A055458 -id:A055458 - cp's OEIS frontend

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

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

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.

A063104 a(0) = 0, a(n) = smallest composite k such that phi(k + 2^n) = phi(k) + 2^n; also cototient(k + 2^n) = cototient(k).

Original entry on oeis.org

0, 6, 12, 24, 39, 84, 69, 75, 213, 1092, 249, 1131, 8736, 13413, 21201, 1275, 2193, 279552, 98337, 968727, 71085, 2783555, 646869, 3145959, 1805781, 5798435, 787605, 27962075, 2073033, 282181709, 1150329, 10380353, 516201, 150807855, 141521295, 860867981

Offset: 0

Labos Elemer, Aug 08 2001

nonn

			n=4, a(4)=39, Phi[39]+16=24+16=40=Phi[55]; a(14) = 21201, Phi(21201) + 2^14 = 13680 + 16384 = 30064 = Phi(37585).

Cf. A000010, A051953, A055458, A063500, A054987.

Do[k = 4; While[ PrimeQ[k] || EulerPhi[k + 2^n] != EulerPhi[k] + 2^n, k++ ]; Print[k], {n, 1, 28} ]

{ n=0; f="b063104.txt"; write(f, "0 0"); for (n=1, 28, k=4; while (isprime(k) || eulerphi(k + 2^n) != eulerphi(k) + 2^n, k++); write(f, n, " ", k) ) } \\ Harry J. Smith, Aug 18 2009

a(n) = Min{x: A000010(n)+2^n = A000010(x+2^n)} = Min{x: A051953(x+2^n) = A051953(n)}

More terms from Robert G. Wilson v, Nov 03 2001

a(29)-a(35) from Donovan Johnson, Aug 18 2011

A063519 Least composite k such that phi(k+12n) = phi(k)+12n and sigma(k+12n) = sigma(k) + 12n where phi is the Euler totient function and sigma is the sum of divisors function.

Original entry on oeis.org

65, 95, 341, 95, 161, 115, 629, 203, 145, 203, 365, 155, 185, 155, 301, 185, 329, 235, 1541, 287, 185, 287, 413, 205, 329, 215, 469, 215, 905, 371, 365, 305, 553, 371, 1037, 235, 1145, 623, 445, 371, 35249, 295, 1133, 371, 497, 515, 749, 413, 305, 671, 565

Offset: 1

Labos Elemer, Aug 01 2001

nonn

No such simultaneous solutions were found if d=12n+6.

			a(97)=10217 because 10217 is composite, phi(10217)+1164 = 9600+1164 = 10764 = phi(11381) and sigma(10217)+1164 = 10836+1164 = 12000 = sigma(11381) with 1164 = 12*97 and there is no smaller composite with these properties.

A000010, A000203, A054904, A054905, A055458.

a(n) = Min{k: phi(k+12n) = phi(k)+12n and sigma(k+12n) = sigma(k)+12n and k is composite} with phi(k) = A000010(k) and sigma(k) = A000203(k).

Name corrected by Sean A. Irvine, Apr 30 2023

cp's OEIS Frontend

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

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

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

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

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A063104 a(0) = 0, a(n) = smallest composite k such that phi(k + 2^n) = phi(k) + 2^n; also cototient(k + 2^n) = cototient(k).

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A063519 Least composite k such that phi(k+12n) = phi(k)+12n and sigma(k+12n) = sigma(k) + 12n where phi is the Euler totient function and sigma is the sum of divisors function.

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