, A001358, A014612, A014613, A014614.

Select[Range[500], PrimeOmega[PolygonalNumber[8, #]] == 5 &] (* Amiram Eldar, Oct 07 2024 *)

Numbers k such that k*(3*k-2) has exactly five prime factors (with multiplicity).
Numbers k such that A000567(k) is a term of A014614.
Numbers k such that A001222(A000567(k)) = 5.
Numbers k such that A001222(k) + A001222(3*k-2) = 5.
Numbers k such that [(3*k-2)*(3*k-1)*(3*k)]/[(3*k-2)+(3*k-1)+(3*k)] is a term of A014614.

12, 63, 99 inserted and 117 removed by R. J. Mathar, Dec 22 2010

A114634 Numbers k such that the k-th octagonal number is 6-almost prime.

Original entry on oeis.org

6, 14, 16, 18, 34, 36, 40, 42, 44, 46, 50, 52, 56, 60, 62, 74, 88, 98, 100, 122, 124, 130, 132, 135, 138, 142, 148, 152, 156, 158, 170, 178, 186, 189, 194, 196, 209, 226, 232, 242, 243, 244, 258, 260, 266, 274, 282, 292, 296, 297, 302, 308, 314, 315, 316, 322

Offset: 1

Jonathan Vos Post, Feb 17 2006

It is necessary but not sufficient that k must be prime (A000040), semiprime (A001358), 3-almost prime (A014612), 4-almost prime (A014613), or 5-almost prime (A014614).

			a(1) = 6 because OctagonalNumber(6) = Oct(6) = 6*(3*6-2) = 96 = 2^5 * 3 has exactly 6 prime factors (five are all equally 2; factors need not be distinct).
a(2) = 14 because Oct(14) = 14*(3*14-2) = 560 = 2^4 * 5 * 7 is 6-almost prime.
a(3) = 16 because Oct(16) = 16*(3*16-2) = 736 = 2^5 * 23.
a(7) = 40 because Oct(40) = 40*(3*40-2) = 4720 = 2^4 * 5 * 59 [also, 4720 = Oct(40) = Oct(Oct(4)), an iterated octagonal number].
a(19) = 100 because Oct(100) = 100*(3*100-2) = 29800 = 2^3 * 5^2 * 149.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Octagonal Number.

Cf. A000040, A000567, A001222, A001358, A014612, A014613, A014614, A046306.

Flatten[Position[Table[n(3n-2),{n,400}],?(PrimeOmega[#]==6&)]] (* _Harvey P. Dale, Jun 17 2013 *)
Select[Range[400],PrimeOmega[PolygonalNumber[8,#]]==6&] (* Harvey P. Dale, Feb 23 2022 *)

is(n)=my(t=bigomega(3*n-2)); t<6 && (t<5 || !isprime(n)) && t+bigomega(n)==6 \\ Charles R Greathouse IV, Feb 01 2017

Numbers k such that k*(3*k-2) has exactly six prime factors (with multiplicity).
Numbers k such that A000567(k) is a term of A046306.
Numbers k such that A001222(A000567(k)) = 6.
Numbers k such that A001222(k) + A001222(3*k-2) = 6.
Numbers k such that [(3*k-2)*(3*k-1)*(3*k)]/[(3*k-2)+(3*k-1)+(3*k)] is a term of A046306.

A114635 Numbers k such that the k-th octagonal number is 7-almost prime.

Original entry on oeis.org

24, 30, 32, 38, 48, 66, 72, 78, 90, 94, 104, 110, 112, 114, 120, 136, 140, 154, 164, 166, 168, 176, 180, 190, 204, 206, 208, 210, 220, 222, 228, 238, 248, 254, 276, 280, 284, 286, 290, 300, 306, 312, 326, 338, 344

Offset: 1

Jonathan Vos Post, Feb 18 2006

It is necessary but not sufficient that k must be prime (A000040), semiprime (A001358), 3-almost prime (A014612), 4-almost prime (A014613), 5-almost prime (A014614), or 6-almost prime (A046308).

			a(1) = 24 because OctagonalNumber(24) = Oct(24) = 24*(3*24-2) = 96 = 1680 = 2^4 * 3 * 5 * 7 has exactly 7 prime factors (four are all equally 2; factors need not be distinct).
a(2) = 30 because Oct(30) = 30*(3*30-2) = 2640 = 2^4 * 3 * 5 * 11 is 7-almost prime.
a(3) = 32 because Oct(32) = 32*(3*32-2) = 3008 = 2^6 * 47 is 7-almost prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Octagonal Number.

Cf. A000040, A000567, A001222, A001358, A014612, A014613, A014614, A046306, A046308.

Select[Range[400],PrimeOmega[PolygonalNumber[8,#]]==7&] (* Harvey P. Dale, Aug 13 2021 *)

Numbers k such that k*(3*k-2) has exactly seven prime factors (with multiplicity).
Numbers k such that A000567(k) is a term of A046308.
Numbers k such that A001222(A000567(k)) = 7.
Numbers k such that A001222(k) + A001222(3*k-2) = 7.
Numbers k such that [(3*k-2)*(3*k-1)*(3*k)]/[(3*k-2)+(3*k-1)+(3*k)] is a term of A046308.

A114636 Numbers k such that the k-th octagonal number is 8-almost prime.

Original entry on oeis.org

22, 70, 80, 84, 102, 108, 118, 126, 134, 160, 174, 184, 200, 230, 240, 250, 252, 262, 264, 272, 318, 330, 334, 336, 350, 368, 378, 400, 408, 420, 430, 434, 444, 450, 454, 459, 462, 464, 484, 494, 500, 502, 510, 518, 520, 522, 540, 560, 564, 566, 570, 574, 582

Offset: 1

Jonathan Vos Post, Feb 18 2006

It is necessary but not sufficient that k must be prime (A000040), semiprime (A001358), 3-almost prime (A014612), 4-almost prime (A014613), 5-almost prime (A014614), 6-almost prime (A046306), or 7-almost prime (A046308).

			a(1) = 22 because OctagonalNumber(22) = Oct(22) = 22*(3*22-2) = 1408 = 2^7 * 11 has exactly 8 prime factors (seven are all equally 2; factors need not be distinct).
a(2) = 70 because Oct(70) = 70*(3*70-2) = 14560 = 2^5 * 5 * 7 * 13 is 8-almost prime.
a(3) = 80 because Oct(80) = 80*(3*80-2) = 19040 = 2^5 * 5 * 7 * 17.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2000 from Harvey P. Dale)
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Octagonal Number.

Cf. A000040, A000567, A001222, A001358, A014612, A014613, A014614, A046306, A046308, A046310.

Select[Range[400],PrimeOmega[PolygonalNumber[8,#]]==8&] (* Harvey P. Dale, Aug 31 2020 *)

Numbers k such that k*(3*k-2) has exactly eight prime factors (with multiplicity).
Numbers k such that A000567(k) is a term of A046310.
Numbers k such that A001222(A000567(k)) = 8.
Numbers k such that A001222(k) + A001222(3*k-2) = 8.
Numbers k such that [(3*k-2)*(3*k-1)*(3*k)]/[(3*k-2)+(3*k-1)+(3*k)] is a term of A046310.

A114989 Numbers whose sum of squares of distinct prime factors is prime.

Original entry on oeis.org

6, 10, 12, 14, 18, 20, 24, 26, 28, 34, 36, 40, 48, 50, 52, 54, 56, 68, 72, 74, 80, 94, 96, 98, 100, 104, 105, 108, 112, 134, 136, 144, 146, 148, 160, 162, 188, 192, 194, 196, 200, 206, 208, 216, 224, 231, 250, 268, 272, 273, 274, 288, 292, 296, 315, 320, 324, 326

Offset: 1

Jonathan Vos Post, Feb 22 2006

A005063 is "sum of squares of primes dividing n." Hence this is the sum of squares of prime factors analog of A114522 "numbers n such that sum of distinct prime divisors of n is prime." Note the distinction between A005063 and A067666 is "sum of squares of prime factors of n (counted with multiplicity)."

			a(1) = 6 because 6 = 2 * 3 and 2^2 + 3^2 = 13 is prime.
a(2) = 10 because 10 = 2 * 5 and 2^2 + 5^2 = 29 is prime.
a(3) = 12 because 12 = 2^2 * 3 and 2^2 + 3^2 = 13 is prime (note that we are not counting the prime factors with multiplicity).
a(4) = 14 because 14 = 2 * 7 and 2^2 + 7^2 = 53 is prime.
a(8) = 26 because 26 = 2 * 3 and 2^2 + 13^2 = 173 is prime.
a(10) = 34 because 34 = 2 * 17 and 2^2 + 17^2 = 293 is prime.

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

Cf. A005063, A067666, A014612, A014613, A069273, A069279, A069281, A114522.

with(numtheory): a:=proc(n) local DPF: DPF:=factorset(n): if isprime(sum(DPF[j]^2,j=1..nops(DPF)))=true then n else fi end: seq(a(n),n=1..400); # Emeric Deutsch, Mar 07 2006

Select[Range[400],PrimeQ[Total[Transpose[FactorInteger[#]][[1]]^2]]&] (* Harvey P. Dale, Jan 16 2016 *)

PARI
```
is(n)=isprime(norml2(factor(n)[,1]))
```

{k such that A005063(k) is prime}. {k such that A005063(k) is an element of A000040}. {k = (for distinct i, j, ... prime(i)^e_1 * prime(j)^e_2 * ...) such that (prime(i)^2 * prime(j)^2 * ...) is prime}.

More terms from Emeric Deutsch, Mar 07 2006

A120045 The (10^n)-th 4-almost prime.

Original entry on oeis.org

16, 88, 693, 5958, 54328, 511725, 4922511, 47997635, 472514554, 4683086217, 46636297326, 466032880556

Offset: 0

Robert G. Wilson v, Feb 15 2006

Cf. A006988, A014613, A114106, A114125, A120044, A120046.

FourAlmostPrimePi[n_] := Sum[ PrimePi[n/(Prime@i*Prime@j*Prime@k)] - k + 1, {i, PrimePi[n^(1/4)]}, {j, i, PrimePi[(n/Prime@i)^(1/3)]}, {k, j, PrimePi@Sqrt[n/(Prime@i*Prime@j)]}];
FourAlmostPrime[n_] := Block[{e = Floor[Log[2, n] +3], a, b}, a = 2^e; Do[b = 2^p; While[FourAlmostPrimePi[a] < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; Do[ Print@FourAlmostPrime[10^n], {n, 0, 11}]

A122093 Product of the first n 4-almost primes, divided by product of the first n primes, rounded down.

Original entry on oeis.org

8, 64, 460, 2633, 12926, 55682, 196527, 837826, 3059886, 9285173, 26956956, 72856639, 184807084, 541527736, 1520886410, 3873955950, 8929796766, 20494615529, 45883467602, 98229395430, 209914872426, 488915652233, 1113313955086, 2451792530303, 5004689907217

Offset: 1

Jonathan Vos Post, Oct 17 2006

This is to 4-almost primes as A122032 is to 3-almost primes and as A122019 is to 2-almost primes (semiprimes). Note that these can nonmonotonic (look at the graphs). What is the asymptotic value of the ratio A114426(n)/A002110(n)?

Probably it can be easily proved that a(n) = 0 for n >= 802. - Giovanni Resta, Jun 13 2016

			a(1) = floor(16/2) = floor(8) = 8.
a(2) = floor((16*24)/(2*3)) = floor(384/6) = floor(64) = 64.
a(3) = floor(13824/30) = floor(460.8) = 460.
a(4) = floor(552960/210) = floor(2633.14286) = 2633.

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A000040, A002110, A014613, A114426, A122019, A122032.

q = Select[Range[1000], PrimeOmega[#] == 4 &]; m = 1; Table[ Floor[ m *= q[[i]]/ Prime[i]], {i, Length@ q}] (* Giovanni Resta, Jun 13 2016 *)

a(n) = floor(A114426(n)/A002110(n)) = floor(Prod(i=1..n)4almostprime(i)/Prod(i=1..n)prime(i)) = floor(Prod(i=1..n)A014613(i)/Prod(i=1..n)A000040(i)) = floor(Prod(i=1..n)(A014613(i)/A000040(i))).

a(11)-a(25) from Giovanni Resta, Jun 13 2016

A138637 Products of prime quadruples.

Original entry on oeis.org

5005, 46189, 121330189, 1445140189, 463236778189, 4862973196189, 12359548828189, 18898278256189, 112254342850189, 144149198626189, 1022657400370189, 7924420639216189, 28604961973900189, 59910402098980189

Offset: 1

Jonathan Vos Post, May 14 2008

Product of numbers n, n+2, n+6 and n+8 when are all prime. Quadruplet analog of A037074. Subset of A014613.

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

Cf. A007530, A014613, A037074.

isA007530 := proc(n) local q; if isprime(n) and n>=5 then q := nextprime(n) ; if q-n = 2 then q := nextprime(q) ; if q -n = 6 then q := nextprime(q) ; RETURN( q-n = 8 ) ; else RETURN(false) ; fi ; else RETURN(false) ; ; fi ; else RETURN(false) ; ; fi ; end: A007530 := proc(n) option remember ; local a; if n = 1 then 5 ; else a := nextprime(A007530(n-1)) ; while not isA007530(a) do a := nextprime(a) ; od: RETURN(a) ; fi ; end: A138637 := proc(n) local p ; p := A007530(n) ; p*(p+2)*(p+6)*(p+8) ; end: seq(A138637(n),n=1..20) ; # R. J. Mathar, May 18 2008

a = {}; For[n = 1, n < 5000, n++, If[{Prime[n+1]-Prime[n], Prime[n+2]-Prime[n+1], Prime[n+3]-Prime[n+2]} == {2, 4, 2}, AppendTo[a, Prime[n]*Prime[n+1]*Prime[n+2]* Prime[n+3]]]]; a (* Stefan Steinerberger, May 18 2008 *)
Times@@@Select[Partition[Prime[Range[2500]],4,1],Differences[#]=={2,4,2}&] (* Harvey P. Dale, Sep 10 2018 *)

a(n) = A007530(n)*A007530(n+2)*A007530(n+6)*A007530(n+8).

More terms from Stefan Steinerberger and R. J. Mathar, May 18 2008

cp's OEIS Frontend

A110226 1 + sum of first n 4-almost primes.

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A114618 Numbers k such that the k-th octagonal number is 4-almost prime.

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A114621 Numbers k such that the k-th octagonal number is 5-almost prime.

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A114634 Numbers k such that the k-th octagonal number is 6-almost prime.

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A114635 Numbers k such that the k-th octagonal number is 7-almost prime.

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A114636 Numbers k such that the k-th octagonal number is 8-almost prime.

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A114989 Numbers whose sum of squares of distinct prime factors is prime.

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