A172367 -id:A172367 - cp's OEIS frontend

A246562 Primes p such that 4+p, 4+p^2, 4+p^3, 4+p^5, and 4+p^7 are all prime.

7, 469363, 2552713, 3378103, 6595597, 6629683, 39837517, 46024063, 46167307, 97371007, 97629403, 105528217, 136983307, 169483033, 202953613, 213792193, 216520987, 216738043, 221705647, 304033927, 317502193, 359133553

Offset: 1

Zak Seidov, Aug 29 2014

nonn

All terms are == {3, 7} mod 10. Subsequence of A246519.

Zak Seidov, Table of n, a(n) for n = 1..60

Cf. A007591, A073573, A125260, A172367, A246519.

Select[Prime[Range[193*10^5]],AllTrue[#^{1,2,3,5,7}+4,PrimeQ]&] (* Harvey P. Dale, Sep 07 2024 *)

forprime(p=1,10^9,if(ispseudoprime(4+p) && ispseudoprime(4+p^2) && ispseudoprime(4+p^3) && ispseudoprime(4+p^5) && ispseudoprime(4+p^7), print1(p,", "))) \\ Derek Orr, Aug 30 2014

A200050 a(2) = 1, then (p-1)*(p-4)/2, with p = prime(n), n > 2.

Original entry on oeis.org

1, 2, 9, 35, 54, 104, 135, 209, 350, 405, 594, 740, 819, 989, 1274, 1595, 1710, 2079, 2345, 2484, 2925, 3239, 3740, 4464, 4850, 5049, 5459, 5670, 6104, 7749, 8255, 9044, 9315, 10730, 11025, 11934, 12879, 13529, 14534, 15575, 15930, 17765, 18144, 18914, 19305

Offset: 2

Arkadiusz Wesolowski, Apr 16 2012

nonn

Record values in A192599. The index sequence of this one is 1, 2, 3, 6, 7, 9, 11, 13, 17, 18, 21, 23, 25, 29, 31, 34, 36, 40, 42, 45, 47, 50, 52, 56, 58, 61.

			A192599(13) = 209 since A192599(17) = 350 is the next record value.

Arkadiusz Wesolowski, Table of n, a(n) for n = 2..10000

Cf. A000096, A014107, A192599.

[(p-1)*Abs(p-4)/2: p in [NthPrime(n+1): n in [1..45]]]

Table[p = Prime[n + 1]; (p - 1)*Abs[p - 4]/2, {n, 45}]
Join[{1},((#-1)(#-4))/2&/@Prime[Range[3,50]]] (* Harvey P. Dale, Aug 04 2020 *)

vector(45, n, p=prime(n+1); (p-1)*abs(p-4)/2)

a(2) = 1, a(n) = A006093(n)*A172367(n-2)/2.

A094069 Prime(p)-4 for primes p such that prime(p) - 4 is prime.

Original entry on oeis.org

7, 13, 37, 79, 349, 397, 457, 613, 769, 1213, 1429, 1783, 2347, 2377, 2473, 3907, 4129, 4513, 4783, 5437, 5647, 7477, 7603, 7879, 8803, 9829, 10429, 10453, 10627, 11443, 11863, 11923, 12109, 13033, 13099, 13327, 14173, 14779, 15679, 16057, 16069

Offset: 1

Cino Hilliard, May 31 2004

nonn

Primes q such that q+4 is in A006450. - Robert Israel, Nov 09 2020

			Prime(13) = 41. 41 - 4 = 37.

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

Cf. A006450.

This is a subsequence of A172367.

R:= NULL: p:= 7:
for i from 5 to 10000 do
  q:= p; p:= nextprime(p);
  if isprime(i) and q=p-4 then R:= R, q fi;
od:
R; # Robert Israel, Nov 09 2020

Prime[#]&/@Select[Prime[Range[2,300]],PrimeQ[Prime[#]-4]&]-4 (* Harvey P. Dale, May 21 2021 *)

a(n) = { forprime(x=2,n, y=prime(x)- 4; if(isprime(y),print1(y",")) ) }

Definition corrected by Robert Israel, Nov 09 2020

A157678 Numbers k such that k + floor(average of digits of k) is prime.

Original entry on oeis.org

1, 12, 16, 27, 30, 34, 38, 41, 56, 63, 67, 70, 74, 89, 92, 96, 101, 102, 105, 107, 112, 125, 128, 130, 136, 146, 147, 154, 161, 164, 168, 175, 186, 188, 190, 193, 208, 210, 219, 226, 229, 231, 236, 237, 247, 254, 258, 265, 273, 276, 278, 280, 290, 305, 308, 309

Offset: 1

Kyle D. Balliet, Mar 04 2009

			n = 89 -> 89 + floor((8+9)/2) = 89 + 8 = 97 (prime).
n = 190 -> 190 + floor((1+9+0)/3) = 190 + 3 = 193 (prime).

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

Cf. A172367.

a := proc (n) local nn: nn := convert(n, base, 10): if isprime(n+floor(add(nn[j], j = 1 .. nops(nn))/nops(nn))) = true then n else end if end proc: seq(a(n), n = 1 .. 350); # Emeric Deutsch, Mar 07 2009

is(n)=isprime(sumdigits(n)\#digits(n)+n) \\ Charles R Greathouse IV, Dec 27 2013

Corrected and extended by Emeric Deutsch, Mar 07 2009

A243095 Least integer m > 1 such that 4 + m^n is prime or 1 if only 4 + 1^n is prime.

Original entry on oeis.org

3, 3, 3, 1, 7, 3, 7, 1, 3, 3, 9, 1, 33, 7, 9, 1, 43, 17, 27, 1, 9, 3, 7, 1, 55, 47, 285, 1, 27, 3, 39, 1, 43, 117, 163, 1, 63, 255, 15, 1, 87, 3, 43, 1, 187, 77, 37, 1, 105, 45, 25, 1, 99, 305, 79, 1, 3, 27, 903, 1, 127, 293, 255, 1, 27, 27, 435, 1, 207, 143, 127, 1, 117, 295, 1159, 1, 477

Offset: 1

Zak Seidov, Aug 29 2014

nonn

If n is a multiple of 4 then 4 + m^n is prime iff m = 1.

4 + m^(4*x) = (m^(2*x)-2*m^x+2) * (m^(2*x)+2*m^x+2). - Jens Kruse Andersen, Sep 02 2014

Zak Seidov and Jens Kruse Andersen, Table of n, a(n) for n = 1..1000 (first 200 terms from Zak Seidov)

Cf. A007591, A073573, A125260, A172367, A246519.

a(n)=if(n%4==0,return(1));m=2;while(!ispseudoprime(4+m^n),m++);return(m)
vector(100,n,a(n)) \\ Derek Orr, Aug 29 2014

A275115 Least prime of the form x^2 + n*y^2 with x>0 and y>0.

Original entry on oeis.org

2, 3, 7, 5, 29, 7, 11, 17, 13, 11, 47, 13, 17, 23, 19, 17, 53, 19, 23, 29, 37, 23, 59, 73, 29, 107, 31, 29, 173, 31, 47, 41, 37, 43, 71, 37, 41, 47, 43, 41, 173, 43, 47, 53, 61, 47, 83, 73, 53, 59, 67, 53, 89, 79, 59, 137, 61, 59, 317, 61, 97, 71, 67, 73, 101, 67, 71, 149, 73, 71

Offset: 1

Zak Seidov and Robert G. Wilson v, Jul 17 2016

nonn

Neither x nor y can be zero because the remaining part of the form would then be composite.
a(n) > n.
The differences, d, between a(n) and n are 1, 4, 9, 16, 24, 25, 36, 49, 64, 81, 100, 121, 132, 144, 169, 196, 225, 256, 258, 289, 324, 361, 400, 441, ..., .
Not all 'd's are squares, such as 24, 132, 258, 1032, 1167, 1518, 2103, 2472, 2652, 2706, 5793. It is conjectured that this list is complete.
d=1 for A006093;
d=4 for A172367;
d=9 for n: 8, 14, 20, 32, 34, 38, 44, 50, 62, 64, 74, 80, 92, 94, 98, 104, 118, 122, 128, 140, 142, 154, 158, ..., ;
d=16 for n: 21, 31, 45, 51, 73, 81, 87, 91, 111, 115, 121, 141, 151, 157, 165, 181, 183, 211, 213, 217, 241, ..., ;
d=25 for n: 48, 54, 76, 84, 114, 124, 132, 168, 174, 186, 204, 208, 216, 244, 246, 252, 258, 286, 288, 324, ..., ;
d=36 for n: 11, 17, 23, 35, 47, 53, 61, 65, 71, 77, 95, 101, 113, 131, 137, 143, 155, 161, 191, 197, 203, 205, ..., ;
d=49 for n: 24, 90, 144, 234, 264, 300, 318, 360, 390, 450, 472, 492, 528, 550, 558, 564, 624, 670, 678, 712, ..., ;
and for the nonsquare differences of 24, 132, 258, 1032, 1167, 1518, 2103, 2472, 2652, 2706 and 5793l, their n's are 5, 41, 59, 341, 314, 479, 626, 749, 881, 755 and 1784, respectively.
Least n that has as its difference k^2: 1, 3, 8, 21, 48, 11, 24, 117, 26, 139, 120, 29, 294, 201, 134, 621, 468, 179, 792, 1269, 356, 1249, 754, 251, 696, ..., .

			a(1) = 2 since it equals 1^2+1*1^2;
a(2) = 3 since it equals 1^2+2*1^2;
a(3) = 7 since it equals 2^2+3*1^2;
a(4) = 5 since it equals 1^2+4*1^2;
a(5) = 29 since it equals 3^2+5*2^2; etc.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Zak Seidov, First ten primes of the form x^2+n*y^2 with x>=0, y>=0, n=1..1000.

Cf. A002350, A232174, A212602, A212603, A212604, A212605, A244030, A244031, A006093, A172367.

f[n_] := Block[{p = NextPrime@ n, y}, While[y = 1; While[p > n*y^2 && !IntegerQ[ Sqrt[p - n*y^2]], y++]; !IntegerQ[ Sqrt[p - n*y^2]], p = NextPrime@ p]; p]; Array[f, 70]

a(n)=if(n==1, return(2)); my(best,x=1+n%2,t); while(!isprime(best=x^2+n), x += 2); for(y=2,sqrtint((best-2)\n), t=best-n*y^2; if(t<1, return(best)); for(x=1,sqrtint(t), if(isprime(t=x^2+n*y^2) && tCharles R Greathouse IV, Jul 17 2016

a(n-1) = n iff n is prime.

A227280 Values of the difference d for 12 primes in geometric-arithmetic progression with the minimal sequence {1313^j + jd}, j = 0 to 11.

Original entry on oeis.org

81647160420, 170655787050, 211212209880, 227961624450

Offset: 1

Sameen Ahmed Khan, Jul 05 2013

nonn

Primality requires d to be multiple of 7# = 2*3*5*7 = 210.

Fifth term is > (1600*10^6)*(210) = 336000000000.

			d = 170655787050 then {13*13^j + j*d}, j = 0 to 11, is {13, 170655787219, 341311576297, 511967389711, 682623519493, 853283762059, 1023997470817, 1195406240071, 1375850795773, 1673760575299, 3498718264537, 25175298780031}, which is 12 primes in geometric-arithmetic progression.

Sameen Ahmed Khan, Primes in Geometric-Arithmetic Progression, arXiv:1203.2083v1 [math.NT], (Mar 09 2012).

Cf. A172367, A209202, A209203, A209204, A209205, A209206, A209207, A209208, A209209, A209210.

Clear[p]; p = 13; gapset12d = {}; Do[If[PrimeQ[{p, p*p + d, p*p^2 + 2*d, p*p^3 + 3*d, p*p^4 + 4*d, p*p^5 + 5*d, p*p^6 + 6*d, p*p^7 + 7*d, p*p^8 + 8*d, p*p^9 + 9*d, p*p^10 + 10*d, p*p^11 + 11*d}] == {True, True, True, True, True, True, True, True, True, True, True, True}, AppendTo[gapset12d, d]], {d, 2, 10^11, 2}]; gapset12d

A303811 Least k such that A006666(k)/A006667(k) = prime(n).

Original entry on oeis.org

159, 6, 10, 40, 640, 2560, 40960, 163840, 2621440, 167772160, 671088640, 42949672960, 687194767360, 2748779069440, 43980465111040, 2814749767106560, 180143985094819840, 720575940379279360, 46116860184273879040, 737869762948382064640, 2951479051793528258560

Offset: 1

Michel Lagneau, Sep 10 2018

nonn

A006666 and A006667 are respectively the number of halving and tripling steps in the '3x+1' problem.

For n > 2, it seems that a(n) is of the form a(n) = 5*2^q with q = 1, 3, 7, 9, 13, 15, 19, 25, 27, 33, 37, 39, 43, 49, 55, 57, 63, 67, 69, ... (Numbers q such that q+4 is prime: A172367)

			a(4) = 40 because A006666(40)/A006667(40) = 7/1 = prime(4).

Cf. A006577, A006666, A006667, A172367, A287798.

nn:=10^20:
for n from 1 to 10 do:
ii:=0:
   for k from 1 to nn while(ii=0) do:
    it0:=0:it1:=0:m:=k:
      for i from 1 to nn while(m<>1) do:
        if irem(m, 2)=0
         then
         m:=m/2:it0:=it0+1:
         else
         m:=3*m+1:it1:=it1+1:
       fi:
     od:
      if it1<>0 and it0/it1 = ithprime(n)
       then
       ii:=1:printf(`%d %d \n`,n,k):
       else
     fi:
od:
od:

cp's OEIS Frontend

A246562 Primes p such that 4+p, 4+p^2, 4+p^3, 4+p^5, and 4+p^7 are all prime.

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A200050 a(2) = 1, then (p-1)*(p-4)/2, with p = prime(n), n > 2.

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Magma

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A094069 Prime(p)-4 for primes p such that prime(p) - 4 is prime.

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Maple

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A157678 Numbers k such that k + floor(average of digits of k) is prime.

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Maple

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Extensions

A243095 Least integer m > 1 such that 4 + m^n is prime or 1 if only 4 + 1^n is prime.

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PARI

A275115 Least prime of the form x^2 + n*y^2 with x>0 and y>0.

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A227280 Values of the difference d for 12 primes in geometric-arithmetic progression with the minimal sequence {13*13^j + j*d}, j = 0 to 11.

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Mathematica

A303811 Least k such that A006666(k)/A006667(k) = prime(n).

A227280 Values of the difference d for 12 primes in geometric-arithmetic progression with the minimal sequence {1313^j + jd}, j = 0 to 11.