A121898 -id:A121898 - cp's OEIS frontend

A276564 Perfect powers k (exponent greater than 1) such that k-1 and k+1 are both semiprime.

144, 216, 900, 1764, 2048, 3600, 10404, 11664, 39204, 97344, 213444, 248832, 272484, 360000, 656100, 685584, 1040400, 1102500, 1127844, 1633284, 2108304, 2214144, 3504384, 3802500, 4112784, 4536900, 4588164, 5475600, 7784100, 7851204, 8388608, 8820900, 9000000, 9734400

Offset: 1

Antonio Roldán, Nov 16 2016

nonn

Intersection of A001597 and A124936. - Michel Marcus, Dec 03 2016

			2048 = 2^11, and both 2047 = 23*89 and 2049 = 3*683 are semiprimes.

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

Cf. A001358, A001597, A108278, A121898, A124936, A167023, A268043, A276565.

Select[Range[10^7], And[GCD @@ FactorInteger[#][[All, 2]] > 1, Union@ # == {2} &@ Map[PrimeOmega, {# - 1, # + 1}]] &] (* Michael De Vlieger, Dec 07 2016, after Ant King at A001597 *)

for(i=2,10^7,if(ispower(i)&&bigomega(i-1)==2&&bigomega(i+1)==2,print1(i,", ")))

A276565 Oblong numbers n such that n - 1 and n + 1 are both semiprime.

Original entry on oeis.org

56, 552, 870, 1056, 1190, 1640, 1892, 2652, 4032, 5256, 5402, 6806, 8372, 9120, 9506, 9702, 10920, 11772, 12656, 12882, 15006, 15252, 15500, 16256, 16770, 17556, 18632, 23256, 24492, 27722, 29070, 30800, 33306, 33672, 34410, 36290, 40200, 40602, 44310, 45582, 46872, 49506

Offset: 1

Antonio Roldán, Nov 16 2016

nonn

Intersection of A002378 and A124936. - Michel Marcus, Nov 26 2016

			1640 is oblong (1640 = 40*41) and 1639 = 11*149, 1641 = 3*547 are both semiprime.

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

Cf. A002378, A124936.

Cf. A001597, A108278, A121898, A167023, A268043, A276564.

select(t -> numtheory:-bigomega(t+1)=2 and numtheory:-bigomega(t-1)=2, [seq(i*(i+1),i=1..1000)]); # Robert Israel, Nov 28 2016

for(i=1,250,n=i*(i+1);if(bigomega(n-1)==2&&bigomega(n+1)==2,print1(n,", ")))

A375384 Triangular numbers that are sandwiched between two squarefree semiprimes.

Original entry on oeis.org

300, 780, 2628, 3240, 3828, 5460, 13530, 18528, 19110, 22578, 25878, 31878, 32640, 37128, 49770, 56280, 64980, 72390, 73920, 78210, 103740, 105570, 115440, 137550, 159330, 161028, 277140, 288420, 316410, 335790, 370230, 386760, 416328, 472878, 541320, 664128

Offset: 1

Massimo Kofler, Aug 13 2024

nonn

All numbers in this sequence are even.
Terms such as 120 and 528 are in A121898 but are not in this sequence.
If they exist, further differences between this sequence and A121898 are > 10^18. - Hugo Pfoertner, Aug 27 2024
If they exist, further terms of A121898 not in this sequence are > 10^7779.  This is based on considering the Diophantine equations x*(x-1) = 2*(y^2-1) and x*(x-1) = 2*(y^2+1). - Robert Israel, Sep 01 2024

			300 (24th triangular number) between 299 = 13 * 23 and 301 = 7 * 43.
780 (39th triangular number) between 779 = 19 * 41 and 781 = 11 * 71.
2628 (72nd triangular number) between 2627 = 37 * 71 and 2629 = 11 * 239.

Cf. A000217, A006881.

Subsequence of A121898.

select(t -> numtheory:-bigomega(t+1)=2 and numtheory:-bigomega(t-1)=2 and numtheory:-issqrfree(t+1) and numtheory:-issqrfree(t-1), [seq(i*(i+1)/2, i=1..2000)]); # Robert Israel, Sep 02 2024

q[n_] := FactorInteger[n][[;; , 2]] == {1, 1}; Select[Accumulate[Range[1100]], And @@ q /@ (# + {-1, 1}) &] (* Amiram Eldar, Aug 13 2024 *)

issp(k) = my(f=factor(k)); (bigomega(f)==2) && issquarefree(f); \\ A006881
lista(nn) = my(list=List()); for (n=2, nn, my(k=n*(n+1)/2); if (issp(k-1) && issp(k+1), listput(list, k))); Vec(list); \\ Michel Marcus, Sep 01 2024

a(n) == 0 (mod 6). - Hugo Pfoertner, Aug 27 2024

cp's OEIS Frontend

A276564 Perfect powers k (exponent greater than 1) such that k-1 and k+1 are both semiprime.

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A276565 Oblong numbers n such that n - 1 and n + 1 are both semiprime.

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A375384 Triangular numbers that are sandwiched between two squarefree semiprimes.

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