A001274 -id:A001274 - cp's OEIS frontend

A332530 Numbers k such that k and k + 1 has the same value of A319445, the equivalent of the Euler totient function in the ring of Eisenstein integers.

34, 51, 152, 679, 1065, 1845, 6525, 12122, 12970, 15656, 38607, 48398, 175473, 272935, 401505, 953342, 1035895, 1210054, 1222988, 1406665, 1589245, 1607095, 2108186, 2116975, 2272425, 2500615, 2751160, 3399591, 4542225, 5298559, 5412986, 6813585, 6898736, 7115553

Offset: 1

Amiram Eldar, Feb 15 2020

nonn

			34 is a term since A319445(34) = A319445(35) = 864.

Amiram Eldar, Table of n, a(n) for n = 1..152

Cf. A001274, A287055, A293184, A319445, A326403, A332316.

f[p_, e_] := If[p == 3, 2*3^(2*e - 1), Switch[Mod[p, 3], 1, (p - 1)^2*p^(2*e - 2), 2, (p^2 - 1)*p^(2*e - 2)]]; eisPhi[1] = 1; eisPhi[n_] := Times @@ f @@@ FactorInteger[n]; seq = {}; e1 = eisPhi[1]; Do[e2 = eisPhi[n]; If[e1 == e2, AppendTo[seq, n - 1]]; e1 = e2, {n, 2, 10^6}]; seq

A336673 Numbers k such that A063659(k) = A063659(k+1).

Original entry on oeis.org

3, 49, 1681, 18490, 23762, 656914, 843637, 5606230, 35558770, 46297822, 59006794, 114594493, 132859642, 138852445, 157906534, 289405462, 299441785, 536671282, 813736930, 1175272581, 1276553470, 1655870629, 5086602202, 5429407657, 6549516022, 8645559934, 10373399185

Offset: 1

Amiram Eldar, Jul 29 2020

nonn

Analogous to A001274 as A063659 is analogous to Euler's phi function (A000010).

			3 is a term since A063659(3) = A063659(4) = 3.

E. K. Haviland, An analogue of Euler's phi-function, Duke Math. J., Vol. 11 (1944), pp. 869-872.

Cf. A000010, A063659, A001274.

f[p_, e_] := If[e == 1, p, p^e - p^(e-2)]; s[n_] := Times @@ f @@@ FactorInteger[n]; s1 = 1; seq = {}; Do[s2 = s[n]; If[s1 == s2, AppendTo[seq, n-1]]; s1 = s2, {n, 2, 10^5}]; seq

A385743 Numbers k such that A384247(k) = A384247(k+1).

Original entry on oeis.org

1, 20, 27, 35, 63, 64, 104, 143, 194, 208, 740, 836, 1220, 1299, 1419, 1803, 1892, 2625, 3255, 3705, 3716, 3843, 4096, 5184, 5186, 5635, 5695, 7868, 10659, 13365, 16904, 17948, 18507, 18914, 21007, 22935, 25388, 25545, 27675, 30380, 31599, 32304, 32864, 34595

Offset: 1

Amiram Eldar, Jul 08 2025

nonn

63 is the only number k below 10^11 such that A384247(k) = A384247(k+1) = A384247(k+2). Are there any other such terms?

			1 is a term since A384247(1) = A384247(2) = 1.
20 is a term since A384247(20) = A384247(21) = 12.

Amiram Eldar, Table of n, a(n) for n = 1..4006 (terms below 10^11)

Cf. A384247.

Similar sequences: A001274, A287055, A293184, A301866, A326403, A349307.

f[p_, e_] := p^e*(1 - 1/p^(2^(IntegerExponent[e, 2]))); iphi[1] = 1; iphi[n_] := iphi[n] = Times @@ f @@@ FactorInteger[n]; Select[Range[35000], iphi[#] == iphi[# + 1] &]

iphi(n) = {my(f = factor(n)); n * prod(i = 1, #f~, (1 - 1/f[i, 1]^(1 << valuation(f[i, 2], 2)))); }
list(lim) = {my(s1 = iphi(1), s2); for(k = 2, lim, s2 = iphi(k); if(s1 == s2, print1(k-1, ", ")); s1 = s2);}

A063100 Compute the cototient function for the g(n) = p(n+1)-p(n)-1 composite numbers between two consecutive primes. Let the number of distinct cototient values be c(n). Then, a(n) = g(n)-c(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Labos Elemer, Aug 07 2001

nonn

a(n) = 0 means that all cototients in the gap are different, while 1, 2, or more means that 1 or more times inside the gap equal cototients occur.

Unlike totient, where phi(n+1) = phi(n) may occur (see A001274), cototients of consecutive numbers are different for n<1000000. At cases x, of A001274, cototient(x+1) = 1+cototient(x).

			Case 1: a(n) = 0; primes = {229, 233}; primes and gap = {229, 230, 231, 232, 233}; cototients = {1, 142, 111, 120, 1}, all cototients inside gap are different, thus a(n) = 0 for p(n) = p(40) = 229 prime.
Case 2: a(n) = 1; primes = {113, 127}; gap = {113, 114, 115, ..., 125, 126, 127}; cototients = {1, 78, 27, 60, 45, 60, 23, 88, 11, 62, 43, 64, 25, 90, 1}; seemingly 60 occurs twice, so a(n) = a(30) = g(n)-c(n) = 13-12 = 1.
Case 3: a(n) = 3, primes = {2861, 2879}, gap = {2861, 2862, ..., 2878, 2879}; cototients = {1, 1926, 415, 1440, 1345, 1434, 107, 1916, 169, 1910, 1191, 1440, 377, 1918, 675, 1440, 1245, 1440, 1}; observe that 1440 occurs four times, so a(n) = 3.

Cf. A051953 (cototient function), A000010, A001274, A061106.

A139766 A number n is included in the sequence if and only if the n-th integer from among those positive integers which are coprime to n+1 = the (n+1)-st integer from among those positive integers which are coprime to n.

Original entry on oeis.org

3, 15, 104, 164, 255, 2625, 2834, 11715, 18315, 48704, 49215, 64004, 65535, 73124, 131144, 215775, 491535, 525986, 546272, 568815, 952575, 1925564, 5781434, 5861583, 13496384, 14409548, 17646615, 17949434, 20171384, 21475124, 22632285

Offset: 1

Leroy Quet, Nov 07 2007

nonn

So far it appears that this is a proper subset of A001274.

Cf. A001274, A069213, A126356, A126357.

Indices n such that A126356(n) = A126357(n). - Ray Chandler

More terms from Stefan Steinerberger, Nov 07 2007

a(8) onwards from Ray Chandler, Jul 01 2009

A303820 a(n) begins the first run of least n consecutive numbers whose phi values have the same set of distinct prime divisors.

Original entry on oeis.org

1, 1, 3, 3, 35, 43570803, 22154517001

Offset: 1

Amiram Eldar, May 01 2018

a(n) is the least k such that rad(phi(k)) = rad(phi(k+1)) = ... = rad(phi(k+n-1)), where rad(n) is the squarefree kernel of n (A007947) and phi(n) is the Euler totient function of n (A000010).

			a(5) = 35 since it is the least number such that
phi(35) = 24 = 2^3 * 3^1,
phi(36) = 12 = 2^2 * 3^1,
phi(37) = 36 = 2^2 * 3^2,
phi(38) = 18 = 2^1 * 3^2,
phi(39) = 24 = 2^3 * 3^1,
all having the same set of prime divisors: 2 and 3.

Cf. A000010, A001274, A007947, A303693.

rad[n_] := Times @@ (First@# & /@ FactorInteger@n); radphi[n_] := rad[ EulerPhi[n] ]; Seq[n_, q_] := Map[radphi, Range[n, n + q - 1]]; findConsec[q_, nmin_, nmax_] := Module[{}, s = Seq[1, q]; n = q + 1; Do[If[CountDistinct[s] == 1, Break[]]; s = Rest[AppendTo[s, radphi[n]]]; n++, {k, nmin, nmax}]; n - q]; seq = {1}; nmax = 10^10; Do[n1 = Last[ seq ]; s1 = findConsec[m, n1, nmax]; AppendTo[seq, s1], {m, 2, 6}]; seq

a(7) from Giovanni Resta, May 08 2018

cp's OEIS Frontend

A332530 Numbers k such that k and k + 1 has the same value of A319445, the equivalent of the Euler totient function in the ring of Eisenstein integers.

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A336673 Numbers k such that A063659(k) = A063659(k+1).

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A385743 Numbers k such that A384247(k) = A384247(k+1).

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A063100 Compute the cototient function for the g(n) = p(n+1)-p(n)-1 composite numbers between two consecutive primes. Let the number of distinct cototient values be c(n). Then, a(n) = g(n)-c(n).

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A139766 A number n is included in the sequence if and only if the n-th integer from among those positive integers which are coprime to n+1 = the (n+1)-st integer from among those positive integers which are coprime to n.

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A303820 a(n) begins the first run of least n consecutive numbers whose phi values have the same set of distinct prime divisors.

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