A152021 -id:A152021 - cp's OEIS frontend

A152022 Numbers > 1 in A000695 which are not in A152021.

16, 20, 64, 65, 68, 80, 84, 85, 256, 260, 272, 273, 276, 320, 324, 325, 336, 340, 1024, 1025, 1028, 1029, 1040, 1044, 1045, 1088, 1092, 1104, 1105, 1108, 1280, 1281, 1284, 1285, 1296, 1300, 1344, 1345, 1348, 1360, 1364, 1365, 4096, 4097, 4100

Offset: 1

Vladimir Shevelev, Nov 20 2008

nonn

Cf. A000695, A152021.

f[n_] := FromDigits[IntegerDigits[n, 2], 4];  s = Array[f, 100, 2]; div[a_, b_] := Divisible[a, b] && a > b; n = 1; s0 = s; While[Length[s] > n, s = Select[s, ! div[#, s[[n]]] &]; n++]; Complement[s0, s] (* Amiram Eldar, Aug 31 2019 *)

A152023 Numbers in A152022 which are not products of terms of A152021.

Original entry on oeis.org

65, 260, 273, 325, 1025, 1029, 1040, 1045, 1092, 1105, 1281, 1300, 1345, 1365, 4097, 4100, 4116, 4160, 4165, 4180, 4368, 4420, 4433, 5124, 5125, 5185, 5200, 5380, 5460, 16385, 16388, 16400, 16401, 16405, 16464, 16465, 16640, 16645, 16660, 16705, 16709, 16720

Offset: 1

Vladimir Shevelev, Nov 20 2008, Jun 11 2009

nonn

Cf. A000695, A152021, A152022.

f[n_] := FromDigits[IntegerDigits[n, 2], 4];  s = Array[f, 300, 2]; div[a_, b_] := Divisible[a, b] && a > b; n = 1; s0 = s; While[Length[s] > n, s = Select[s, ! div[#, s[[n]]] &]; n++]; s1 = Complement[s0, s]; aQ[n_, s_] := Times @@ (s^(IntegerExponent[n, #] & /@ s)) == n; Select[s1, ! aQ[#, s] &] (* Amiram Eldar, Aug 31 2019 *)

a(12)-a(14) inserted and more terms added by Amiram Eldar, Aug 31 2019

A152079 Primes p such that A000695(p) are also prime.

Original entry on oeis.org

3, 5, 17, 23, 29, 43, 47, 53, 61, 83, 107, 139, 149, 151, 173, 179, 181, 191, 229, 233, 239, 257, 263, 269, 277, 307, 353, 379, 383, 389, 409, 439, 443, 449, 479, 503, 563, 569, 631, 647, 653, 661, 673, 743, 787, 823, 863, 877, 929, 967, 983

Offset: 1

Vladimir Shevelev, Nov 23 2008

nonn

Numbers a(n) are in A077718. Conjecture: The sequence contains infinitely many terms.

Supporting the conjecture, there are 8398 terms up to a million, 56915 up to 10 million, 407875 terms up to 100 million, and 3135962 terms up to a billion. - Charles R Greathouse IV, Apr 28 2015

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000695, A077718, A152021, A152022, A152023, A152078.

A000695(n)=n=binary(n); sum(i=1, #n, n[i]*4^(#n-i))
is(n)=isprime(A000695(n)) && isprime(n) \\ Charles R Greathouse IV, Apr 28 2015

More terms derived from b000695.txt by Michel Marcus, Jun 02 2013

A152078 Numbers a(n) for which A000695(a(n)) = A077718(n).

Original entry on oeis.org

3, 5, 17, 23, 29, 43, 47, 53, 55, 61, 77, 83, 87, 91, 107, 115, 117, 121, 139, 149, 151, 171, 173, 179, 181, 185, 191, 203, 213, 229, 233, 239, 253, 257, 263, 269, 277, 299, 307, 327, 329, 353, 369, 379, 383, 389, 405, 409, 415, 425, 439, 443, 449, 471, 475, 477

Offset: 1

Vladimir Shevelev, Nov 23 2008

nonn

			If n=1, then a(1)=3, A000695(3)=5=A077718(1).

Cf. A000695, A077718, A152021, A152022, A152023.

f[n_] := FromDigits[IntegerDigits[n, 2], 4]; Select[Range[500], PrimeQ[f[#]] &] (* Amiram Eldar, Aug 31 2019 *)

More terms from Amiram Eldar, Aug 31 2019

A152080 Composite numbers c for which A000695(c) is prime.

Original entry on oeis.org

55, 77, 87, 91, 115, 117, 121, 171, 185, 203, 213, 253, 299, 327, 329, 369, 405, 415, 425, 471, 475, 477, 481, 493, 501, 539, 583, 625, 637, 665, 675, 687, 721, 755, 789, 803, 805, 815, 817, 833, 835, 841, 855, 861, 913, 935, 955, 973, 979, 981

Offset: 1

Vladimir Shevelev, Nov 23 2008

nonn

Cf. A000695, A077718, A152021, A152022, A152023, A152078, A152079.

f[n_] := FromDigits[IntegerDigits[n, 2], 4]; Select[Range[1000], CompositeQ[#] && PrimeQ[f[#]] &] (* Amiram Eldar, Dec 13 2018 *)

a000695(n) = fromdigits(binary(n), 4);
lista(nn) = forcomposite(n=1, nn, if (isprime(a000695(n)), print1(n, ", "))); \\ Michel Marcus, Dec 13 2018

More terms derived from b000695.txt by Michel Marcus, Jun 02 2013

A216240 Composite numbers arising in Eratosthenes sieve with removing the multiples of every other remaining numbers after 2 (see comment).

Original entry on oeis.org

9, 21, 33, 49, 51, 77, 87, 119, 121, 123, 141, 177, 187, 201, 203, 219, 237, 287, 289, 291, 309, 319, 327, 329, 357, 393, 413, 417, 447, 451, 469, 471, 493, 501, 511, 517, 543, 553, 573, 591, 633, 649, 669, 679, 687, 697, 721, 723, 737, 763, 771, 799, 803, 807

Offset: 1

Vladimir Shevelev, Mar 14 2013

nonn

We remove even numbers except for 2. The first two remaining numbers are 3,5. Further we remove all remaining numbers multiple of 5,except for 5. The following two remaining numbers are 7,9. Now we remove all remaining numbers multiple of 9, except for 9, etc. The sequence lists the remaining composite numbers.

Conjecture. There exists x_0 such that for every x>=x_0, the number of a(n)<=x is more than pi(x).

Peter J. C. Moses, Table of n, a(n) for n = 1..10000

Cf. A003309, A038179, A004280, A054103, A055398, A055399, A066680, A083140, A145592, A152021, A152114.

Module[{a=Insert[Range[1,1000,2], 2, 2], k=4}, While[Length[a] >= 2k, a = Flatten[{Take[a,k], Select[Take[a,-Length[a]+k], Mod[#,a[[k]]] != 0 &]}]; k+=2]; Rest[Select[a,!PrimeQ[#]&]]] (* Peter J. C. Moses, Mar 27 2013 *)

A152081 Least prime divisor of terms of A152080.

Original entry on oeis.org

5, 7, 3, 7, 5, 3, 11, 3, 5, 7, 3, 11, 13, 3, 7, 3, 3, 5, 5, 3, 5, 3, 13, 17, 3, 7, 11, 5, 7, 5, 3, 3, 7, 5, 3, 11, 5, 5, 19, 7, 5, 29, 3, 3, 11, 5, 5, 7, 11, 3, 7, 29, 5, 3, 3, 19, 3, 3, 3, 13, 5, 7, 29, 7, 31, 3, 23, 17, 13, 13, 3, 19, 3, 3, 7, 3, 5, 7, 3, 11, 11, 29, 37

Offset: 1

Vladimir Shevelev, Nov 23 2008

nonn

Cf. A020639, A152080, A000625, A152021, A152022, A152023, A152078.

a000695(n) = fromdigits(binary(n), 4);
lista(nn) = forcomposite(n=1, nn, if (isprime(a000695(n)), print1(factor(n)[1,1], ", "))); \\ Michel Marcus, Dec 18 2018

a(n) = A020639(A152080(n)). - Michel Marcus, Sep 24 2023

More terms from Michel Marcus, Jun 02 2013

More terms from Michel Marcus, Dec 18 2018

A152082 Largest prime divisors of the terms of A152080.

Original entry on oeis.org

11, 11, 29, 13, 23, 13, 11, 19, 37, 29, 71, 23, 23, 109, 47, 41, 5, 83, 17, 157, 19, 53, 37, 29, 167, 11, 53, 5, 13, 19, 5, 229, 103, 151, 263, 73, 23, 163, 43, 17, 167, 29, 19, 41, 83, 17, 191, 139, 89, 109, 149, 37, 43, 73, 53, 59, 5, 379, 383, 89, 233, 167

Offset: 1

Vladimir Shevelev, Nov 23 2008

nonn

Cf. A000695, A152021, A152022, A152023, A152078, A152080.

f[n_] := FromDigits[IntegerDigits[n, 2], 4]; pmax[n_] := FactorInteger[n][[-1, 1]]; pmax /@ Select[Range[1300], CompositeQ[#] && PrimeQ[f[#]] &] (* Amiram Eldar, Aug 31 2019 *)

More terms from Amiram Eldar, Aug 31 2019

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A152022 Numbers > 1 in A000695 which are not in A152021.

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A152023 Numbers in A152022 which are not products of terms of A152021.

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A152079 Primes p such that A000695(p) are also prime.

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A152078 Numbers a(n) for which A000695(a(n)) = A077718(n).

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A152080 Composite numbers c for which A000695(c) is prime.

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A216240 Composite numbers arising in Eratosthenes sieve with removing the multiples of every other remaining numbers after 2 (see comment).

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A152081 Least prime divisor of terms of A152080.

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A152082 Largest prime divisors of the terms of A152080.

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