A015913 -id:A015913 - cp's OEIS frontend

A055009 Smallest composite number x such that sigma(x + prime(n)#) = sigma(x) + prime(n)#, where prime(n)# = A002110(n) and sigma is A000203.

434, 104, 44, 176, 2924, 34256, 83509, 539081, 254963216, 14600541172, 201346999808

Offset: 1

Labos Elemer, May 31 2000

a(12) <= 14841476269604. a(13) <= 314064788156864. - Donovan Johnson, Mar 17 2013

			a(7) = 83509 = 37*37*61, sigma(83509)+510510 = 87234+510510 = sigma(83509+510510) = sigma(594019) = 597744.

The prime solutions for particular sigma(x+primorial) = sigma(x)+primorial equations are in A049481-A049485.

A000203, A002110, A015913-A015917, A049902-A049905, A049481-A049485.

a(n)=my(P=prod(i=1,n,prime(i)),x=4); while(isprime(x) || sigma(x+P) != sigma(x)+P, x++); x \\ Charles R Greathouse IV, Feb 14 2013

a(9)-a(10) from Donovan Johnson, Oct 15 2008

a(11) from Donovan Johnson, Mar 08 2013

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

A056774 Composite n such that phi(n+2) = phi(n)+2.

Original entry on oeis.org

6, 12, 14, 18, 20, 44, 62, 92, 116, 164, 212, 254, 332, 356, 452, 524, 692, 716, 764, 932, 956, 1004, 1124, 1172, 1436, 1676, 1724, 1772, 1964, 2036, 2372, 2564, 2612, 2636, 2732, 2876, 2972, 3044, 3236, 3644, 3812, 4052, 4076, 4124, 4196, 4412, 4892

Offset: 1

Labos Elemer, Aug 17 2000

nonn

Below 100000 no common composite solutions with sigma(n+2)=sigma(n)+2, while prime solutions are common.

phi(x)+2=phi(x+2) is equivalent to cototient(x+2)=cototient(x), so also defines closest numbers with identical value of cototients (A051953), either primes or composites.

			n=254, phi(254+2) = phi(256) = 128 = phi(254)+2 = 126+2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A001838, A055458, A015913-A015917, A050507, A054799.

Select[Range[5000],CompositeQ[#]&&EulerPhi[#]+2==EulerPhi[#+2]&] (* Harvey P. Dale, Jul 10 2017 *)

isok(n) = !isprime(n) && (eulerphi(n+2) == eulerphi(n)+2); \\ Michel Marcus, Aug 30 2019

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

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.

A056773 Composite n such that phi(n+4) = phi(n)+4.

Original entry on oeis.org

12, 18, 24, 28, 36, 40, 66, 88, 124, 184, 232, 328, 424, 508, 664, 712, 904, 1048, 1384, 1432, 1528, 1864, 1912, 2008, 2248, 2344, 2586, 2872, 3352, 3448, 3544, 3928, 4072, 4744, 5128, 5224, 5272, 5464, 5752, 5944, 6088, 6472, 7288, 7624, 8104, 8152, 8248

Offset: 1

Labos Elemer, Aug 17 2000

nonn

Are all terms even? - Robert Israel, Apr 28 2020

			24 is in the sequence because 24 is composite and phi(24)+4 = 12 = phi(24+4).

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

A001838, A015913, A055458. Composites in A056772. Primes in A056772 are A023200.

filter:= n -> not isprime(n) and numtheory:-phi(n+4)=numtheory:-phi(n)+4:
select(filter, [$1..10000]); # Robert Israel, Apr 28 2020

Select[Range[9000],CompositeQ[#]&&EulerPhi[#]+4==EulerPhi[#+4]&] (* Harvey P. Dale, Feb 12 2015 *)

is(n)=!isprime(n) && eulerphi(n+4)==eulerphi(n)+4 \\ Charles R Greathouse IV, Apr 28 2020

Edited by Robert Israel, Apr 28 2020

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.

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

A055009 Smallest composite number x such that sigma(x + prime(n)#) = sigma(x) + prime(n)#, where prime(n)# = A002110(n) and sigma is A000203.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A056774 Composite n such that phi(n+2) = phi(n)+2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A056773 Composite n such that phi(n+4) = phi(n)+4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica