A054906 -id:A054906 - cp's OEIS frontend

A020483 Least prime p such that p+2n is also prime.

2, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 7, 5, 3, 3, 7, 5, 3, 5, 3, 3, 5, 3, 7, 5, 3, 7, 5, 3, 3, 7, 5, 3, 5, 3, 3, 7, 5, 3, 5, 3, 7, 5, 3, 13, 7, 5, 3, 5, 3, 3, 5, 3, 3, 5, 3, 19, 13, 11, 13, 7, 5, 3, 5, 3, 7, 5, 3, 3, 11, 11, 7, 5, 3, 3, 7, 5, 3, 7, 5, 3, 5, 3, 7, 5, 3, 7, 5, 3, 3, 11, 11, 7, 5, 3, 3, 5, 3, 3, 13, 11, 31, 7

Offset: 0

David W. Wilson

nonn

It is conjectured that a(n) always exists. a(n) has been computed for n < 5 * 10^11, with largest value a(248281210271) = 3307. - Jens Kruse Andersen, Nov 28 2004

If a(n) = a(n+1) = k, then 2*n + k and 2*(n+1) + k are twin primes. - Ya-Ping Lu, Sep 22 2020

			Given n = 2, we see that 2 + 2n = 6 = 2 * 3, but 3 + 2n = 7, which is prime, so a(2) = 3.
Given n = 3, we see that 2 + 2n = 8 = 2^3 and 3 + 2n = 9 = 3^2, but 5 + 2n = 11, which is prime, so a(3) = 5.

T. D. Noe, Table of n, a(n) for n = 0..10000
Jens Kruse Andersen, Prime gaps (not necessarily consecutive), Yahoo! group "primenumbers", Nov 26 2004.
Jens Kruse Andersen, Mike Oakes, Ed Pegg Jr, Prime gaps (not necessarily consecutive), digest of 5 messages in primenumbers Yahoo group, Nov 26 - Nov 27, 2004.

Cf. A087711, A101042, A101043, A101044, A101045, A101046.
Cf. A101045, A239392 (record values).
Cf. A000040, A010051, A020484, A237055.
It is likely that A054906 is an identical sequence, although this seems to have not yet been proved. - N. J. A. Sloane, Feb 06 2017

P:=Filtered([1..10000],IsPrime);;
a:=List(List([0..110],n->Filtered(P,i->IsPrime(i+2*n))),Minimum); # Muniru A Asiru, Mar 26 2018

a020483 n = head [p | p <- a000040_list, a010051' (p + 2 * n) == 1]
-- Reinhard Zumkeller, Nov 29 2014

A020483 := proc(n)
    local p;
    p := 2;
    while true do
        if isprime(p+2*n) then
            return p;
        end if;
        p := nextprime(p) ;
    end do:
end proc:
seq(A020483(n),n=0..40); # R. J. Mathar, Sep 23 2016

Table[j = 1; found = False; While[!found, j++; found = PrimeQ[Prime[j] + 2i]]; Prime[j], {i, 200}]
leastPrimep2n[n_] := Block[{k = 1, p, q = 2 n}, While[p = Prime@k; !PrimeQ[p + q], k++]; p]; Array[leastPrimep2n, 102] (* Robert G. Wilson v, Mar 26 2008 *)

a(n)=forprime(p=2,,if(isprime(p+2*n), return(p))) \\ Charles R Greathouse IV, Mar 19 2014

If a(n) exists, a(n) < 2n, which of course is a great overestimate. - T. D. Noe, Jul 16 2002
a(n) = A087711(n) - n. - Zak Seidov, Nov 28 2007
a(n) = A020484(n) - 2n. - Zak Seidov, May 29 2014
a(n) = 2 if and only if n = 0. - Alonso del Arte, Mar 14 2018

a(0)=2 added by N. J. A. Sloane, Apr 25 2015

A015886 a(n) = smallest number k such that sigma(k + n) = sigma(k) + n, or -1 if no such number exists.

Original entry on oeis.org

1, 2, 3, 2, 3, 2, 5, 74, 3, 2, 3, 2, 5, 4418, 3, 2, 3, 2, 5, 6, 3, 2, 7

Offset: 0

Robert G. Wilson v

There are solutions to sigma(k)+n=sigma(k+n) whenever n is the difference between two primes (A030173), e.g. k and k+n are primes. There are other values of n that have solutions (see example).
a(23) > 4292000000, if it exists. - Jud McCranie, Jan 05 2000
The sequence begins: 1, 2, 3, 2, 3, 2, 5, 74, 3, 2, 3, 2, 5, 4418, 3, 2, 3, 2, 5, 6, 3, 2, 7, ?, 5, ?, 3, 2, 3, 2, 7, ?, 5, 18, 3, 2, 5, 44, 3, 2, 3, 2, 5, ?, 3, 2, 7, ?, 5, 3315, 3, 2, 7, ?, 5, ?, 3, 2, 3, 2, 7, ?, 5, ?, 3, 2, 5, ?, 3, 2, 3, 2, 7, 18, 5, ?, 3, 2, 5, ?, 3, 2, 7, ?, 5, ?, 3, 2, 13, ?, 7, ?, 5, 32, 3, 2, 5 where the other missing terms (designated by "?") are > 10^9, if they exist. - Jud McCranie, Jan 08 2000
The "other" values of n are the odd n such that n+2 is not prime. For these n, in order for sigma(k) or sigma(n+k) to be odd, either k or n+k must be a square or twice a square. Examples: for n=7, n+k=9^2; for n=13, k=2*47^2 and for n=19, n+k=5^2. Using this idea, it is easy to show that if a(23) exists it is greater than 10^12. - T. D. Noe, Sep 24 2007

			sigma(74+7) = 121 = sigma(74)+7, so a(7) = 74.

Cf. A000203, A020483, A030173, A054906.

Table[k=1; While[DivisorSigma[1,k+n] != DivisorSigma[1,k]+n, k++ ]; k, {n,0,22}] (* T. D. Noe, Sep 24 2007 *)

a(n) = {my(k=1); while(sigma(k+n) != sigma(k) + n, k++); k;} \\ Michel Marcus, May 23 2018

a(2n) = A020483(n) = A054906(n) - T. D. Noe, Sep 24 2007

A055458 a(n) = smallest composite solution x to the equation phi(x+2n) = phi(x)+2n.

Original entry on oeis.org

6, 12, 21, 24, 36, 45, 48, 39, 63, 72, 72, 95, 60, 57, 224, 84, 15, 135, 1058, 45, 301, 144

Offset: 1

Labos Elemer, Jun 26 2000

Sivaramakrishnan (1989) quotes Makowski, who gave solutions for phi(x+d) = phi(x)+d with d = 2^a and d = 2*3^a. Compare also A007694 and A049237.
Smallest prime solutions appear to be identical with A054906.
a(23) is presently unknown.
The sequence continues as (with ? for unknown values): ?, 95, 162, 63, 189, 69, 156, 161, 180, 69, 260, 150, ?, 115, 204, 129, 400, 75, 180, 165, 35, 117, 476, 7105, 288, 195, ?, 324, 620, 240, 81, 145, 14531, 153, 644, 309, ?, 203, ?, 63, 640, 75, 372, 285, 2312, 33, 343, 642, 336, 155, ?, 147, 728, 396, 1564, 185, 564, 87, 567, 360, 360, 155, 492, 510, 560, 516, 516, 301, 4232, 261, 860, 387, 576, 185, 564, 309, 1000, 225 ... - Don Reble, Apr 29 2015

			a(19) = 1058 because phi(1058 + 38) = phi(1096) = 544 = 506 + 38 = phi(1058) + 38.
a(100) = 225, phi(225 + 200) = phi(425) = 320 = 120 + 200 = phi(225) + 200.

Sivaramakrishnan, R. (1989): Classical theory of Arithmetical Functions. Marcel Dekker, Inc., New York-Basel. Chapter V, Problem 20, page 113.

Cf. A000010, A054906, A050472, A050473, A007694, A049237.

A055458 := proc(n)
    local x;
    for x from 0 do
        if not isprime(x) then
        if numtheory[phi](x+2*n) = numtheory[phi](x)+2*n then
            return x;
        end if;
        end if;
    end do:
end proc: # R. J. Mathar, Sep 23 2016

Table[k = 4; While[Nand[CompositeQ@ k, EulerPhi[k + 2 n] == EulerPhi[k] + 2 n], k++]; k, {n, 22}] (* Michael De Vlieger, Dec 17 2016 *)

a(n)=forcomposite(x=4,, if(eulerphi(x+2*n) == eulerphi(x)+2*n, return(x))) \\ does not handle -1s; Charles R Greathouse IV, Apr 28 2015

More terms from Michel ten Voorde Jun 14 2003

Entry revised by N. J. A. Sloane, Apr 28 2015

A084293 a(n) = 2n + A054905(n).

Original entry on oeis.org

436, 305635361, 110, 35, 195566, 77, 26, 55, 38, 76, 938, 104, 212308, 85, 74, 106677, 86, 161

Offset: 1

Labos Elemer, May 26 2003

The sequence begins 436, 305635361, 110, 35, 195566, 77, 26, 55, 38, 76, 938, 104, 212308, 85, 74, 106677, 86, 161, ?, 91, 87, 92, 122, 111, 1585396, 145, 94, 76627, 10283, 159, 772, 133, 122, 412, 194, 142, 964, 205, 374, 925, 6725, 209, ?, 1015, 178, ?, ?, 206, 146, ?, ..., where the other missing terms (designated by "?") are unknown, if they exist (see also A206768).

			To terms of A054905, where sigma(x+2n)=sigma(x)+2n replacing x+2n=y,x=y-2n, we get sigma(y)-2n=sigma(y-2n);
For several analogous sequences, the corresponding "mirror-solutions" can be easily constructed. See: e.g. A015913-A015918; A050507, A054799, A054903-A054906; A054982-A054987; A059118; A055009, A055458, A063500, etc.

Cf. A054905.

Composite x satisfying sigma(x-2n) = sigma(x) - 2n.

A084292 a(n) = 6n + A054904(n).

Original entry on oeis.org

110, 77, 38, 104, 74, 161, 87, 111, 94, 159, 122, 142, 374, 209, 178, 206, 206, 253, 326, 302, 206, 302, 471, 249, 519, 341, 346, 303, 354, 481, 542, 377, 2057, 533, 386, 411, 5138, 662, 846, 527, 386, 437, 1034, 519, 794, 689, 626, 493, 566, 629, 873, 527, 638

Offset: 1

Labos Elemer, May 26 2003

nonn

Composite solutions y to sigma(y-6n) = sigma(y) - 6n. For terms x of A054904, where sigma(x+6n) = sigma(x) + 6n, replacing x+6n = y, x = y-6n, we get sigma(y) - 6n = sigma(y-6n).

Amiram Eldar, Table of n, a(n) for n = 1..724

Cf. A000203 (sigma), A054904, A084293.

For several analogous sequences such corresponding "mirror-solutions" can be easily constructed. See, e.g., A015913-A015918, A050507, A054799, A054903-A054906, A054982-A054987, A059118, A055009, A055458, A063500, etc.

A206770 Smallest number k such that sigma(k-2n)=sigma(k)-2n.

Original entry on oeis.org

5, 7, 11, 11, 13, 17, 17, 19, 23, 23, 29, 29, 21, 31, 37, 37, 37, 41, 28, 33, 47, 47, 53, 53, 53, 59, 59, 44, 61, 67, 67, 67, 71, 57, 73, 79, 79, 79, 83, 83, 69, 89, 74, 101, 68, 97, 97, 85, 101, 103, 107, 107, 109, 113, 93, 131, 127, 127, 131, 127, 127, 127

Offset: 1

Paolo P. Lava, Jan 10 2013

nonn

Note all k>=1 are considered, even if k-2n<0. If the search space is k>=2n, variants of A020484 and A060264 appear. - R. J. Mathar, Jan 12 2013

			a(15)=37 because 37 is the minimum number for which sigma(37-2*15)=sigma(7)=8 and sigma(37)-2*15=38-30=8.

Cf. A054906

A206770:=proc(q)
local k,n;
for n from 1 to q do
for k from 1 to q do
  if sigma(k-2*n)=sigma(k)-2*n then print(k); break; fi;
od; od; end:
A206770(1000000000);
A206770 := proc(n)
    local k ;
    for k from 1 do
        if numtheory[sigma](k-2*n) = numtheory[sigma](k)-2*n then
            return k;
        end if;
    end do:
end proc: # R. J. Mathar, Jan 12 2013

cp's OEIS Frontend

A020483 Least prime p such that p+2n is also prime.

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A015886 a(n) = smallest number k such that sigma(k + n) = sigma(k) + n, or -1 if no such number exists.

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A055458 a(n) = smallest composite solution x to the equation phi(x+2n) = phi(x)+2n.

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A084293 a(n) = 2n + A054905(n).

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A084292 a(n) = 6n + A054904(n).

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A206770 Smallest number k such that sigma(k-2*n)=sigma(k)-2*n.

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Maple

A206770 Smallest number k such that sigma(k-2n)=sigma(k)-2n.