A268043 -id:A268043 - cp's OEIS frontend

A169635 Integers m such that sigma_2(m) = sigma_2(m + 2) where sigma_2(m) is the sum of squares of divisors of m (A001157).

24, 215, 280, 1079, 947519, 1362239, 2230271, 14939999, 19720007, 32509439, 45581759, 45841247, 49436927, 78436511, 82842911, 101014631, 166828031, 225622151, 225757799, 250999559, 377129087, 554998751, 619606439, 846765431, 1204092287, 1302170687, 1710035711

Offset: 1

Michel Lagneau, Apr 04 2010

nonn

The equation sigma_2(m) = sigma_2(m + k) has infinitely many solutions where k >= 2 and k is even (J.-M. De Koninck).
From Amiram Eldar, Apr 19 2024: (Start)
De Koninck's proof is based on the assumption of Schinzel's hypothesis H. If q, r = q + 2, s = q^2 + q + 1, and p = q^2 + 3*q + 3 are all primes, then p*q is a term (the values of q+1 are the terms of A268043).
The equation sigma_2(m) = sigma_2(m + 1) has only one solution: m = 6. (End)

			For m=24, sigma_2(24) = sigma_2(26) = 850.

Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, p. 118, entry 1079.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B13, pp. 103-104.

Amiram Eldar, Table of n, a(n) for n = 1..156
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Jean-Marie De Koninck, On the solutions of sigma_2(n) = sigma_2(n + l), Ann. Univ. Sci. Budapest Sect. Comput. 21 (2002), 127-133.
Wikipedia, Schinzel's hypothesis H.

Cf. A001157, A002961, A007373, A175199, A268043.

with(numtheory):for n from 1 to 500000000 do:liste:= divisors(n) : s2 :=sum(liste[i]^2, i=1..nops(liste)):liste:=divisors(n+2):s3:=sum(liste[i]^2, i=1..nops(liste)):if s2 = s3 then print(n):else fi:od:

Select[Range[10^9], DivisorSigma[2,#] == DivisorSigma[2,#+2]&]

is(n) = sigma(n, 2) == sigma(n + 2, 2); \\ Amiram Eldar, Apr 19 2024

lista(mmax) = {my(s1 = sigma(1, 2), s2 = sigma(2, 2), s3, s4); forstep(m = 3, mmax, 2, s3 = sigma(m, 2); s4 = sigma(m+1, 2); if(s1 == s3, print1(m - 2, ", ")); if(s2 == s4, print1(m - 1, ", ")); s1 = s3; s2 = s4);} \\ Amiram Eldar, Apr 19 2024

a(25)-a(27) from Donovan Johnson, Apr 14 2013

A268186 Numbers n such that n^2 + 2, n^2 - 2, n + 2 and n - 2 are all semiprimes.

Original entry on oeis.org

12, 53, 84, 204, 207, 251, 379, 413, 456, 471, 483, 631, 687, 705, 765, 783, 1079, 1135, 1140, 1167, 1269, 1335, 1347, 1395, 1475, 1515, 1587, 1641, 1709, 1767, 1851, 1855, 1943, 1959, 2049, 2157, 2319, 2325, 2575, 2843, 2865, 3099, 3153, 3225, 3267, 3601, 3779

Offset: 1

K. D. Bajpai, Jan 28 2016

			12 appears in the sequence because:
  12^2 + 2 = 146 = 2*73
  12^2 - 2 = 142 = 2*71
  12 + 2   = 14  = 2*7
  12 - 2   = 10  = 2*5 are all semiprimes.

K. D. Bajpai, Table of n, a(n) for n = 1..3379

Subsequence of A105571.

Cf. A001358, A086005, A096173, A096175, A105571, A124936, A237037, A268043.

IsSemiprime:=func;[ n : n in [2..10000] | IsSemiprime(n^2 + 2) and  IsSemiprime(n^2 - 2) and  IsSemiprime(n + 2) and  IsSemiprime(n - 2)];

Maple

with(numtheory): select(n -> (bigomega(n^2 + 2)=2 and bigomega(n^2 - 2)=2 and bigomega(n + 2)=2 and bigomega(n - 2)=2), [seq(n, n=1..10000)]);

Mathematica

Select[Range[10000], PrimeOmega[#^2 + 2] == PrimeOmega[#^2 - 2] == PrimeOmega[# + 2] == PrimeOmega[# - 2] == 2 &]

PARI

for(n = 1, 10000,if(bigomega(n^2 + 2) == 2 && bigomega(n^2 - 2) == 2 && bigomega(n + 2) == 2 && bigomega(n - 2) == 2, print1(n, ", ")))

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,", ")))

A329727 Numbers k such that k^3 +- 2 and k +- 2 are prime.

Original entry on oeis.org

129, 1491, 1875, 2709, 5655, 6969, 10335, 14325, 14421, 17319, 26559, 35109, 37509, 43719, 50229, 52629, 101871, 102795, 104325, 105501, 120429, 127599, 132699, 136395, 137829, 157521, 172425, 173685, 179481, 186189, 191829, 211371, 219681, 221199, 229215, 234195

Offset: 1

K. D. Bajpai, Nov 19 2019

nonn

All terms in this sequence are divisible by 3.

			a(1) = 129:
  129^3 + 2 = 2146691;
  129^3 - 2 = 2146687;
  129   + 2 =     131;
  129   - 2 =     127; all four results are prime.
a(2) = 1491:
  1491^3 + 2 = 3314613773;
  1491^3 - 2 = 3314613769;
  1491   + 2 =       1493;
  1491   - 2 =       1489; all four results are prime.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Intersection of A038599, A067200, and A087679.

Cf. A040976, A052147, A090121, A268043, A268186.

[k:k in [1..250000]|forall{m:m in [-2,2]|IsPrime(k+m) and IsPrime(k^3+m)}]; // Marius A. Burtea, Nov 20 2019

Select[Range[500000], PrimeQ[#^3 + 2] && PrimeQ[#^3 - 2] && PrimeQ[# + 2] && PrimeQ[# - 2] &]

isok(k) = isprime(k-2) && isprime(k+2) && isprime(k^3-2) && isprime(k^3+2); \\ Michel Marcus, Nov 24 2019

list(lim)=my(v=List(),p=127,k); forprime(q=131,lim+2,if(q-p==4 && isprime((k=p+2)^3-2) && isprime(k^3+2), listput(v,k)); p=q); Vec(v) \\ Charles R Greathouse IV, May 06 2020

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

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

cp's OEIS Frontend

A169635 Integers m such that sigma_2(m) = sigma_2(m + 2) where sigma_2(m) is the sum of squares of divisors of m (A001157).

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A268186 Numbers n such that n^2 + 2, n^2 - 2, n + 2 and n - 2 are all semiprimes.

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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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PARI

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

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Maple

PARI

A329727 Numbers k such that k^3 +- 2 and k +- 2 are prime.

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Magma

Mathematica

PARI

PARI

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

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Keywords

Crossrefs

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Mathematica

PARI

Extensions

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

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Magma