A321167 -id:A321167 - cp's OEIS frontend

A349308 Numbers k such that A321167(k) = A321167(k+1) > 1.

80, 135, 296, 343, 375, 624, 728, 1160, 1431, 1592, 1624, 2240, 2295, 2456, 2511, 2624, 2727, 2888, 3429, 3591, 3624, 3752, 3992, 4023, 4184, 4671, 4887, 4913, 5048, 5144, 5264, 5319, 5480, 5696, 6183, 6344, 6375, 6591, 6615, 6776, 6858, 6859, 7479, 7624, 7640

Offset: 1

Amiram Eldar, Nov 14 2021

nonn

Without the restriction that A321167(k) > 1, all the terms of A340152 would be in this sequence.

In contrast to A001274, which has only one known pair of consecutive terms (5186 and 5187), this sequence seems to have many pairs of consecutive terms. The smaller members of these pairs are 6858, 13375, 22625, ...

			80 is a term since A321167(80) = A321167(81) = 3.

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

Cf. A321167, A340152.
Subsequence of A068140.
Similar sequences: A001274, A287055, A293184, A326403, A349307.

f[p_, e_] := p^e - 1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; fe[p_, e_] := uphi[e]; euphi[n_] := Times @@ fe @@@ FactorInteger[n]; Select[Range[8000], euphi[#] == euphi[# + 1] > 1 &]

A358658 Decimal expansion of the asymptotic mean of the e-unitary Euler function (A321167).

Original entry on oeis.org

1, 3, 0, 7, 3, 2, 1, 3, 7, 1, 7, 0, 6, 0, 7, 2, 3, 6, 9, 2, 9, 6, 4, 2, 2, 8, 0, 4, 2, 5, 3, 9, 8, 8, 3, 9, 1, 4, 2, 7, 4, 3, 4, 6, 8, 6, 0, 8, 2, 3, 9, 4, 0, 9, 8, 0, 1, 5, 3, 6, 3, 5, 6, 9, 8, 1, 7, 0, 0, 9, 7, 0, 8, 9, 0, 0, 8, 4, 9, 7, 3, 2, 2, 0, 0, 7, 2, 0, 2, 5, 4, 0, 4, 5, 4, 8, 4, 4, 8, 1, 2, 9, 7, 2, 9

Offset: 1

Amiram Eldar, Nov 25 2022

			1.307321371706072369296422804253988391427434686082394...

Nicuşor Minculete and László Tóth, Exponential unitary divisors, Annales Univ. Sci. Budapest., Sect. Comp. Vol. 35 (2011), pp. 205-216.

Cf. A047994, A321167, A327838.

f[p_, e_] := p^e - 1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; $MaxExtraPrecision = 500; m = 500; fun[x_] := Log[1 + Sum[x^e*(uphi[e] - uphi[e - 1]), {e, 3, m}]]; c = Rest[CoefficientList[Series[fun[x], {x, 0, m}], x]*Range[0, m]]; RealDigits[Exp[fun[1/2] + NSum[Indexed[c, k]*(PrimeZetaP[k] - 1/2^k)/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

Equals lim_{m->oo} (1/m) Sum_{k=1..m} A321167(k).

Equals Product_{p prime} (1 + Sum_{e >= 3} (uphi(e) - uphi(e-1))/p^e), where uphi is the unitary totient function (A047994).

A384423 The number of prime powers (not including 1) p^e that divide n such that e is unitarily coprime to the p-adic valuation of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 4, 2, 2, 2, 2, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 4, 3, 2, 1, 3, 2, 2, 2

Offset: 1

Amiram Eldar, May 28 2025

A number k is unitarily coprime to m if the largest divisor of k that is a unitary divisor of m is 1.

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

Cf. A047994, A077761, A085548, A321167.

f[p_, e_] := p^e - 1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n];
ff[p_, e_] := uphi[e]; a[1] = 0; a[n_] := Plus @@ ff @@@ FactorInteger[n]; Array[a, 100]

uphi(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^f[i, 2]-1);}
a(n) = vecsum(apply(uphi, factor(n)[, 2]));

Additive with a(p^e) = uphi(e), where uphi is the unitary totient function (A047994).

Sum_{k=1..n} a(k) ~ n*(log(log(n)) + B - C + D), where B is Mertens's constant (A077761), C = Sum_{p prime} 1/p^2 (A085548), and D = Sum_{p prime, e>=2} (1-1/p)*A047994(e)/p^e = 0.74335242036929441969... .

cp's OEIS Frontend

A349308 Numbers k such that A321167(k) = A321167(k+1) > 1.

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A358658 Decimal expansion of the asymptotic mean of the e-unitary Euler function (A321167).

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A384423 The number of prime powers (not including 1) p^e that divide n such that e is unitarily coprime to the p-adic valuation of n.

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Author

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Mathematica

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