A023202 -id:A023202 - cp's OEIS frontend

A054987 Smallest composite x such that sigma(x+2^n) = sigma(x) + 2^n.

434, 305635357, 27, 39, 106645, 69, 2275, 63, 6475, 249, 7735, 3703, 10803, 16383, 58869, 51181, 87951, 1695, 9579, 105237, 98829, 1143369, 789609, 11625, 14038691, 178975, 48627929, 1881333, 402373721, 2667945, 82915599, 353195221, 70106601

Offset: 1

Labos Elemer, May 29 2000

nonn

The sequence is initiated by smallest of A015915. Special primes of A023202, A049488-A049491 also satisfy the Sigma[p+2^w]=Sigma[p]+2^w relation

			For the term 69: Sigma[69+2^6] = Sigma[133] = 1+7+19+133 = Sigma[69]+64 = (1+3+23+69)+64 = 160.

Cf. A049488-A049491, A001359, A054799, A015913, A015915, A023200, A023202, A054905.

Table[ Select[ Range[ 1, 110000 ], Equal[ EulerPhi[ #+2^k ]-EulerPhi[ # ]-2^k, 0 ] &&!PrimeQ[ # ]& ], {k, 1, 22} ]

a(n)=my(N=2^n,x=3); while(isprime(x++) || sigma(x+N) != sigma(x)+N,); x \\ Charles R Greathouse IV, Mar 11 2014

More terms from Labos Elemer, Aug 14 2003

a(21) corrected and a(27)-a(33) from Donovan Johnson, Nov 30 2008

A156320 List of prime pairs of the form (p, p+8).

Original entry on oeis.org

3, 11, 5, 13, 11, 19, 23, 31, 29, 37, 53, 61, 59, 67, 71, 79, 89, 97, 101, 109, 131, 139, 149, 157, 173, 181, 191, 199, 233, 241, 263, 271, 269, 277, 359, 367, 389, 397, 401, 409, 431, 439, 449, 457, 479, 487, 491, 499, 563, 571, 569, 577, 593, 601, 599, 607, 653, 661, 683, 691, 701

Offset: 1

Vincenzo Librandi, Feb 08 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Fernando Neres de Oliveira, On the Solvability of the Diophantine Equation p^x + (p + 8)^y = z^2 when p > 3 and p + 8 are Primes, Annals of Pure and Applied Mathematics (2018) Vol. 18, No. 1, 9-13.

Cf. A023202, A092402.

Flatten[Select[{#, # + 8} &/@Prime[Range[1000]], PrimeQ[Last[#]]&]] (* Vincenzo Librandi, Nov 01 2012 *)

from sympy import isprime, primerange
for pn in primerange(1,300):
    if isprime(pn+8):
        print(pn, pn+8)
# Stefano Spezia, Dec 06 2018

a(2n+1) = A023202(n+1). a(2n+2) = A092402(n+1). [R. J. Mathar, Feb 09 2009]

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}]

A262085 Numbers n such that phi(n + 8) = phi(n) + 8 where phi(n) = A000010(n) is Euler's totient function.

Original entry on oeis.org

3, 5, 11, 23, 24, 29, 36, 42, 48, 50, 53, 56, 59, 71, 72, 80, 89, 101, 102, 125, 131, 132, 149, 173, 176, 191, 230, 233, 248, 263, 269, 359, 368, 389, 401, 431, 449, 464, 479, 491, 563, 569, 593, 599, 638, 653, 656, 683, 701, 719, 743, 761, 821, 848, 911, 929, 983

Offset: 1

Kevin J. Gomez, Sep 10 2015

Sequence includes numbers n such that n and n + 8 are both prime (A023202).
Sequence also includes numbers n equal to 8*(a Mersenne prime) (cf A000668).
Sequence also includes n such that n/16 and n/8 + 1 are both odd primes.
Contains more composites than sequences A262084 and A262086. This is most likely due to the fact that 8 is a power of 2, as in A001838.

			3 since phi(11) = phi(3) + 8 (3 and 11 are both prime).
24 is a solution since phi(32) = phi(24) + 8 (24 is 8 * 3; 3 is a Mersenne prime).

Robert Israel, Table of n, a(n) for n = 1..10000
Wikipedia, Euler's totient function

Cf. A000010.

Cf. A001838 (k=2), A056772 (k=4), A262084 (k=6), A262086 (k=10).

[n: n in [1..1000] | EulerPhi(n+8) eq EulerPhi(n)+8]; // Vincenzo Librandi, Sep 11 2015

select(t -> numtheory:-phi(t+8) = numtheory:-phi(t)+8, [$1..1000]); # Robert Israel, Mar 04 2016

Select[Range@1000, EulerPhi@(# + 8)== EulerPhi[#] + 8 &] (* Vincenzo Librandi, Sep 11 2015 *)

is(n)=eulerphi(n + 8) == eulerphi(n) + 8 \\ Anders Hellström, Sep 11 2015

[n for n in (1..1000) if euler_phi(n+8) == euler_phi(n)+8] # Bruno Berselli, Mar 04 2016

A361483 Primes p such that p + 256 is also prime.

Original entry on oeis.org

7, 13, 37, 61, 97, 103, 127, 163, 193, 211, 223, 307, 313, 331, 337, 397, 421, 463, 487, 541, 571, 601, 607, 631, 673, 691, 727, 757, 853, 907, 937, 967, 1021, 1033, 1051, 1063, 1117, 1153, 1171, 1231, 1237, 1297, 1303, 1327, 1381, 1453, 1531, 1567, 1621, 1657, 1693, 1723

Offset: 1

Elmo R. Oliveira, Mar 13 2023

All terms are == 1 (mod 6).

			61 and 61 + 256 = 317 are both prime.

Cf. A000040.

Cf. sequences of the type p + k are primes: A001359 (k = 2), A023200 (k = 4), A023202 (k = 8), A049488 (k = 16), A049489 (k = 32), A049490 (k = 64), A049491 (k = 128), this sequence (k = 256), A361484 (k = 512), A361485 (k = 1024).

A307563 Numbers k such that both 6k - 1 and 6k + 7 are prime.

Original entry on oeis.org

1, 2, 4, 5, 9, 10, 12, 15, 17, 22, 25, 29, 32, 39, 44, 45, 60, 65, 67, 72, 75, 80, 82, 94, 95, 99, 100, 109, 114, 117, 120, 124, 127, 137, 152, 155, 164, 169, 172, 177, 185, 194, 199, 204, 205, 214, 215, 220, 229, 240, 242, 247, 254, 260, 262, 267, 269, 270, 289, 304, 312, 330, 334, 347, 355, 359, 369, 374, 379, 389

Offset: 1

Sally Myers Moite, Apr 14 2019

nonn

There are 140 such numbers between 1 and 1000.
These numbers correspond to all the prime pairs which differ by 8 except 3 and 11.
Numbers in this sequence are those which are not 6cd - c - d - 1, 6cd + c - d, 6cd - c + d or 6cd + c + d - 1, that is, they are not (6c - 1)d - c - 1, (6c - 1)d + c, (6c + 1)d - c or (6c + 1)d + c - 1.

			a(4) = 5, so 6(5) - 1 = 29 and 6(5) + 7 = 37 are both prime.

Robert Israel, Table of n, a(n) for n = 1..10000
Sally M. Moite, “Primeless” Sieves for Primes and for Prime Pairs Which Differ by 2m, vixra:1903.0353 (2019).

The primes are A023202, A092402, A031926.
Similar sequences for twin primes are A002822, A067611, for "cousin" primes A056956, A186243.
Intersection of A024898 and A153218.
Cf. also A307561, A307562.

select(t -> isprime(6*t-1) and isprime(6*t+7), [$1..500]); # Robert Israel, May 27 2019

isok(n) = isprime(6*n-1) && isprime(6*n+7); \\ Michel Marcus, Apr 16 2019

A361484 Primes p such that p + 512 is also prime.

Original entry on oeis.org

11, 29, 59, 89, 101, 107, 131, 149, 179, 197, 227, 239, 257, 311, 317, 347, 479, 509, 521, 557, 617, 641, 659, 701, 719, 809, 887, 911, 941, 947, 971, 977, 1019, 1031, 1097, 1109, 1151, 1181, 1187, 1229, 1277, 1289, 1319, 1361, 1367, 1439, 1481, 1487, 1499, 1571, 1601

Offset: 1

Elmo R. Oliveira, Mar 13 2023

All terms are == 5 (mod 6).

			59 and 59 + 512 = 571 are both prime.

Cf. A000040.

Cf. sequences of the type p + k are primes: A001359 (k = 2), A023200 (k = 4), A023202 (k = 8), A049488 (k = 16), A049489 (k = 32), A049490 (k = 64), A049491 (k = 128), A361483 (k = 256), this sequence (k = 512), A361485 (k = 1024).

A361679 A(n,k) is the n-th prime p such that p + 2^k is also prime; square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

3, 3, 5, 3, 7, 11, 3, 5, 13, 17, 5, 7, 11, 19, 29, 3, 11, 13, 23, 37, 41, 3, 7, 29, 31, 29, 43, 59, 7, 11, 19, 41, 37, 53, 67, 71, 11, 13, 23, 37, 47, 43, 59, 79, 101, 7, 29, 37, 29, 43, 71, 67, 71, 97, 107, 5, 37, 59, 61, 53, 67, 107, 73, 89, 103, 137

Offset: 1

Alois P. Heinz, Mar 20 2023

			Square array A(n,k) begins:
    3,   3,   3,   3,   5,   3,   3,   7,  11,   7, ...
    5,   7,   5,   7,  11,   7,  11,  13,  29,  37, ...
   11,  13,  11,  13,  29,  19,  23,  37,  59,  67, ...
   17,  19,  23,  31,  41,  37,  29,  61,  89,  73, ...
   29,  37,  29,  37,  47,  43,  53,  97, 101,  79, ...
   41,  43,  53,  43,  71,  67,  71, 103, 107, 127, ...
   59,  67,  59,  67, 107,  73,  83, 127, 131, 139, ...
   71,  79,  71,  73, 131, 103, 101, 163, 149, 157, ...
  101,  97,  89,  97, 149, 109, 113, 193, 179, 163, ...
  107, 103, 101, 151, 167, 127, 149, 211, 197, 193, ...

Alois P. Heinz, Antidiagonals n = 1..200, flattened

Columns k=1-10 give: A001359, A023200, A023202, A049488, A049489, A049490, A049491, A361483, A361484, A361485.
Row n=1 gives A056206.
Main diagonal gives A361680.
Cf. A000040.

A:= proc() option remember; local f; f:= proc() [] end;
      proc(n, k) option remember; local p;
        p:= `if`(nops(f(k))=0, 1, f(k)[-1]);
        while nops(f(k))

A046141 p, p+8 and p+12 are primes.

Original entry on oeis.org

5, 11, 29, 59, 71, 89, 101, 269, 389, 431, 449, 479, 491, 761, 929, 1289, 1439, 1481, 1559, 1571, 1601, 2129, 2339, 2381, 2531, 2609, 2699, 2741, 2789, 3011, 3209, 3449, 3911, 4721, 5471, 5519, 5639, 5849, 6569, 6899, 6959

Offset: 1

Eric W. Weisstein

nonn

R. J. Mathar, Table of n, a(n) for n = 1..3766
Eric Weisstein's World of Mathematics, Prime Triplet.

Cf. A023202, A046133.

Select[Range@ 7000, AllTrue[{#, # + 8, # + 12}, PrimeQ] &] (* Michael De Vlieger, Jul 24 2015, Version 10 *)
Select[Prime[Range[1000]],AllTrue[#+{8,12},PrimeQ]&] (* Harvey P. Dale, Oct 07 2023 *)

lista(nn) = forprime(p=2, nn, if (isprime(p+8) && isprime(p+12), print1(p, ", "))); \\ Michel Marcus, Jul 24 2015

A136208 Primes p such that p-8 or p+8 is prime.

Original entry on oeis.org

3, 5, 11, 13, 19, 23, 29, 31, 37, 53, 59, 61, 67, 71, 79, 89, 97, 101, 109, 131, 139, 149, 157, 173, 181, 191, 199, 233, 241, 263, 269, 271, 277, 359, 367, 389, 397, 401, 409, 431, 439, 449, 457, 479, 487, 491, 499, 563, 569, 571, 577, 593, 599, 601, 607, 653

Offset: 1

Carlos Alves, Dec 21 2007, Dec 22 2007

nonn

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

Cf. A136207, A023202.

Select[Prime[Range[150]], PrimeQ[ # - 8] || PrimeQ[ # + 8] &] (* Stefan Steinerberger, Dec 22 2007 *)
Select[Prime[Range[150]],AnyTrue[#+{8,-8},PrimeQ]&] (* Harvey P. Dale, Jul 12 2022 *)

More terms from Stefan Steinerberger, Dec 22 2007

cp's OEIS Frontend

A054987 Smallest composite x such that sigma(x+2^n) = sigma(x) + 2^n.

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A156320 List of prime pairs of the form (p, p+8).

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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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A262085 Numbers n such that phi(n + 8) = phi(n) + 8 where phi(n) = A000010(n) is Euler's totient function.

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A361483 Primes p such that p + 256 is also prime.

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A307563 Numbers k such that both 6k - 1 and 6k + 7 are prime.

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A361484 Primes p such that p + 512 is also prime.

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A361679 A(n,k) is the n-th prime p such that p + 2^k is also prime; square array A(n,k), n>=1, k>=1, read by antidiagonals.

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