A023221 -id:A023221 - cp's OEIS frontend

A126329 Primes of the form 6p+5 where p is a prime.

17, 23, 47, 71, 83, 107, 179, 191, 227, 251, 263, 359, 431, 443, 479, 503, 587, 647, 659, 683, 827, 839, 911, 947, 983, 1091, 1151, 1163, 1187, 1367, 1439, 1451, 1511, 1583, 1619, 1667, 1847, 1871, 1907, 2027, 2087, 2099, 2207, 2243, 2339, 2411, 2459, 2531, 2591

Offset: 1

J. M. Bergot, Mar 09 2007

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

For the corresponding primes p, see A023221.

lst={};Do[p=Prime[n];p=6*p+5;If[PrimeQ[p],AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 04 2009 *)
6#+5&/@Select[Prime[Range[100]],PrimeQ[6#+5]&] (* Harvey P. Dale, Dec 31 2015 *)

Corrected and extended by N. J. A. Sloane, Mar 10 2007

A352331 Numbers k for which phi(k) = phi(k''), where phi is the Euler totient function (A000010) and k'' the second arithmetic derivative of k (A068346).

Original entry on oeis.org

4, 27, 104, 260, 296, 405, 525, 740, 910, 945, 1460, 1806, 1818, 2504, 3125, 3140, 3176, 3656, 3860, 4563, 5540, 6056, 6930, 7016, 8420, 8636, 9224, 10820, 12573, 13256, 14024, 15140, 15464, 15944, 16136, 19940, 20456, 21690, 21860, 22856, 23336, 24020, 24260

Offset: 1

Marius A. Burtea, Apr 09 2022

nonn

If m is a term in A051674, then m'' = m, phi(m'') = phi(m) so the sequence is infinite.
If p > 3 is at the intersection of A023208 and A005383 then m = 8*p is a term. Indeed, m'' = (8*p)'' = (12*p + 8)' = (4*(3*p + 2))' = 12*(p + 1) and phi(m'') = phi(12*(p + 1)) = phi(24*(p + 1)/2) = 8*(p - 1)/2 = 4*(p - 1) and phi(m) = phi(8*p) = 4*(p - 1).
If p > 5 is at the intersection of A023221 and A005383 then m = 20*p is a term. Indeed, m'' = (20*p)'' = (24*p + 20)' = (4*(6*p + 5))' = 4*(6*p + 6)  = 24*(p + 1) and phi(m'') = phi(24*(p + 1)) = phi(48*(p + 1)/2) = 16*(p - 1)/2 = 8*(p - 1) and phi(m) = phi(20*p) = 8*(p - 1).

			phi(4'') = phi(4) because 4'' = 4, so 4 is a term.
phi (27'') = phi(27) because 27'' = 27, so 27 is a term.
phi(104'') = phi(164') = phi(168) = phi (8*3*7) = 4*2*6 = 48 and phi(104) = phi(8*13) = 4*12 = 48, so 104 is a term.

Cf. A000010, A005383, A051674, A068346, A023208, A023221, A190402,

f:=func; [n:n in [2..24300]| not IsPrime(n) and EulerPhi(n) eq EulerPhi(Floor(f(Floor(f(n))))) ];

d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[25000], EulerPhi[#] == EulerPhi[d[d[#]]] &] (* Amiram Eldar, Apr 10 2022 *)

A352332 Numbers k for which k = phi(k') + phi(k''), where phi is the Euler totient function (A000010), k' the arithmetic derivative of k (A003415) and k'' the second arithmetic derivative of k (A068346).

Original entry on oeis.org

4, 260, 294, 740, 1460, 3140, 3860, 5540, 8420, 10820, 15140, 19940, 21860, 24020, 24260, 27620, 37460, 40340, 46820, 49460, 55940, 61220, 70340, 85460, 101540, 114020, 124340, 132740, 133220, 144260, 148340, 149540, 155060, 162020, 164420, 167060, 170420, 173540

Offset: 1

Marius A. Burtea, Apr 09 2022

nonn

If p > 3 is at the intersection of A023221 and A005383, then m = 20*p is a term. Indeed, m' = (20*p)' = 24*p + 20 = 4*(6*p + 5), m'' = (4*(6*p + 5))' = 4*(6*p + 6) = 24*(p + 1), phi(m') + phi(m'') = phi(4*(6*p + 5)) + phi(24*(p + 1)) = 2*(6*p + 4) + phi(48*(p + 1)/2) = 2*(6*p + 4) + 16*(p - 1)/2 = 12*p + 8 + 8*p - 8 = 20*p = m.

			phi(4') + phi(4'') = phi(4) + phi(4) = 2 + 2 = 4, so 4 is a term.
phi(260') + phi(260'') = phi(332) + phi(336) = 164 + 96 = 260, so 260 is a term.

Cf. A000010, A023221, A003415, A005383, A068346.

f:=func; [n:n in [2..174000]|not IsPrime(n) and n-EulerPhi(Floor(f(n))) eq EulerPhi(Floor(f(Floor(f(n)))))];

d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[200000], CompositeQ[#] && EulerPhi[d[#]] + EulerPhi[d[d[#]]] == # &] (* Amiram Eldar, Apr 10 2022 *)

ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok(k) = my(adk=ad(k)); !isprime(k) && (k == eulerphi(adk) + eulerphi(ad(adk))); \\ Michel Marcus, Apr 30 2022

A359330 Composite k for which phi(k) + phi(k') = k, where k' is the arithmetic derivative of k (A003415).

Original entry on oeis.org

4, 6, 8, 10, 12, 18, 22, 28, 34, 58, 60, 72, 82, 84, 88, 108, 112, 118, 124, 132, 140, 142, 202, 204, 214, 216, 220, 228, 260, 274, 298, 324, 340, 358, 372, 382, 394, 444, 454, 478, 492, 508, 538, 562, 564, 572, 580, 620, 622, 644, 694, 708, 740, 804, 812, 820

Offset: 1

Marius A. Burtea, Jan 28 2023

nonn

Composite numbers k for which phi(k') = cototient(k) (A051953).
The sequence refers only to composite numbers because for any prime number p we obtain phi(p) + phi(p') = p - 1 + phi(1) = p.
If p = 2^k - 1 is a Mersenne prime (A000668), then m = 4*p is a term. Indeed, m' = 4*(p + 1) = 4*2^k = 2^(k + 2) and phi(m) + phi(m') = phi(4*p) + phi(2^(k + 2)) = 2*(p-1) + 2^(k+1) = 2*(p - 1) + 2*(p + 1) = 4*p = m, so m is a term.
If p, q and p*q + p + q are prime numbers then m = 4*p*q is a term. Indeed, m'= 4*(p*q + p + q) and phi(m) + phi(m') = phi(4*p*q) + phi(4*(p*q + p + q)) = 2*(p - 1)*(q - 1) + 2*(p*q + p + q - 1) = 4*p*q.
If p is in A023221 then m = 20*p is a term. Indeed, m' = 24*p + 20 = 4*(6*p + 5) and phi(m) + phi(m') = phi(20*p) + phi(4*(6*p + 5)) = 8*(p-1) + 2*(6*p + 4) = 20*p = m, so m is a term.

			If m = 4 then m' = 4 and phi(m) + phi(m') = phi(4) + phi(4) = 2 + 2 = 4, so 4 is a term.
If m = 8 then m' = 12 and phi(m) + phi(m') = phi(8) + phi(12) = 4 + 4 = 8, so 8 is a term.
14 is not a term because phi(14) + phi(14') = 6 + phi(9) = 6 + 6 = 12 <> 14.

Cf. A000010, A001359, A002808, A003415, A051953, A066938, A023221, A190402.

f:=func;  [n:n in [2..850]|not IsPrime(n) and n eq EulerPhi(Floor(f(n))) + EulerPhi(n)];

d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
q:= n-> not isprime(n) and (p-> p(n)+p(d(n))=n)(numtheory[phi]):
select(q, [$4..1000])[];  # Alois P. Heinz, Jan 29 2023

d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[1000], CompositeQ[#] && EulerPhi[#] + EulerPhi[d[#]] == # &] (* Amiram Eldar, Jan 29 2023 *)

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A126329 Primes of the form 6p+5 where p is a prime.

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A352331 Numbers k for which phi(k) = phi(k''), where phi is the Euler totient function (A000010) and k'' the second arithmetic derivative of k (A068346).

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A352332 Numbers k for which k = phi(k') + phi(k''), where phi is the Euler totient function (A000010), k' the arithmetic derivative of k (A003415) and k'' the second arithmetic derivative of k (A068346).

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A359330 Composite k for which phi(k) + phi(k') = k, where k' is the arithmetic derivative of k (A003415).

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