A180150 -id:A180150 - cp's OEIS frontend

A180151 Numbers k such that k and k + 2 are both divisible by exactly five primes (counted with multiplicity).

270, 592, 700, 750, 918, 1168, 1240, 1638, 1648, 1672, 1710, 1750, 2070, 2310, 2392, 2548, 2550, 2608, 2728, 2860, 2862, 2896, 2898, 3184, 3330, 3568, 3630, 3822, 3848, 3850, 3942, 3976, 4230, 4264, 4648, 4662, 5070, 5080, 5236, 5238, 5390, 5550, 5560

Offset: 1

Jonathan Vos Post, Aug 12 2010

"5-almost primes" that keep that property when incremented by 2. This sequence is to 5 as 4 is to A180150, as 3 is to A180117, as A092207 is to 2, and as A001359 is to 1. That is, this sequence is the 5th row of the infinite array A[k,n] = n-th natural number m such that m and m+2 are both divisible by exactly k primes (counted with multiplicity). The first row is the lesser of twin primes. The second row is the sequence such that m and m+2 are both semiprimes.

			a(1) = 270 because 270 = 2 * 3^3 * 5 is divisible by exactly 5 primes (counted with multiplicity), and so is 270+2 = 272 = 2^4 * 17.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A001222, A001358, A001359, A014614, A033987, A101637, A180117, A180150.

f[n_] := Plus @@ (Last@# & /@ FactorInteger@n); fQ[n_] := f[n] == 5 == f[n + 2]; Select[ Range@ 10000, fQ] (* Robert G. Wilson v, Aug 15 2010 *)

for(x=2,10^4,if(bigomega(x)==5&&bigomega(x+2)==5,print1(x", "))) \\ Zak Seidov, Aug 12 2010

{m in A014614 and m+2 in A014614} = {m such that bigomega(m) = bigomega(m+2) = 5} = {m such that A001222(m) = A001222(m+2) = 5}.

Corrected and extended by Zak Seidov and R. J. Mathar, Aug 12 2010

A190898 Least odd prime p>n^2 with (n/p) = 1, where ( / ) is the Legendre symbol.

Original entry on oeis.org

3, 7, 11, 17, 29, 43, 53, 71, 83, 107, 127, 157, 173, 199, 229, 257, 293, 337, 379, 401, 457, 499, 541, 577, 631, 683, 733, 787, 857, 911, 967, 1031, 1091, 1163, 1229, 1297, 1373, 1447, 1553, 1601, 1697, 1787, 1867, 1973, 2029, 2129, 2213, 2339, 2411, 2503, 2617, 2707, 2819, 2927, 3041, 3137, 3251, 3457, 3491, 3607

Offset: 1

Zhi-Wei Sun, Dec 29 2012

Conjecture: a(n)<(n+1)^2 for all n>0. (See also A185150.)

This conjecture implies that a(1),a(2),a(3),... are pairwise distinct.

			a(2)=7 since 7 is the first prime p>2^2 with (2/p) = 1.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Cf. A180150, A014085, A185636.

Do[Do[If[n^2+k>2&&PrimeQ[n^2+k]==True&&JacobiSymbol[n,n^2+k]==1,Print[n," ",n^2+k];Goto[aa]],{k,1,2n}];
Label[aa];Continue,{n,1,100}]
js[n_]:=Module[{p=NextPrime[n^2]},While[JacobiSymbol[n,p]!=1,p= NextPrime[ p]];p]; Join[{3},Array[js,60,2]] (* Harvey P. Dale, Jan 29 2023 *)

A180245 n-th natural number m such that m and m+2 are both divisible by exactly n primes (counted with multiplicity).

Original entry on oeis.org

3, 33, 42, 196, 918, 6640, 24750, 246078, 781248, 6565374, 25227774, 165009150, 673932798, 5268548608, 25737162750, 179511912448, 818179991550, 4228689854464, 26455088693248, 104384041582590, 820632501420030

Offset: 1

Jonathan Vos Post, Aug 19 2010

nonn

Main diagonal A[n,n] of A[k,n] = n-th natural number m such that m and m+2 are both divisible by exactly k primes (counted with multiplicity).
This is the main diagonal of the array mentioned in A180117, A180150, and A180151.
Row 1 = A001359 = the lesser of twin primes.
Row 2 = A092207 = Numbers n such that n and n+2 are semiprimes.
Row 3 = A180117 = m and m+2 are both divisible by exactly 3 primes (counted with multiplicity).
Row 4 = A180150 = m and m+2 are both divisible by exactly 4 primes (counted with multiplicity).
Row 5 = A180151 = m and m+2 are both divisible by exactly 5 primes (counted with multiplicity).

			a(1) = 3 because 3 is the first natural number m such that m and m+2 are both divisible by exactly 1 prime (i.e., the first of the lesser of twin primes).
a(2) = 33 because that is the 2nd natural number m such that m and m+2 are both divisible by exactly 2 primes (i.e. 33 = 3 * 11 is semiprime and when 2 is added becomes 35 = 5 * 7 which is also semiprimes) the 1st such being 4.

Cf. A001359, A092207, A180117, A180150, A180151.

Corrected and extended by Jack Brennen, D. S. McNeil and Ray Chandler, Aug 19 2010

a(16)-a(21) from Donovan Johnson, Aug 27 2010

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A180151 Numbers k such that k and k + 2 are both divisible by exactly five primes (counted with multiplicity).

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A190898 Least odd prime p>n^2 with (n/p) = 1, where ( / ) is the Legendre symbol.

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A180245 n-th natural number m such that m and m+2 are both divisible by exactly n primes (counted with multiplicity).

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