A054905 -id:A054905 - cp's OEIS frontend

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

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.

A054906 Smallest number x such that sigma(x+2n) = sigma(x)+2n (first definition).

Original entry on oeis.org

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

Offset: 1

Labos Elemer, May 23 2000

nonn

Least (prime) solutions for phi(x+2n)=phi(x)+2n seems to be identical to this sequence, while prime solutions are indeed identical to this sequence.
2nd definition = smallest number x such that phi(x+2n)=phi(x)+2n.
3rd definition = smallest primes p such that p+2n=q prime (A020483).
The 3 definitions are identical or conjectured to be identical.
The definitions are not identical if we do not take the smallest numbers. These smallest solutions are believed to be always prime numbers.
Duplicate of A020483, assuming that the 3rd definition is also correct. - R. J. Mathar, Apr 26 2015
If it can be proved that all these definitions are identical, then this entry should be merged with A020483. - N. J. A. Sloane, Feb 06 2017

			n-th primes 2,3,5,7,11,13, are solutions to sigma(x+2n)=2n+sigma(x) at 2n=2,6,22,116,88.

Sivaramakrishnan,R.(1989):Classical Theory of Arithmetical Functions. Marcel Dekker,Inc., New York.

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

Cf. A023200-A023203, A015913-A015917, A000203, A000010, A020483.

A054906 := proc(n)
    local x;
    for x from 0 do
        if numtheory[sigma](x+2*n) = numtheory[sigma](x)+2*n then
            return x;
        end if;
    end do:
end proc:
seq(A054906(n),n=1..40); # R. J. Mathar, Sep 23 2016

Table[x = 1; While[DivisorSigma[1, x + 2 n] != DivisorSigma[1, x] + 2 n, x++]; x, {n, 100}] (* Michael De Vlieger, Feb 05 2017 *)

a(n) = my(x = 1); while(sigma(x+2*n) != sigma(x)+2*n, x++); x; \\ Michel Marcus, Dec 17 2013

Minimal solutions to A000203(x+2n)=A000203(x)+2n or to A000010(x+2n)=A000010(x)+2n or to p+2n=q; p, q primes, a(n)=p.

a(n) <= A054905(n). - R. J. Mathar, Apr 28 2015

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.

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

Original entry on oeis.org

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

A054982 a(n) = least composite number such that sigma(a(n)+n!) = sigma(a(n))+n! where sigma() = A000203.

Original entry on oeis.org

434, 104, 80, 182, 427, 1727, 4147, 7163, 42031, 165841, 569257, 2683909, 10040081, 39094849, 155533969, 717519401, 3041377519, 16076525809, 71749935913

Offset: 2

Labos Elemer, May 29 2000

a(21) <= 328823468719, a(22) <= 1542201899569, a(23) <= 9325753929619. - Donovan Johnson, Sep 22 2013

			a(7) = 1727 = 11*157, 4 divisors, sigma(1727)+5040 = 1896+5040 = 6936, sigma(1727+5040) = sigma(6767) = 1+67+101+6767 = 6936.
a(2) = A054799(24) = 434, a(3) = A015914(19) = 104, the first composites in that series.

Cf. A054904, A054905, A054799, A015914, A033560.

L = {}; Do[i = 1; While[ ! ((Plus @@ Divisors[i + j! ] == j! + Plus @@ Divisors[i]) && ! PrimeQ[i]), i++ ]; L = Append[L, i], {j, 2, 13}]; L (from Vit Planocka)

More terms from Vit Planocka (planocka(AT)mistral.cz), Sep 22 2003
a(14)-a(19) from Donovan Johnson, Nov 30 2008
a(20) from Donovan Johnson, Sep 19 2013

A063500 a(n) is the least composite solution of phi(k + 6n) = phi(k) + 6n.

Original entry on oeis.org

21, 45, 63, 95, 224, 135, 301, 95, 189, 161, 260, 115, 400, 165, 476, 195, 620, 145, 644, 203, 640, 285, 343, 155, 728, 185, 567, 155, 560, 301, 860, 185, 1000, 329, 1892, 235

Offset: 1

Labos Elemer, Jul 30 2001

Next term (a(37)) exceeds 10^5. - Matthew Conroy, Sep 13 2001
Next term (a(37)), if it exists, exceeds 10^7. - David Wasserman, Jul 03 2002
If a(37) exists, it is larger than 7.6 * 10^11. - Jud McCranie, Aug 29 2013
a(37) > 10^13, if it exists. - Giovanni Resta, Feb 25 2020

			n=100, d=600=6n, a(100)=671=11*61, phi(671)=600, phi(671+600)=phi(1271)=(31-1)*(41-1)=600+600=phi(671)+d.

Cf. A000010, A054904, A054905.

for (n=1, 36, a=1; while (isprime(a) || eulerphi(a + 6*n) != eulerphi(a) + 6*n, a++); write("b063500.txt", n, " ", a) ) \\ Harry J. Smith, Aug 24 2009

Smallest values satisfying A000010(a(n)+6n)=A000010(a(n))+6n relation.

A063679 Numbers k such that (3^k - 7)/2 is prime.

Original entry on oeis.org

4, 12, 18, 26, 106, 164, 246, 956, 2554, 3350, 6496, 8706, 9008, 15398, 15490, 20408, 39240, 41060, 41842, 58358, 60346, 82214, 134972, 194014, 344204, 587712, 778070

Offset: 1

Jud McCranie, Jul 28 2001

x = 3^k is a solution to sigma(x - 7) = sigma(x) - 7 when (3^k - 7)/2 is prime.

a(28) > 10^6

			(3^4 - 7)/2 = 37 is prime, so 4 is in the sequence.

Cf. A000203, A063680, A015913-A015917, A054905, A116970.

with(numtheory):i := 0:x := 1:while i < 1000 do i := i+1:x := 3*x: if isprime((x-7)/2) then print(i):fi:od:

Do[ If[ PrimeQ[ (3^n - 7)/2 ], Print[n] ], {n, 2, 5500} ]
Select[Range[2, 10000], PrimeQ[((3^# - 7)/2)] &] (* Vincenzo Librandi, Sep 30 2012 *)

is(n)=ispseudoprime((3^n-7)/2) \\ Charles R Greathouse IV, May 22 2017

More terms from Robert G. Wilson v, Aug 02 2001
0, 1 removed and a(11)-a(13) added from Vincenzo Librandi, Sep 30 2012
a(14)-a(17) from Seth A. Troisi, Oct 17 2022
a(17) corrected, a(18)-a(25) from Seth A. Troisi, Oct 29 2022
a(26)-a(27) from Seth A. Troisi, Nov 28 2022

A063680 Solutions to sigma(k) + 7 = sigma(k+7).

Original entry on oeis.org

74, 531434, 387420482, 2541865828322

Offset: 1

Jud McCranie, Jul 28 2001

No other solutions < 4290000000. Sequence A063679 shows how to generate more solutions, but there may be solutions other than those produced by A063679.
No others < 10^17. - Seth A. Troisi, Oct 25 2022
k or k+7 must be a square or twice a square (A028982). See comment in A015886. - Seth A. Troisi, Oct 26 2022
From Jon E. Schoenfield, Oct 26 2022: (Start)
Each of the first 4 terms of the sequence is of the form k = 9^j - 7:
             74 = 9^2  - 7,
         531434 = 9^6  - 7,
      387420482 = 9^9  - 7,
  2541865828322 = 9^13 - 7.
The next terms of this form are 9^53 - 7 and 9^82 - 7.
Does the sequence contain any terms that are not of this form?
(End)
No other terms < 2.7*10^15. - Jud McCranie, Jul 27 2025

			sigma(74) + 7 = 121 = sigma(74+7), so 74 is in the sequence.

Cf. A000203, A063679.

Cf. A054799, A015913, A015914, A015915, A015916, A015917, A054905.

isok(k) = sigma(k) + 7 == sigma(k+7); \\ Michel Marcus, Oct 25 2022

a(4) from Seth A. Troisi, Oct 24 2022

A099642 a(n) is the smallest nonprime k such that tau(k + n) = tau(k) + n , where tau(n) is the number of divisors of n (A000005).

Original entry on oeis.org

1, 10, 9, 26, 25, 74, 169, 82, 441, 170, 133, 348, 6889, 166, 3025, 344, 559, 1602, 9981, 820, 9979, 986, 4333, 1236, 9191, 694, 3249, 1652, 3481, 9378, 34969, 3118, 859329, 5636, 36829, 3324, 51947, 3994, 6561, 5000, 15835, 16806, 944741, 6436, 119025

Offset: 1

Labos Elemer, Nov 02 2004

nonn

			n=5: a(5)=25 because tau(25)+5 = 3+5 = 8 = tau(25+5) = tau(30).

Cf. A000005, A054905, A305196.

ds[x_, de_] :=DivisorSigma[0, x+de]-DivisorSigma[0, x]-de; a[n_] :=Block[{m=1, s=ds[m, n]}, While[(s!=0||PrimeQ[m])&&!Greater[m, 10000000], m++ ];m];

A055036 Min[x] composite zero site for sigma(x+6^n) - sigma(x) - 6^n.

Original entry on oeis.org

104, 125, 195, 415, 2743, 2935, 3535, 19735, 22645, 108703, 977353, 1921033, 2523433, 2425175, 4227575, 85969345, 32606935, 224917033, 1362833713, 716210677, 1557843865, 6226853857, 20369543065

Offset: 1

Labos Elemer, Jun 01 2000

			n = 6: d = 6^6 = 46656, a(n) = a(6) = 2935 because sigma(2935) + 46656 = 1 + 5 + 587 + 2935 + 46656 = sigma(2935 + 46656) = sigma(49591) = 1 + 101 + 491 + 49591 = 50184.

Subsequence of A054904.

Cf. A054902-A054905, A054799, A054982-A054985, A054987, A055009, A015913-A015518.

L = {}; Do[i = 1; While[ ! ((Plus @@ Divisors[i + 6^j] == 6^j + Plus @@ Divisors[i]) && ! PrimeQ[i]), i++ ]; L = Append[L, i], {j, 1, 11}]; L (from Vit Planocka)

a(n) = Min(x) solution for A000203(x+A000400(n)) = A000203(x) + A000400(n) Diophantine equation.

One more term from Vit Planocka (planocka(AT)mistral.cz), Sep 23 2003

a(12)-a(23) from Donovan Johnson, Nov 30 2008

cp's OEIS Frontend

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

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A054906 Smallest number x such that sigma(x+2n) = sigma(x)+2n (first definition).

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

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A054982 a(n) = least composite number such that sigma(a(n)+n!) = sigma(a(n))+n! where sigma() = A000203.

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A063500 a(n) is the least composite solution of phi(k + 6n) = phi(k) + 6n.

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A063679 Numbers k such that (3^k - 7)/2 is prime.

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A063680 Solutions to sigma(k) + 7 = sigma(k+7).

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