A001358 -id:A001358 - cp's OEIS frontend

A102913 Take characteristic function of the semiprimes A001358, interpret it as a binary fraction and convert to a decimal fraction.

0, 4, 0, 5, 7, 3, 5, 0, 0, 2, 0, 1, 3, 9, 8, 0, 6, 8, 6, 7, 4, 3, 1, 1, 2, 6, 6, 4, 2, 3, 5, 3, 5, 7, 5, 0, 6, 9, 3, 6, 2, 7, 5, 8, 2, 1, 9, 4, 0, 0, 2, 3, 5, 8, 6, 0, 8, 3, 3, 4, 0, 6, 9, 4, 6, 3, 3, 3, 6, 2, 5, 2, 4, 7, 3, 5, 1, 3, 5, 1, 3, 9, 1, 0, 5, 4, 4, 2, 5, 2, 5, 8, 2, 3, 8, 0, 5, 8, 6, 4, 3, 3, 4, 5, 2

Offset: 0

Jonathan Vos Post, Jan 17 2005

Eric Weisstein's World of Mathematics, Prime Constant.
Eric Weisstein's World of Mathematics, Semiprime.
Eric Weisstein et al., Characteristic Function.

For the continued fraction form of the semiprime constant, see A102914. For the equivalent characteristic function for primes, see A010051; interpreted as a binary fraction see A051006; for the continued fraction form of that see A051007.

Cf. A001358, A010051, A051006, A064911, A102914.

Semiprime[n_] := If[Plus @@ Last[ Transpose[ FactorInteger[n]]] == 2, 1, 0]; RealDigits[ FromDigits[{Table[ Semiprime[n], {n, 2, 350}], -2}, 2], 10, 111][[1]] (* Ed Pegg Jr *)

The characteristic function of the semiprimes is the function f(n) = 1 iff n is semiprime, 0 otherwise. This begins, for n = 0, 1, 2, 3, ... f(n) = 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1... If we concatenate these bits and interpret them as the binary fraction 0.0000101001100011000001... (base 2) we have, expressed as a decimal fraction, 0.0405735002013980686743112664235357506936275821940023586083340694633362...

The characteristic function of A001358 is A064911 (for n >= 1, starting with 0, 0, 0, 1 ...). The binary constant here has an additional 0 after the binary point. - Georg Fischer, Aug 04 2021

More terms from Robert G. Wilson v, Jan 24 2005

A109403 Examine the sequence of all (even or odd) semiprimes, A001358, and record the averages of any pair of successive terms of the same parity.

Original entry on oeis.org

5, 12, 18, 50, 53, 56, 60, 67, 89, 92, 113, 120, 126, 131, 144, 160, 173, 184, 186, 204, 211, 216, 220, 236, 242, 248, 251, 266, 276, 288, 290, 293, 300, 304, 307, 320, 322, 328, 337, 340, 368, 374, 379, 384, 392

Offset: 1

Zak Seidov, Jun 27 2005

nonn

			5 is OK because sp(1)=4, sp(2)=6 and (4+6)/2=5; 12 is OK because sp(4)=10, sp(5)=14 and (10+14)/2=12; sp(n)=n-th semiprime.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

Cf. A001358, A100484.

A001358 := proc(n) option remember; local a; if n = 1 then 4; else for a from procname(n-1)+1 do if numtheory[bigomega](a) = 2 then return a; end if; end do: end if; end proc:
# To produce a b-file
M:=50000; c:=0;
for n from 1 to M do
if type(A001358(n)+A001358(n+1),even) then c:=c+1;
m:=(A001358(n)+A001358(n+1))/2;
lprint(c,m);
fi; od; # N. J. A. Sloane, Aug 27 2020

 Select[Mean/@Partition[Select[Range[400],PrimeOmega[#]==2&],2,1],IntegerQ] (* Harvey P. Dale, Aug 27 2020 *)

Definition corrected by N. J. A. Sloane, Aug 26 2020. The original definition appeared to be based on A100484, but that was incorrect. Thanks to Harvey P. Dale for pointing out that something was wrong.

A113475 a(1)=1 and a(n) for n>1 has the smallest positive value such that Sum_{i=1..n} a(i)^a(n-i+1) is semiprime (A001358).

Original entry on oeis.org

1, 3, 5, 2, 4, 2, 2, 4, 2, 4, 3, 2, 3, 4, 2, 2, 1, 1, 2, 1, 5, 1, 7, 1, 5, 4, 2, 2, 3, 3, 2, 11, 5, 10, 4, 2, 2, 6, 14, 4, 6, 2, 3, 9, 14, 10, 3, 3, 4, 2, 1, 5, 4, 16, 8, 9, 5, 8, 14, 6, 2, 2, 26, 8, 30, 4, 5, 1, 4, 2, 22, 36, 20, 2, 10, 2, 15, 3, 18, 6, 15

Offset: 1

Jonathan Vos Post, Jan 08 2006

nonn

Previous name was: Least integers so ascending descending base exponent transforms all semiprime.
Semiprime analogy to A113320. The sequence is probably infinite, but it is hard to characterize the asymptotic cost of adding an n-th term. The ascending descending base exponent transform of semiprimes is A113173.
The sequence is infinite because a(n) is the minimum k such that a(1)^k + k^a(1) + Sum_{i=2..n-1} a(i)^a(n-i+1) is semiprime, and since a(1)=1 this is equal to 1+k+T where T does not depend on k, thus k is the smallest positive value that makes 1+k+T semiprime, which exists because semiprimes are infinite. - Giovanni Resta, Jan 03 2020

			a(1) = 1 by definition.
a(2) = 3 because 3 is the min x such that 1^x + x^1 is semiprime, i.e., 1^3 + 3^1 = 4 = 2*2.
a(3) = 5 because 1^5 + 3^3 + 5^1 = 33 = 3 * 11 is semiprime.
a(4) = 2 because 1^2 + 3^5 + 5^3 + 2^1 = 371 = 7 * 53.
a(5) = 4 because 1^4 + 3^2 + 5^5 + 2^3 + 4^1 = 3147 = 3 * 1049.
a(6) = 2 because 1^2 + 3^4 + 5^2 + 2^5 + 4^3 + 2^1 = 205 = 5 * 41.
a(7) = 2 because 1^2 + 3^2 + 5^4 + 2^2 + 4^5 + 2^3 + 2^1 = 1673 = 7 * 239.
a(8) = 4 because 1^4 + 3^2 + 5^2 + 2^4 + 4^2 + 2^5 + 2^3 + 4^1 = 111 = 3 * 37.

Cf. A001358, A005408, A113122, A113153, A113154, A113336, A113320, A113271, A113258, A113257, A113231, A087316, A113208.

semipQ[n_] := PrimeOmega[n] == 2; inve[w_] := Total[w^Reverse[w]]; a[1] = 1; a[n_] := a[n] = Block[{k = 0}, While[! semipQ[ inve@ Append[ Array[a, n - 1], ++k]]]; k]; Array[a, 81] (* Giovanni Resta, Jun 13 2016 *)

lista(n)={my(a=vector(n)); a[1]=1; print1(1, ", "); for(n=2, #a, my(t=sum(i=2, n-1, a[i]^a[n-i+1])); my(k=1); while(2!=bigomega(t+1+k), k++); a[n]=k; print1(k, ", "))} \\ Andrew Howroyd, Jan 03 2020

a(1) = 1. For n>1, a(n) = min {k>0: a(1)^k + k^a(1) + Sum_{i=2..n-1} a(i)^a(n-i+1) is in A001358}.

Corrected and extended by Giovanni Resta, Jun 13 2016

A115665 Brilliant numbers (A078972) whose digit reversal is a semiprime (A001358).

Original entry on oeis.org

4, 6, 9, 15, 49, 121, 143, 169, 187, 221, 289, 319, 323, 341, 437, 493, 533, 551, 559, 589, 629, 667, 737, 767, 781, 817, 851, 871, 893, 899, 913, 923, 949, 961, 979, 989, 1027, 1067, 1079, 1121, 1219, 1247, 1273, 1343, 1357, 1387, 1411, 1513, 1577

Offset: 1

Giovanni Resta, Jan 31 2006

			551=19*29 is brilliant and 155=5*31 is semiprime.

Cf. A001358, A078972, A115666.

A115666 Semiprimes (A001358) whose digit reversal is a brilliant number (A078972).

Original entry on oeis.org

4, 6, 9, 51, 94, 121, 122, 143, 155, 158, 169, 178, 187, 319, 323, 329, 335, 341, 394, 398, 718, 734, 737, 766, 767, 781, 913, 926, 949, 955, 961, 979, 982, 985, 989, 998, 1141, 1202, 1211, 1273, 1322, 1343, 1405, 1415, 1655, 1691, 1703, 1714, 1807

Offset: 1

Giovanni Resta, Jan 31 2006

			155=5*31 is semiprime and 551=19*29 is brilliant.

Cf. A001358, A078972, A115665.

A115712 Semiprimes (A001358) whose digit reversal is a cube.

Original entry on oeis.org

10, 46, 215, 3194, 9586, 27845, 35605, 40393, 52651, 63379, 70597, 76121, 84601, 98342, 119753, 189622, 211591, 234413, 291149, 376219, 485038, 616571, 778841, 886877, 946711, 2134493, 2198998, 2579365, 3405221, 3735467, 4430089

Offset: 1

Giovanni Resta, Jan 31 2006

			46=2*23 is semiprime and 64=4^3.

Cf. A001358, A115740.

Select[Range[5*10^6],PrimeOmega[#]==2&&IntegerQ[Surd[ IntegerReverse[ #],3]]&] (* The program uses the IntegerReverse function from Mathematica version 10 *) (* Harvey P. Dale, Jan 18 2016 *)

A115740 Cubes whose digit reversal is a semiprime (A001358).

Original entry on oeis.org

64, 512, 4913, 6859, 10648, 12167, 15625, 24389, 39304, 50653, 54872, 64000, 79507, 97336, 117649, 148877, 175616, 195112, 226981, 314432, 357911, 512000, 778688, 830584, 912673, 941192, 1092727, 1124864, 1191016, 1225043, 1295029

Offset: 1

Giovanni Resta, Jan 31 2006

			64=4^3 and 46=2*23 is semiprime.

Cf. A001358, A115712.

A115741 Semiprimes (A001358) whose digit reversal is a triangular number.

Original entry on oeis.org

6, 10, 51, 55, 82, 87, 309, 501, 649, 694, 1198, 1207, 1473, 1765, 1803, 1837, 1959, 1981, 3043, 3057, 3561, 3574, 5158, 5221, 5721, 6139, 6193, 6267, 6535, 6711, 6814, 6843, 10473, 10497, 11553, 12477, 12693, 13123, 13701, 13951, 14169

Offset: 1

Giovanni Resta, Jan 31 2006

			51=3*17 is semiprime and 15=T(5).

Cf. A001358, A115742.

A116000 phi(n) + sigma(n) gives a semiprime (A001358).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 10, 11, 13, 16, 17, 19, 22, 23, 25, 27, 29, 31, 37, 40, 41, 43, 46, 47, 52, 53, 58, 59, 61, 64, 67, 68, 71, 72, 73, 79, 80, 82, 83, 89, 97, 98, 101, 103, 106, 107, 109, 113, 117, 127, 128, 131, 136, 137, 139, 144, 149, 151, 157, 162, 163, 166, 167

Offset: 1

Giovanni Resta, Feb 13 2006

nonn

			phi(101)+sigma(101)=202=2*101.

Cf. A001358, A116016, A116001.

Select[Range[200],PrimeOmega[EulerPhi[#]+DivisorSigma[1,#]]==2&] (* Harvey P. Dale, Feb 05 2013 *)

A116023 The n-th prime plus n gives a semiprime (A001358).

Original entry on oeis.org

10, 12, 14, 15, 16, 19, 20, 21, 23, 25, 30, 31, 36, 37, 38, 39, 40, 44, 52, 54, 56, 57, 58, 60, 62, 67, 74, 75, 77, 80, 83, 84, 86, 88, 90, 99, 107, 111, 115, 120, 124, 136, 140, 145, 146, 154, 156, 160, 162, 164, 165, 166, 168, 174, 175, 182, 189, 192, 195, 196

Offset: 1

Giovanni Resta, Feb 13 2006

nonn

			p(30)+30=143=11*13.

Cf. A001358, A116012, A116024, A014688

cp's OEIS Frontend

A102913 Take characteristic function of the semiprimes A001358, interpret it as a binary fraction and convert to a decimal fraction.

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A109403 Examine the sequence of all (even or odd) semiprimes, A001358, and record the averages of any pair of successive terms of the same parity.

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A113475 a(1)=1 and a(n) for n>1 has the smallest positive value such that Sum_{i=1..n} a(i)^a(n-i+1) is semiprime (A001358).

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A115665 Brilliant numbers (A078972) whose digit reversal is a semiprime (A001358).

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A115666 Semiprimes (A001358) whose digit reversal is a brilliant number (A078972).

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A115712 Semiprimes (A001358) whose digit reversal is a cube.

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A115740 Cubes whose digit reversal is a semiprime (A001358).

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A115741 Semiprimes (A001358) whose digit reversal is a triangular number.

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A116000 phi(n) + sigma(n) gives a semiprime (A001358).

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Mathematica

A116023 The n-th prime plus n gives a semiprime (A001358).

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