A236537 -id:A236537 - cp's OEIS frontend

A238538 A fourth-order linear divisibility sequence: a(n) = (2^n + 1)(2^(3n) - 1)/ ( (2 + 1)*(2^3 - 1) ).

1, 15, 219, 3315, 51491, 811395, 12882499, 205321155, 3278747331, 52408827075, 838132189379, 13406842675395, 214483303960771, 3431523432591555, 54902699475185859, 878429788032676035, 14054769379960303811, 224875452250864496835, 3598000373385828511939

Offset: 1

Peter Bala, Feb 28 2014

This is a fourth-order linear divisibility sequence, that is, the sequence satisfies a linear recurrence of order 4 and if n | m then a(n) | a(m). This is a consequence of the following more general result: The polynomials P(n,x,y) := (x^n + y^n)*(x^(3*n) - y^(3*n)) form a fourth-order linear divisibility sequence in the polynomial ring Z[x,y]. See the Bala link.
Hence, for a fixed integers M and N, the normalized sequence (M^n + N^n)*(M^(3*n) - N^(3*n))/ ( (M + N)*(M^3 - N^3) ) for n = 1,2,3,... is a linear divisibility sequence of order 4. It has the rational o.g.f. x*(1 - 2*M*N*(M^2 - M*N + N^2)*x + (M*N)^4*x^2)/( (1 - M^4*x)*(1 - M^3*N*x)*(1 - M*N^3*x)*(1 - N^4*x) ). This is the case M = 2, N = 1. For other cases see A238539(M = 2, N = -1), A238540(M = 3, N = 1) and A238541(M = 3, N = 2). See also A238536, A238537 and A215466.
Note, these sequences do not belong to the family of linear divisibility sequences of the fourth order studied by Williams and Guy, which have o.g.f.s of the form x*(1 - q*x^2)/Q(x), Q(x) a quartic polynomial and q an integer parameter.

Michael De Vlieger, Table of n, a(n) for n = 1..831
Peter Bala, A family of linear divisibility sequences of order four
E. L. Roettger and H. C. Williams, Appearance of Primes in Fourth-Order Odd Divisibility Sequences, J. Int. Seq., Vol. 24 (2021), Article 21.7.5.
Wikipedia, Divisibility sequence
Wikipedia, Lucas Sequence
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume
Index entries for linear recurrences with constant coefficients, signature (27,-202,432,-256).

Cf. A215466, A236536, A236537, A238539, A238540, A238541.

seq(1/21*(2^n + 1)*(2^(3*n) - 1), n = 1..20);

LinearRecurrence[{27,-202,432,-256},{1,15,219,3315},20] (* Harvey P. Dale, Jul 04 2019 *)

a(n) = (1/21)*(2^n + 1)*(2^(3*n) - 1) = A000051(n)*A024088(n)/21.
a(n) = (1/21)*(4^n - 1)*(8^n - 1)/(2^n - 1).
O.g.f.: x*(1 - 12*x + 16*x^2)/((1 - x)*(1 - 2*x)*(1 - 8*x)*(1 - 16*x)).
Recurrence equation: a(n) = 27*a(n-1) - 202*a(n-2) + 432*a(n-4) - 256*a(n-4).

A242677 Semiprimes whose binary representation, read in decimal, is also semiprime.

Original entry on oeis.org

15, 33, 55, 57, 65, 69, 77, 87, 115, 121, 129, 143, 169, 205, 209, 265, 299, 305, 321, 339, 361, 415, 417, 447, 451, 481, 493, 505, 517, 519, 535, 551, 553, 581, 611, 623, 667, 687, 695, 721, 737, 779, 789, 799, 865, 871, 893, 901, 905, 923, 943, 949, 955, 973

Offset: 1

K. D. Bajpai, May 20 2014

			a(2) = 33 = 3 * 11 is semiprime. Binary representation of 33 = 100001 = 11 * 9091 which is also semiprime.
a(4) = 57 = 3 * 19 is semiprime. Binary representation of 57 = 111001 = 11 * 10091 which is also semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A001358, A236365, A236537, A241764.

with(numtheory):A242677 := proc() if bigomega(n)=2 and bigomega(convert(n, binary))=2 then RETURN (n) ; fi; end:  seq(A242677(), n=1..2000);

c = 0; Do[If [PrimeOmega[n] == 2 && PrimeOmega[FromDigits[IntegerDigits[n, 2]]] == 2, c ++;  Print[c, "  ", n]], {n, 1, 3*10^5}];

A278931 Semiprimes whose ternary representations are also semiprime when read as a decimal number.

Original entry on oeis.org

25, 49, 65, 82, 106, 115, 118, 121, 142, 143, 155, 187, 209, 235, 254, 259, 262, 265, 274, 289, 299, 314, 319, 326, 334, 335, 341, 355, 361, 382, 398, 415, 445, 451, 454, 458, 469, 493, 511, 515, 538, 551, 562, 566, 583, 586, 589, 614, 622, 634, 649, 667, 679

Offset: 1

K. D. Bajpai, Dec 04 2016

			65 is in the sequence because 5*13 = 65 (semiprime) and its ternary representation, 2102 = 2*1051, when read as a decimal number, is also semiprime.
115 is in the sequence because 5*23 = 115 (semiprime) and its ternary representation, 11021 = 103*107, when read as a decimal number, is also semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..9646
Wikipedia, Ternary numeral system

Subsequence of A001358.

Cf. A001358, A005935, A007089, A065721, A089981, A236537.

Select[Range[5000], PrimeOmega[#] == 2 && PrimeOmega[FromDigits[ IntegerDigits[ #, 3]]] == 2 &]

A279052 Semiprimes whose binary and ternary representations are prime when read in decimal.

Original entry on oeis.org

295, 1189, 2515, 4399, 4897, 5137, 7045, 7261, 7999, 8065, 9019, 9637, 10579, 10951, 10963, 11035, 11233, 12679, 13315, 13603, 13849, 16279, 18295, 20065, 20467, 20497, 23089, 23419, 23551, 23983, 26359, 27007, 27301, 27787, 29647, 33127, 33253, 33763, 34189, 34411

Offset: 1

K. D. Bajpai, Dec 05 2016

			295 is in the sequence because 295 = 5*59 (semiprime), 295_10 = 100100111_2 = 101221_3, and both 100100111_10 and 101221_10 are prime.
1189 is in the sequence because 1189 = 29*41 (semiprime), and both its binary representation 10010100101 and its ternary representation 1122001, if read as decimal numbers, are prime.

K. D. Bajpai and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 4075 terms from K. D. Bajpai)

Subsequence of A001358.

Cf. A000040, A007089, A036952, A065720, A236365, A236537.

Select[Range[50000], PrimeOmega[#] == 2 && PrimeQ[FromDigits[IntegerDigits[#, 2]]] && PrimeQ[FromDigits[IntegerDigits[#, 3]]] &]

has(n,b)=isprime(fromdigits(digits(n,b),10))
list(lim)=my(v=List(),t); forprime(p=2,lim\2, forprime(q=2,min(lim\p,p), if(has(t=p*q,2) && has(t,3), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Dec 05 2016

A316479 a(n) is the smallest prime whose base-b expansion, read as a base-10 number, is a prime for every b in 2, 3, ..., n. (For n > 10, each base-b expansion for 10 < b <= n must contain no digit larger than 9.)

Original entry on oeis.org

3, 157, 157, 9241, 9241, 48404791, 18172964503, 50006393431, 50006393431, 181395559296673

Offset: 2

Jon E. Schoenfield, Jul 16 2018

a(2)=3, the smallest term in A065720, primes whose binary representation is also the decimal representation of a prime;
a(3)=157, the smallest integer in both A065720 and A065721, primes p whose base-3 expansion is also the decimal expansion of a prime;
similarly, a(4)=157 is the smallest integer in A065720, A065721, and A065722.
Is this sequence infinite?
a(12) > 10^16. - Giovanni Resta, Aug 01 2018

			a(2)=3 because 3 is prime, 3_10 = 11_2, and 11 is prime, and 3 is the smallest such number.
a(3)=157 because 157 is prime, 157_10 = 10011101_2, 157_10 = 12211_3, and 10011101 and 12211 are prime, and 157 is the smallest such number. a(4)=157 as well, since 157_10 = 2131_4 and 2131 is also prime.

Cf. A065720, A065721, A065722, A065723, A065724, A065725, A065726, A065727.

Cf. A084482, A236537.

a(8)-a(10) from Giovanni Resta, Jul 17 2018

a(11) from Giovanni Resta, Jul 24 2018

cp's OEIS Frontend

A238538 A fourth-order linear divisibility sequence: a(n) = (2^n + 1)*(2^(3*n) - 1)/ ( (2 + 1)*(2^3 - 1) ).

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A242677 Semiprimes whose binary representation, read in decimal, is also semiprime.

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A278931 Semiprimes whose ternary representations are also semiprime when read as a decimal number.

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A279052 Semiprimes whose binary and ternary representations are prime when read in decimal.

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A316479 a(n) is the smallest prime whose base-b expansion, read as a base-10 number, is a prime for every b in 2, 3, ..., n. (For n > 10, each base-b expansion for 10 < b <= n must contain no digit larger than 9.)

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A238538 A fourth-order linear divisibility sequence: a(n) = (2^n + 1)(2^(3n) - 1)/ ( (2 + 1)*(2^3 - 1) ).