A108278 -id:A108278 - cp's OEIS frontend

A189117 Conjectured number of pairs of consecutive perfect powers (A001597) differing by n.

1, 1, 2, 3, 1, 0, 2, 1, 3, 1, 2, 1, 3, 0, 2, 1, 5, 2, 3, 1, 1, 0, 1, 2, 1, 2, 1, 3, 0, 1, 0, 1, 1, 0, 2, 1, 1, 1, 3, 1, 1, 0, 1, 0, 1, 0, 3, 1, 2, 0, 1, 0, 2, 0, 2, 1, 1, 0, 1, 2, 1, 0, 1, 0, 3, 0, 2, 2, 1, 0, 2, 0, 2, 1, 1, 1, 1, 0, 3, 1, 1, 0, 1, 0, 1, 0, 1, 0, 3, 0, 0, 1, 1, 1, 2, 0, 2, 0, 1, 5

Offset: 1

T. D. Noe, Apr 16 2011

nonn

Only a(1) is proved. Perfect powers examined up to 10^21. This is similar to A076427, but more restrictive.
Hence, through 10^21, there is only one value in the sequence: Semiprimes which are both one more than a perfect power and one less than another perfect power. This is to perfect powers A001597 approximately as A108278 is to squares. A more exact analogy would be to the set of integers such as 30^2 = 900 since 900-1 = 899 = 29 * 31, and 900+1 = 901 = 17 * 53. A189045 INTERSECTION A189047. a(1) = 26 because 26 = 2 * 13 is semiprime, 26-1 = 25 = 5^2, and 26+1 = 27 = 3^3. - Jonathan Vos Post, Apr 16 2011
Pillai's conjecture is that a(n) is finite for all n. - Charles R Greathouse IV, Apr 30 2012

			1 = 3^2 - 2^3;
2 = 3^3 - 2^5;
3 = 2^2 - 1^2 = 2^7 - 5^3;
4 = 2^3 - 2^2 = 6^2 - 2^5 = 5^3 - 11^2.

Cf. A023056 (least k such that k and k+n are consecutive perfect powers).

Cf. A023057 (conjectured n such that a(n)=0).

nn = 10^12; pp = Join[{1}, Union[Flatten[Table[n^i, {i, 2, Log[2, nn]}, {n, 2, nn^(1/i)}]]]]; d = Select[Differences[pp], # <= 100 &]; Table[Count[d, n], {n, 100}]

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

Original entry on oeis.org

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,", ")))

A276905 Numbers k such that k^5-1 and k^5+1 are semiprimes.

Original entry on oeis.org

12, 1452, 11352, 79398, 146520, 281622, 352110, 536778, 643302, 680988, 723492, 739200, 878988, 992112, 1115268, 1189650, 1397022, 1698378, 1698510, 1728540, 1806222, 2486220, 2873178, 3031578, 3571458, 3946140, 4467012, 4983858, 5064510, 5135658, 5567562, 5753352

Offset: 1

Gary E. Davis, Sep 21 2016

nonn

Cf. A001358, A108278, A124936, A268043.

Intersection of A104238 and A261435.

upper=600000;
Select[Range[upper],
PrimeOmega[#^5 - 1] == PrimeOmega[#^5 + 1] == 2 &]

isok(n) = (bigomega(n^5-1)==2) && (bigomega(n^5+1)==2); \\ Michel Marcus, Sep 22 2016

More terms from Altug Alkan, Sep 30 2016

A373209 Numbers k such that k^2 - 1 and k^2 + 1 have 8 divisors each.

Original entry on oeis.org

68, 112, 128, 162, 200, 212, 252, 294, 318, 336, 338, 372, 448, 450, 498, 502, 542, 578, 592, 598, 612, 648, 672, 678, 708, 752, 762, 808, 812, 852, 878, 888, 938, 952, 992, 996, 1012, 1038, 1098, 1102, 1116, 1122, 1188, 1202, 1212, 1248, 1258, 1328, 1362, 1380

Offset: 1

Jon E. Schoenfield, Jun 21 2024

nonn

Among the first 10000 terms (from a(1) = 68 through a(10000) = 697578), k^2 - 1 and k^2 + 1 are each the product of three distinct primes, except for
     125 terms for which k^2 + 1 =  5^3 times a prime
       6 terms for which k^2 + 1 = 13^3 times a prime
       1 terms for which k^2 + 1 = 17^3 times a prime
       1 terms for which k^2 + 1 = 29^3 times a prime, and
       4 terms for which k^2 - 1 =  p^3 * (p^3 +/- 2) (with p = 19, 29, 37, 83, respectively).
The first term for which both k^2 - 1 and k^2 + 1 are of the form p^3 * q is k = 41457661182: k^2 - 1 = 3461^3 * 41457661183, while k^2 + 1 = 5^3 * 13749901365452077097.

			68 is a term: both 68^2 - 1 = 4623 = 3 * 23 * 67 and 68^2 + 1 = 4625 = 5^3 * 37 have 8 divisors.

Cf. A000005, A002522, A005563, A069062, A108278, A193432, A347191.

{ k : tau(k^2 - 1) = tau(k^2 + 1) = 8}, where tau() is the number of divisors function, A000005.

A279272 Numbers k such that k^7 - 1 and k^7 + 1 are semiprimes.

Original entry on oeis.org

72, 282, 9000, 13932, 19212, 22158, 49920, 65538, 72228, 78888, 144408, 169320, 201492, 201828, 218460, 234540, 270030, 296478, 325080, 355008, 365748, 411000, 448872, 461052, 484152, 504618, 555522, 558252, 586362, 622620, 674058, 981810, 1067490, 1095792

Offset: 1

Vincenzo Librandi, Dec 09 2016

nonn

Since k^7 - 1 = (k-1)*(k^6 + k^5 + k^4 + k^3 + k^2 + k + 1) and k^7 + 1 = (k+1)*(k^6 - k^5 + k^4 - k^3 + k^2 - k + 1) (and since there is no term less than 3, so k-1 must have at least one prime factor), this sequence lists the numbers k such that k-1, k+1, k^6 + k^5 + k^4 + k^3 + k^2 + k + 1, and k^6 - k^5 + k^4 - k^3 + k^2 - k + 1 are all prime. - Jon E. Schoenfield, Dec 14 2016

Cf. A105041, A108278, A261436, A268043, A276905.

IsSemiprime:=func; [n: n in [4..10000] | IsSemiprime(n^7-1)and IsSemiprime(n^7+1)];

Select[Range[100000], PrimeOmega[#^7 - 1] == PrimeOmega[#^7 + 1]== 2 &]

More terms from Jon E. Schoenfield, Dec 14 2016

A373213 Numbers k such that k^2 - 1 and k^2 + 1 have 6 divisors each.

Original entry on oeis.org

168, 1368, 97968, 10374840, 16104168, 44049768, 68674368, 100741368, 281803368, 486775968, 1177381968, 1262878368, 1336852968, 2321986968, 2404627368, 3476635368, 4374102768, 5102102040, 5142754368, 5182128168, 5385651768, 6035269968, 9218496168, 10657878168

Offset: 1

Jon E. Schoenfield, Jun 21 2024

nonn

Each term is a number of the form k = sqrt(p^2 * q + 1) such that q = p^2 - 2 and k^2 + 1 = r^2 * s, where p, q, r, and s are distinct primes.

			168 is a term: both 168^2 - 1 = 28223 = 13^2 * 167 and 168^2 + 1 = 28225 = 5^2 * 1129 have 6 divisors.

Cf. A000005, A002522, A005563, A069062, A108278, A193432, A347191, A373209.

{ k : tau(k^2 - 1) = tau(k^2 + 1) = 6}, where tau() is the number of divisors function, A000005.

cp's OEIS Frontend

A189117 Conjectured number of pairs of consecutive perfect powers (A001597) differing by n.

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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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A276905 Numbers k such that k^5-1 and k^5+1 are semiprimes.

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A373209 Numbers k such that k^2 - 1 and k^2 + 1 have 8 divisors each.

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A279272 Numbers k such that k^7 - 1 and k^7 + 1 are semiprimes.

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Magma

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A373213 Numbers k such that k^2 - 1 and k^2 + 1 have 6 divisors each.

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