A015917 -id:A015917 - cp's OEIS frontend

A054984 Composite numbers k such that sigma(k + 6!) = sigma(k + 720) = sigma(k) + 720.

427, 553, 595, 623, 737, 871, 913, 923, 1199, 1207, 1241, 1507, 1582, 1817, 1848, 2193, 2226, 2337, 2398, 2407, 2553, 2561, 2728, 2758, 2929, 3016, 3115, 3248, 3346, 3502, 3503, 3598, 3705, 3762, 4171, 4293, 4343, 4462, 4587, 4633, 4841, 4867, 4984

Offset: 1

Labos Elemer, May 29 2000

nonn

			553 is a term because sigma(553) + 720 = 640 + 720 = 1360 = sigma(553 + 720) = sigma(1273) = 1 + 19 + 67 + 1273.

Amiram Eldar, Table of n, a(n) for n = 1..6000

Cf. A015914, A015915, A015916, A015917, A033560, A033561, A054799.

Select[Range[5000], CompositeQ[#] && Differences@ DivisorSigma[1, {#, # + 720}] == {720} &] (* Amiram Eldar, Mar 09 2025 *)

isok(k) = !isprime(k) && sigma(k + 720) == sigma(k) + 720; \\ Amiram Eldar, Mar 09 2025

A054985 Composite numbers x such that sigma(x+120) = sigma(x)+120.

Original entry on oeis.org

182, 203, 287, 350, 407, 558, 611, 731, 779, 803, 963, 1424, 1643, 2627, 2747, 3431, 3806, 4187, 4223, 5063, 6767, 7946, 8927, 9047, 11904, 12707, 12878, 15794, 18923, 20567, 27263, 31175, 32111, 34427, 43139, 43811, 45854, 50165, 52592, 57479

Offset: 1

Labos Elemer, May 29 2000

nonn

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

A054983 Composite numbers n such that sigma(n+24) = sigma(n) + 24.

Original entry on oeis.org

80, 95, 119, 299, 527, 962, 1247, 1479, 1739, 2783, 4307, 4958, 5240, 6015, 7878, 8342, 10379, 11639, 16967, 20687, 21439, 29294, 34547, 36917, 49022, 51959, 54707, 59807, 76127, 97319, 153242, 181427, 203318, 203822, 213419, 363302, 423999, 494882, 582902

Offset: 1

Labos Elemer, May 29 2000

nonn

A056775 Numbers k such that phi(k+12) = phi(k) + 12.

Original entry on oeis.org

5, 7, 11, 17, 19, 29, 31, 41, 45, 47, 59, 61, 65, 67, 71, 80, 89, 97, 99, 101, 112, 117, 127, 135, 137, 139, 151, 167, 171, 176, 179, 181, 196, 199, 207, 209, 211, 227, 229, 239, 251, 257, 269, 271, 272, 279, 281, 294, 304, 310, 312, 337, 347, 367, 369, 389

Offset: 1

Labos Elemer, Aug 17 2000

nonn

Prime solutions are in A046133, common with primes in A015917.

			65 is a term since phi(65) = 48, phi(65+12) = phi(77) = 60 = 48 + 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A000010, A001838, A015917, A054902, A046133, A046133.

Select[Range[400],EulerPhi[#]+12==EulerPhi[#+12]&] (* Harvey P. Dale, Jan 21 2013 *)

A056776 Composite numbers k such that phi(k+12) = phi(k) + 12.

Original entry on oeis.org

45, 65, 80, 99, 112, 117, 135, 171, 176, 196, 207, 209, 272, 279, 294, 304, 310, 312, 369, 406, 429, 477, 496, 531, 592, 656, 657, 711, 752, 801, 909, 927, 944, 981, 1014, 1072, 1078, 1179, 1251, 1359, 1424, 1557, 1611, 1629, 1712, 1719, 1744, 1786, 1791

Offset: 1

Labos Elemer, Aug 17 2000

nonn

There are common cases with A054902.

			656 is a term since it is composite and phi(656) = 320, phi(656+12) = phi(668) = 332 = 320 + 12.
657 is a term since it is composite and phi(657) = 432, phi(657+12) = phi(669) = 444 = 432 + 12.

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

Cf. A000010, A001838, A015917, A054902, A046133, A046133.

Select[Range[1800], CompositeQ[#] && EulerPhi[# + 12] == EulerPhi[#] + 12 &] (* Amiram Eldar, Mar 01 2020 *)

A056777 Composite numbers k such that both phi(k+12) = phi(k) + 12 and sigma(k+12) = sigma(k) + 12.

Original entry on oeis.org

65, 209, 11009, 38009, 680609, 2205209, 3515609, 4347209, 10595009, 12006209, 31979009, 89019209, 169130009, 244766009, 247590209, 258084209, 325622009, 357777209, 377330609, 441630209, 496175609, 640343009, 1006475609

Offset: 1

Labos Elemer, Aug 17 2000

nonn

It is easy to show that if p, p+2, p+6 and p+8 are all prime (a prime quadruple as defined in A007530, which lists the values of p) with x=p(p+8), x+12=(p+2)(p+6), then x is in the sequence. I conjecture that all members of the sequence are of this form. - Jud McCranie, Oct 11 2000

Numbers so far are all congruent to 65 (mod 72). - Ralf Stephan, Jul 07 2003

			k = 209 = 11*19, k + 12 = 221 = 13*17, phi(k + 12) = 192 = 180 + 12 = phi(k) + 12, also sigma(221) = 252 = sigma(209) + 12 = 240 + 12.
phi(65) + 12 = 60 = phi(65 + 12), sigma(65) + 12 = 96 = sigma(65 + 12), 65 is composite.

Cf. A000010, A001838, A015917, A054902, A046133.

isok(n) = !isprime(n) && (sigma(n+12) == sigma(n)+12) && (eulerphi(n+12)==eulerphi(n)+12); \\ Michel Marcus, Jul 14 2017

More terms from Jud McCranie, Oct 11 2000

cp's OEIS Frontend

A054984 Composite numbers k such that sigma(k + 6!) = sigma(k + 720) = sigma(k) + 720.

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A054985 Composite numbers x such that sigma(x+120) = sigma(x)+120.

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

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

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A054983 Composite numbers n such that sigma(n+24) = sigma(n) + 24.

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A056775 Numbers k such that phi(k+12) = phi(k) + 12.

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A056776 Composite numbers k such that phi(k+12) = phi(k) + 12.

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A056777 Composite numbers k such that both phi(k+12) = phi(k) + 12 and sigma(k+12) = sigma(k) + 12.

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