A068811 -id:A068811 - cp's OEIS frontend

A359121 a(n) = number of terms of A068811 that are <= n.

0, 0, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12

Offset: 1

N. J. A. Sloane, Dec 18 2022

nonn

This is to A068811 as pi = A000720 is to the primes.

			A068811 begins 3, 5, 7, ..., so the present sequence begins 0, 0, 1, 1, 2, 2, 3, ..., with an increment as each term of A068811 is reached.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000720, A068811.

c = 0; Reap[Do[If[AllTrue[{n, 10^(1 + Floor@ Log10[n]) - n}, PrimeQ], c++]; Sow[c], {n, 86}]][[-1, -1]] (* Michael De Vlieger, Dec 18 2022 *)

A359122 Index of prime(n) in A068811, or -1 if prime(n) is missing from A068811.

Original entry on oeis.org

-1, 1, 2, 3, 4, -1, 5, -1, -1, 6, -1, -1, 7, -1, 8, 9, 10, -1, -1, 11, -1, -1, 12, 13, 14, -1, -1, -1, -1, 15, -1, -1, 16, -1, -1, -1, -1, -1, -1, 17, 18, -1, 19, -1, -1, -1, -1, -1, 20, -1, -1, 21, -1, -1, 22, -1, -1, -1, -1, 23, -1, -1, -1, -1, -1, 24, -1, -1, 25, -1, 26, 27, -1

Offset: 1

N. J. A. Sloane, Dec 18 2022

sign

			Prime(4) = 7 is the third term in A068811, so a(4) = 3.
Prime(6) = 13 is missing from A068811, so a(6) = -1.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Raster showing a(n) > 0 in black else white, n = 1..10^6, in rows of 1000 pixels.

Cf. A068811, A359121.

c = 1; Reap[Do[Sow@ If[PrimeQ[10^(1 + Floor@ Log10[#]) - #], c++, -1] &[Prime[n]], {n, 73}]][[-1, -1]] (* or, generate nn <= 10^6 terms of this sequence from the bitmap: *)
Block[{nn = 10^4, s = Flatten@ ImageData[Import["https://oeis.org/A359122/a359122.png"]] /. {1. -> 0, 0. -> 1}, c}, c = 1; Map[If[# == 0, -1, c++] &, s[[1 ;; nn]]]] (* Michael De Vlieger, Dec 18 2022 *)

A359123 First differences of A068811, halved.

Original entry on oeis.org

1, 1, 2, 3, 6, 6, 3, 3, 3, 6, 6, 3, 4, 8, 12, 18, 3, 6, 18, 6, 9, 12, 18, 15, 3, 3, 12, 9, 15, 6, 18, 6, 9, 6, 18, 6, 15, 9, 12, 3, 3, 15, 18, 12, 9, 6, 18, 6, 3, 18, 12, 12, 9, 6, 3, 3, 9, 3, 3, 7, 17, 9, 51, 6, 9, 6, 12, 21, 21, 3, 6, 27, 27, 30, 6, 27, 9, 21, 12, 36, 84, 6

Offset: 1

N. J. A. Sloane, Dec 18 2022

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A068811, A359121, A359122.

1/2*Differences@ Select[Prime@ Range[300], PrimeQ[10^(1 + Floor@ Log10[#]) - #] &] (* Michael De Vlieger, Dec 18 2022 *)

A092564 Duplicate of A068811.

Original entry on oeis.org

3, 5, 7, 11, 17, 29, 41, 47, 53, 59, 71, 83, 89, 97, 113, 137, 173, 179, 191, 227, 239, 257

Offset: 1

dead

A145987 Duplicate of A068811.

Original entry on oeis.org

3, 5, 7, 11, 17, 29, 41, 47, 53, 59, 71, 83, 89, 97, 113, 137, 173, 179, 191, 227, 239, 257, 281, 317, 347, 353, 359, 383, 401, 431, 443, 479, 491, 509, 521, 557, 569, 599, 617, 641, 647, 653, 683, 719, 743, 761, 773, 809, 821, 827, 863, 887, 911, 929, 941, 947, 953, 971, 977, 983, 997

Offset: 1

dead

A086081 Numbers m such that m and its 2's complement are both primes. In other words, m and 2^k - m (where k is the smallest power of 2 such that 2^k > m) are primes.

Original entry on oeis.org

2, 5, 11, 13, 19, 29, 41, 47, 53, 59, 61, 67, 97, 109, 149, 167, 173, 197, 227, 233, 239, 251, 271, 283, 313, 331, 349, 373, 409, 433, 439, 499, 509, 521, 557, 563, 593, 641, 677, 743, 761, 773, 797, 827, 857, 887, 911, 941, 953, 971, 977, 983, 1013, 1019, 1021

Offset: 1

Chuck Seggelin, Jul 08 2003

In the first 672509 primes, 64894 of them (about 9.65%) are 2's-complement primes.

			19 is a term because 19 is prime and (2^5 - 19) = (32 - 19) = 13 which is prime.
1777 is a term because 1777 is prime and (2^11 - 1777) = (2048 - 1777) = 271 which is prime.

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

Cf. A068811.

Join[{2}, Select[Prime[Range[250]], PrimeQ[BitXor[#, 2^Ceiling[Log[2, #]] - 1] + 1] &]] (* Alonso del Arte, Feb 12 2013 *)

select(m->isprime((2<<(log(m+.5)\log(2)))-m), primes(100)) \\ Charles R Greathouse IV, Feb 13 2013

If isPrime(p) And isPrime(2^(floor(Log(p, 2)) + 1) - p) then sequence.add(p)

If A(x) is the counting function of the terms a(n) <= x, then A(x) = O(xloglogx/(logx)^2) [From Vladimir Shevelev, Dec 04 2008]

A145985 Primes resulting from subtracting primes from 10^n in order (see Comments for precise definition).

Original entry on oeis.org

7, 5, 3, 89, 83, 71, 59, 53, 47, 41, 29, 17, 11, 3, 887, 863, 827, 821, 809, 773, 761, 743, 719, 683, 653, 647, 641, 617, 599, 569, 557, 521, 509, 491, 479, 443, 431, 401, 383, 359, 353, 347, 317, 281, 257, 239, 227, 191, 179, 173, 137, 113, 89, 71, 59, 53, 47, 29, 23, 17, 3, 8969, 8951, 8849, 8837, 8819, 8807, 8783

Offset: 1

Enoch Haga, Oct 27 2008

Comments from N. J. A. Sloane, Dec 18 2022 (Start)
A more precise definition is the following.
Start with k=1; let N=10^k, let i run from 10^(k-1)-1 to N-1, let j = N-i, if i and j are both primes, append j to the sequence; increment k.
This is derived from A068811 via a(n) = 10^d - A068811(n) where d is the number of digits in A068811(n). A068811 is more fundamental, for there the primes appear in order and there are no duplicates. (End)
Primes may appear more than once.

			887 is a term because 1000-887 = 113 and both 887 and 113 are prime.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 [First 1000 terms from _Harvey P. Dale_]
N. J. A. Sloane, Table of n, a(n) for n = 1..80338

Cf. A068811, A358310, A359121, A359122, A359123.

See A359120 for the length of the n-th block of decreasing terms.

a:=[];
for k from 1 to 6 do
N := 10^k;
   for i from 10^(k-1)+1 to N-1 do
      j:=N-i;
      if isprime(i) and isprime(j) then a:=[op(a),j]; fi;
   od:
od;
a; # N. J. A. Sloane, Dec 16 2022

Select[Table[10^IntegerLength[p]-p,{p,Prime[Range[200]]}],PrimeQ] (* Harvey P. Dale, Dec 16 2022 *)

Corrected and edited by Harvey P. Dale and N. J. A. Sloane, Dec 16 2022

A087325 Numbers k such that k and its 10's complement both have the same prime signature.

Original entry on oeis.org

3, 5, 7, 11, 14, 15, 17, 26, 29, 30, 35, 38, 41, 47, 50, 53, 59, 62, 65, 70, 71, 74, 83, 85, 86, 89, 94, 97, 110, 111, 113, 122, 129, 132, 134, 137, 140, 150, 153, 158, 170, 173, 174, 179, 183, 185, 186, 187, 191, 195, 201, 206, 209, 212, 215, 219, 221, 227, 236

Offset: 1

Amarnath Murthy, Sep 04 2003

Conjecture: (1) Sequence is infinite. (2) For every prime signature there corresponds a term in this sequence.
From Robert Israel, Jul 02 2024: (Start)
Conjecture (2) is false: k and its 10's complement can't both have prime signature p^m where m is even.
If k is a term, then so is 10 * k.
It appears that the first term with m prime factors, counted with multiplicity, is 3 * 10^((m-1)/2) if m is odd and 132 * 10^((m-4)/2) if m >= 4 is even. (End)

			35 is a member as 35= 5*7 and its 10's complement (100-35) = 65 = 13*5 both have the prime signature p*q.
35 is a member as 35 = 5*7 and its 10's complement (100-35) = 65 = 13*5 both have the prime signature p*q.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A087324, A089186. Contains A068811.

ps:= n -> sort(ifactors(n)[2][..,2]):
tc:= n -> 10^(1+ilog10(n))-n:
select(n -> ps(n) = ps(tc(n)), [$1..1000]); # Robert Israel, Jul 02 2024

More terms from David Wasserman, May 06 2005

A228075 Numbers n whose 10's complement is prime, i.e., 10^k-n, where k is the number of digits of n, is prime.

Original entry on oeis.org

3, 5, 7, 8, 11, 17, 21, 27, 29, 33, 39, 41, 47, 53, 57, 59, 63, 69, 71, 77, 81, 83, 87, 89, 93, 95, 97, 98, 113, 117, 119, 123, 137, 141, 143, 147, 161, 171, 173, 177, 179, 189, 191, 203, 213, 227, 231, 239, 243, 249, 257, 261, 267, 273, 281, 291, 299

Offset: 1

Jayanta Basu, Aug 09 2013

A068811 is a subset.

			8 is a term since 10^1 - 8 = 2 is a prime.
Similarly, 39 is a term as 10^2 - 39 = 61 is prime.

Jayanta Basu, Table of n, a(n) for n = 1..1000

Cf. A068811.

Select[Range[300], PrimeQ[10^(IntegerLength[#]) - #] &]

A086082 Numbers m such that m and all of its even complements from 2 to 10 are primes. In other words, m and j^k - m (where k is the smallest power of j such that j^k > m) are prime for all of the following values of j: 2, 4, 6, 8, 10.

Original entry on oeis.org

53, 59, 557, 773, 887, 2207, 2273, 2543, 2789, 3209, 3449, 3677, 33347, 33893, 36887, 41927, 54323, 61547, 131303, 131687, 136217, 138143, 139493, 140177, 150083, 150533, 153353, 153437, 154277, 157007, 158303, 161333, 162263, 163847, 166157

Offset: 1

Chuck Seggelin, Jul 08 2003

nonn

Primes meeting the requirements to be members of this sequence are fairly rare. The 653rd prime in this sequence is the 672448th prime in the sequence of all primes (i.e., 0.0971% of the first 672448 primes belong to this sequence). Primes which need only be j-complement for one value of j (such as 6-complement primes) are relatively common (in the first 672509 primes, 122932 are 6-complement primes, or about 18.28%).

Odd complement primes are very rare, simply because any odd number raised to a power yields an odd number. Subtracting from this an odd prime yields an even number that cannot be prime unless it is 2. As a result, odd-complement primes are either 2 or of the form j^k-2 - for example, the first few 7's complement primes are 2, 5 (7^1-2), 47 (7^2-2), 2399 (7^4-2), 823541 (7^7-2), 5764799 (7^8-2), 13841287199 (7^12-2), 4747561509941 (7^15-2) and so forth. This is a natural result of the fact that most primes are odd and so are odd numbers when raised to any power > 0.

			887 is a term because i: 887 is prime. ii: (2^10 - 887) = (1024 - 887) = 137 which is prime. iii: (4^5 - 887) = (1024 - 887) = 137 which is prime. iv: (6^4 - 887) = (1296 - 887) = 409 which is prime. v: (8^4 - 887) = (4096 - 887) = 3209 which is prime. vi: (10^3 - 887) = (1000 - 887) = 113 which is prime.

Cf. A068811, A086081.

If isPrime(p) And isPrime(2^(floor(Log(p, 2))+1)-p) And isPrime(4^(floor(Log(p, 4))+1)-p) And isPrime(6^(floor(Log(p, 6))+1)-p) And isPrime(8^(floor(Log(p, 8))+1)-p) And isPrime(10^(floor(Log(p, 10))+1)-p) then sequence.add(p)

cp's OEIS Frontend

A359121 a(n) = number of terms of A068811 that are <= n.

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A359122 Index of prime(n) in A068811, or -1 if prime(n) is missing from A068811.

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A359123 First differences of A068811, halved.

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A092564 Duplicate of A068811.

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A145987 Duplicate of A068811.

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A086081 Numbers m such that m and its 2's complement are both primes. In other words, m and 2^k - m (where k is the smallest power of 2 such that 2^k > m) are primes.

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A145985 Primes resulting from subtracting primes from 10^n in order (see Comments for precise definition).

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A087325 Numbers k such that k and its 10's complement both have the same prime signature.

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Maple

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A228075 Numbers n whose 10's complement is prime, i.e., 10^k-n, where k is the number of digits of n, is prime.

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