A023203 -id:A023203 - cp's OEIS frontend

A104719 Concatenations of pairs of primes that differ by 10.

313, 717, 1323, 1929, 3141, 3747, 4353, 6171, 7383, 7989, 97107, 103113, 127137, 139149, 157167, 163173, 181191, 223233, 229239, 241251, 271281, 283293, 307317, 337347, 349359, 373383, 379389, 409419, 421431, 433443, 439449, 457467, 499509

Offset: 1

Jonathan Vos Post and Parthasarathy Nambi, Mar 20 2005

There are no primes in this sequence after a(1) = 313, as all values thereafter are divisible by 3. Semiprimes in this sequence include: a(2) = 717 = 3 * 239, a(4) = 1929 = 3 * 643, a(6) = 3747 = 3 * 1249, a(7) = 4353 = 3 * 1451, a(10) = 7989 = 3 * 2663, a(11) = 97107 = 3 * 32369, a(13) = 127137 = 3 * 42379, a(17) = 181191 = 3 * 60397, a(18) = 223233 = 3 * 74411, a(29) = 421431 = 3 * 140477, a(30) = 433443 = 3 * 144481, a(34) = 547557 = 3 * 182519, a(35) = 577587 = 3 * 192529, a(40) = 691701 = 3 * 230567, a(41) = 709719 = 3 * 236573, a(49) = 919929 = 3 * 306643, a(52) = 10091019 = 3 * 3363673.

			Primes 3 and 13 differ by 10.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A001358, A023202, A103195, A103206, A104718.

FromDigits[Join[IntegerDigits[#],IntegerDigits[#+10]]]&/@Select[ Prime[ Range[ 100]], PrimeQ[ #+10]&] (* Harvey P. Dale, Jun 14 2015 *)

a(n) = A023203(n) concatenated with A023203(n)+10.

A248855 a(n) is the smallest positive integer m such that if k >= m then a(k+1,n)^(1/(k+1)) <= a(k,n)^(1/k), where a(k,n) is the k-th term of the sequence {p | p and p+2n are primes}.

Original entry on oeis.org

1, 1, 1, 1, 3556, 1, 34, 3, 4, 1, 2, 1, 11285, 5, 2, 124, 569, 1, 290, 3, 1, 165, 2, 1, 1, 2, 1, 316, 1, 2, 58957, 1, 3, 58617, 522, 2, 1, 1, 4, 1, 2, 1, 1, 2, 1, 7932, 4, 1, 5875, 1679, 4, 4, 3, 3, 1, 2, 307, 1, 1, 1, 1, 1, 4, 3206, 2, 1, 1, 3, 2, 1, 1, 1, 1, 5, 2, 11170, 1, 2, 4245, 1, 1, 81, 2, 1, 1, 2, 58, 1, 3, 4, 7303, 1, 1, 5, 1, 3, 3, 3, 383, 111408, 1

Offset: 0

Farideh Firoozbakht and Jahangeer Kholdi, Nov 25 2014

nonn

All terms conjecturally are found. Note that according to the definition a(k,0) is the k-th term of the sequence {p | p is prime} namely for every positive integer k, a(k,0) = prime(k). Hence if Firoozbakht's conjecture is true then a(0)=1.

			a(0)=a(1)=a(2)=a(3)=1 conjecturally states that the four sequences A000040, A001359, A023200 and A023201 have this property: For every positive integer n, b(n) exists and b(n+1) < b(n)^(1+1/n). Namely b(n)^(1/n) is a strictly decreasing function of n.
If in the definition instead of the sequence {p | p and p+2n are primes} we set {p | p is prime and nextprime(p)=p+2n} then it seems that except for n=3 all terms of the new sequence {c(n)} are equal to 1 and for n=3, c(3)=7746. Note that c(3)=7746 means that the sequence {p | p is prime and nextprime(p)=p+6} = A031924 has this property: For all k >= 7746, A031924(k+1)^(1/(k+1)) < A031924(k)^(1/k).

Wikipedia, Firoozbakht's conjecture

Cf. A000040, A001359, A023200, A023201, A023202, A023203, A031924.

A054902 Composite numbers n such that sigma(n)+12 = sigma(n+12).

Original entry on oeis.org

65, 170, 209, 1394, 3393, 4407, 4556, 11009, 13736, 27674, 38009, 38845, 47402, 76994, 157994, 162393, 184740, 186686, 209294, 680609, 825359, 954521, 1243574, 2205209, 3515609, 4347209, 5968502, 6539102, 6916241, 8165294, 10352294, 10595009, 10786814

Offset: 1

Labos Elemer, May 23 2000

nonn

			n = 65, sigma(65)+12 = 84+12 = 96 = sigma(65+12) = sigma(77).
n = 11009, sigma(11009)+12 = 11220+12 = 11232 = sigma(11009+12) = sigma(11021).

Donovan Johnson, Table of n, a(n) for n = 1..500

Complement of A046133 with respect to A015917.

Cf. A015913-A015917, A023200, A023203, A046133, A001359, A054799.

isok(n) = !isprime(n) && ((sigma(n)+12) == sigma(n+12)); \\ Michel Marcus, Dec 18 2013

More terms from Jud McCranie, May 24 2000

Three more terms from Michel Marcus, Dec 18 2013

A054904 x = a(n) is the smallest composite number such that sigma(x+6n) = sigma(x)+6n, where sigma = A000203.

Original entry on oeis.org

104, 65, 20, 80, 44, 125, 45, 63, 40, 99, 56, 70, 296, 125, 88, 110, 104, 145, 212, 182, 80, 170, 333, 105, 369, 185, 184, 135, 180, 301, 356, 185, 1859, 329, 176, 195, 4916, 434, 612, 287, 140, 185, 776, 255, 524, 413, 344, 205, 272, 329, 567, 215, 320, 469

Offset: 1

Labos Elemer, May 23 2000

nonn

If sigma(x+d) = sigma(x)+d and d = 6k, then composite solutions seem to be more frequent and arise sooner.

a(725) > 3*10^11 (if it exists). - Donovan Johnson, Sep 23 2013

			n = 20, 6n = 120, a(20) = 182, sigma(182)+120 = 336+120 = 456 = sigma(182+120) = sigma(302).

Donovan Johnson, Table of n, a(n) for n = 1..724

Cf. A015914-A015917, A023200-A023203, A054799, A054903-A054905.

Table[x = 4; While[Nand[CompositeQ@ x, DivisorSigma[1, x + 6 n] == DivisorSigma[1, x] + 6 n], x++]; x, {n, 54}] (* Michael De Vlieger, Feb 18 2017 *)

/* finds first 696 terms */ mx=7695851; s=vector(mx); for(j=4, mx, if(isprime(j)==0, s[j]=sigma(j))); for(n=1, 696, n6=n*6; for(x=4, 7691753, if(s[x]>0, if(s[x+n6]==s[x]+n6, write("b054904.txt", n " " x); next(2))))) /* Donovan Johnson, Sep 23 2013 */

sigma(x+6n) = sigma(x)+6n, a(n) = min(x) and it is composite.

A252089 Primes p such that p + 26 is prime.

Original entry on oeis.org

3, 5, 11, 17, 41, 47, 53, 71, 83, 101, 113, 131, 137, 167, 173, 197, 251, 257, 281, 311, 347, 353, 383, 431, 461, 521, 587, 593, 617, 647, 683, 701, 743, 761, 797, 827, 857, 881, 911, 941, 971, 983, 1013, 1061, 1091, 1097, 1103, 1187, 1223, 1277, 1301, 1373

Offset: 1

Vincenzo Librandi, Dec 14 2014

			17 is in this sequence because 17+26 = 43 is prime.
431 is in this sequence because 431+26 = 457 is prime.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. sequences of the type p+n are primes: A001359 (n=2), A023200 (n=4), A023201 (n=6), A023202 (n=8), A023203 (n=10), A046133 (n=12), A153417 (n=14), A049488 (n=16), A153418 (n=18), A153419 (n=20), A242476 (n=22), A033560 (n=24), this sequence (n=26), A252090 (n=28), A049481 (n=30), A049489 (n=32), A252091 (n=34), A156104 (n=36); A062284 (n=50), A049490 (n=64), A156105 (n=72), A156107 (n=144).

[NthPrime(n): n in [1..250] | IsPrime(NthPrime(n)+26)];

Select[Prime[Range[200]], PrimeQ[# + 26] &]

A054903 Composite numbers n such that sigma(n)+6 = sigma(n+6), where sigma=A000203.

Original entry on oeis.org

104, 147, 596, 1415, 4850, 5337, 370047, 1630622, 35020303, 120221396, 3954451796, 742514284703

Offset: 1

Labos Elemer, May 23 2000

Complement of A023201 with respect to A015914.
Intersection of A015914 and A018252.
Below 1000000 there are only 7 such composite numbers, compared with more than 16000 primes.
a(13) > 10^13. - Giovanni Resta, Jul 11 2013

			n=104, sigma(104)+6 = 210+6 = 216 = sigma(104+6) = sigma(110).
a(4) = 1415 = 5*283, 1415+6 = 1421 = 7*7*29:
sigma(1415) = 1+5+283+1415 = 1704,
sigma(1421) = 1+7+29+49+203+1421 = 1710 = sigma(1415)+6.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 104, p. 37, Ellipses, Paris 2008.

Cf. A015913-A015917, A023200-A023203, A046133, A001359, A054799.

forcomposite(n=9,1e7,if(sigma(n)+6==sigma(n+6),print1(n", "))) \\ Charles R Greathouse IV, Feb 14 2013

More terms from Jud McCranie, May 25 2000
New definition from Reinhard Zumkeller, Jan 27 2009
Edited by N. J. A. Sloane, Jan 31 2009 at the suggestion of R. J. Mathar.
a(12) from Giovanni Resta, Jul 11 2013

A231608 Table whose n-th row consists of primes p such that p + 2n is also prime, read by antidiagonals.

Original entry on oeis.org

3, 3, 5, 5, 7, 11, 3, 7, 13, 17, 3, 5, 11, 19, 29, 5, 7, 11, 13, 37, 41, 3, 7, 13, 23, 17, 43, 59, 3, 5, 11, 19, 29, 23, 67, 71, 5, 7, 17, 17, 31, 53, 31, 79, 101, 3, 11, 13, 23, 19, 37, 59, 37, 97, 107, 7, 11, 13, 31, 29, 29, 43, 71, 41, 103, 137

Offset: 1

T. D. Noe, Nov 26 2013

			The following sequences are read by antidiagonals
{3, 5, 11, 17, 29, 41, 59, 71, 101, 107,...}
{3, 7, 13, 19, 37, 43, 67, 79, 97, 103,...}
{5, 7, 11, 13, 17, 23, 31, 37, 41, 47,...}
{3, 5, 11, 23, 29, 53, 59, 71, 89, 101,...}
{3, 7, 13, 19, 31, 37, 43, 61, 73, 79,...}
{5, 7, 11, 17, 19, 29, 31, 41, 47, 59,...}
{3, 5, 17, 23, 29, 47, 53, 59, 83, 89,...}
{3, 7, 13, 31, 37, 43, 67, 73, 97, 151,...}
{5, 11, 13, 19, 23, 29, 41, 43, 53, 61,...}
{3, 11, 17, 23, 41, 47, 53, 59, 83, 89,...}
...

T. D. Noe, Rows n = 1..100 of triangle, flattened

Cf. A001359, A023200, A023201, A023202, A023203.
Cf. A046133, A153417, A049488, A153418, A153419.
Cf. A020483 (numbers in first column).
Cf. A086505 (numbers on the diagonal).

A231608 := proc(n,k)
    local j,p ;
    j := 0 ;
    p := 2;
    while j < k do
        if isprime(p+2*n ) then
            j := j+1 ;
        end if;
        if j = k then
            return p;
        end if;
        p := nextprime(p) ;
    end do:
end proc:
for n from 1 to 10 do
    for k from 1 to 10 do
        printf("%3d ",A231608(n,k)) ;
    end do;
    printf("\n") ;
end do: # R. J. Mathar, Nov 19 2014

nn = 10; t = Table[Select[Range[100*nn], PrimeQ[#] && PrimeQ[# + 2*n] &, nn], {n, nn}]; Table[t[[n-j+1, j]], {n, nn}, {j, n}]

A262086 Numbers k such that phi(k + 10) = phi(k) + 10, where phi(k) = A000010(k) is Euler's totient function.

Original entry on oeis.org

3, 7, 13, 19, 31, 36, 37, 43, 61, 73, 79, 97, 103, 127, 139, 157, 163, 181, 223, 229, 241, 271, 283, 307, 337, 349, 373, 379, 409, 421, 433, 439, 457, 499, 547, 577, 607, 631, 643, 673, 691, 709, 733, 751, 787, 811, 829, 853, 877, 919, 937, 967

Offset: 1

Kevin J. Gomez, Sep 10 2015

The only composite term less than 10^11 is 36. - Giovanni Resta, Sep 14 2015

			3 is in the sequence since phi(13) = phi(3) + 10.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Seiichi Manyama)
Wikipedia, Euler's totient function.

Cf. A000010, A023203.

Cf. A001838 (k=2), A056772 (k=4), A262084 (k=6), A262085 (k=8), this sequence (k=10).

[n: n in [1..1000] | EulerPhi(n+10) eq EulerPhi(n)+10]; // Vincenzo Librandi, Sep 11 2015

Select[Range@1000, EulerPhi@(# + 10) == EulerPhi[#] + 10 &] (* Vincenzo Librandi, Sep 11 2015 *)

is(n)=eulerphi(n + 10) == eulerphi(n) + 10 \\ Anders Hellström, Sep 11 2015

A288021 Prime p1 of consecutive primes p1, p2, where p2 - p1 = 4, and p1, p2 are in different decades.

Original entry on oeis.org

7, 19, 37, 67, 79, 97, 109, 127, 229, 277, 307, 349, 379, 397, 439, 457, 487, 499, 739, 757, 769, 859, 877, 907, 937, 967, 1009, 1087, 1279, 1297, 1429, 1447, 1489, 1549, 1567, 1579, 1597, 1609, 1867, 1999, 2137, 2239, 2269, 2347, 2377, 2389, 2437, 2539, 2617, 2659, 2689, 2707, 2749, 2797, 2857

Offset: 1

Hartmut F. W. Hoft, Jun 04 2017

nonn

The unit digits of the numbers in the sequence are 7's or 9's.

			7 is in this sequence since pair (7,11) is the first with difference 4 spanning a multiple of 10.

Cf. A001359, A023201, A023203, A031924, A031925, A031928, A046117, A046132, A158277, A158861, A160370, A160440, A160500, A287049, A287050, A060229, A288022, A288024.

a288021[n_] := Map[Last, Select[Map[{NextPrime[#, 1], NextPrime[#, -1]}&, Range[10, n, 10]], First[#]-Last[#]==4&]]
a288021[3000] (* data *)

A288022 Prime p1 of consecutive primes p1, p2, where p2 - p1 = 6, and p1, p2 are in different decades.

Original entry on oeis.org

47, 157, 167, 257, 367, 557, 587, 607, 647, 677, 727, 947, 977, 1097, 1117, 1187, 1217, 1367, 1657, 1747, 1777, 1907, 1987, 2207, 2287, 2417, 2467, 2677, 2837, 2897, 2957, 3307, 3407, 3607, 3617, 3637, 3727, 3797, 4007, 4357, 4457, 4507, 4597, 4657, 4937, 4987

Offset: 1

Hartmut F. W. Hoft, Jun 04 2017

nonn

The unit digits of the numbers in the sequence are 7's.

Number of terms < 10^k: 0, 0, 1, 13, 81, 565, 4027, 30422, 237715, ... - Muniru A Asiru, Jan 09 2018

			47 is in the sequence since pair (47,53) is the first with difference 6 spanning a multiple of 10.

Muniru A Asiru, Table of n, a(n) for n = 1..10000

Cf. A001359, A023201, A023203, A031924, A031925, A031928, A046117, A046132, A158277, A158861, A160370, A160440, A160500, A287049, A287050, A060229, A288021, A288024.

P:=Filtered([1..20000], IsPrime);
P1:=List(Filtered(Filtered(List([1..Length(P)-1],n->[P[n],P[n+1]]),i->i[2]-i[1]=6),j->j[1] mod 5=2),k->k[1]); # Muniru A Asiru, Jul 08 2017

for n from 1 to 2000 do if [ithprime(n+1)-ithprime(n), ithprime(n) mod 5] = [6,2] then print(ithprime(n)); fi; od; # Muniru A Asiru, Jan 19 2018

a288022[n_] := Map[Last, Select[Map[{NextPrime[#, 1], NextPrime[#, -1]}&, Range[10, n, 10]], First[#]-Last[#]==6&]]
a288022[3000] (* data *)

cp's OEIS Frontend

A104719 Concatenations of pairs of primes that differ by 10.

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A248855 a(n) is the smallest positive integer m such that if k >= m then a(k+1,n)^(1/(k+1)) <= a(k,n)^(1/k), where a(k,n) is the k-th term of the sequence {p | p and p+2n are primes}.

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A054902 Composite numbers n such that sigma(n)+12 = sigma(n+12).

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A054904 x = a(n) is the smallest composite number such that sigma(x+6n) = sigma(x)+6n, where sigma = A000203.

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A252089 Primes p such that p + 26 is prime.

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Magma

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A054903 Composite numbers n such that sigma(n)+6 = sigma(n+6), where sigma=A000203.

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A231608 Table whose n-th row consists of primes p such that p + 2n is also prime, read by antidiagonals.

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Maple

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A262086 Numbers k such that phi(k + 10) = phi(k) + 10, where phi(k) = A000010(k) is Euler's totient function.

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