A054902 -id:A054902 - cp's OEIS frontend

A015917 Numbers k such that sigma(k) + 12 = sigma(k+12).

5, 7, 11, 17, 19, 29, 31, 41, 47, 59, 61, 65, 67, 71, 89, 97, 101, 127, 137, 139, 151, 167, 170, 179, 181, 199, 209, 211, 227, 229, 239, 251, 257, 269, 271, 281, 337, 347, 367, 389, 397, 409, 419, 421, 431, 449, 467, 479, 487, 491, 509, 557, 587

Offset: 1

Robert G. Wilson v

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000203.

Union of A046133 and A054902.

Select[Range[600],DivisorSigma[1,#]+12==DivisorSigma[1,#+12]&] (* Harvey P. Dale, Oct 14 2012 *)

is(n)=sigma(n)+12==sigma(n+12) \\ Charles R Greathouse IV, Mar 09 2014

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 *)

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

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

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A015917 Numbers k such that sigma(k) + 12 = sigma(k+12).

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

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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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