A085651 -id:A085651 - cp's OEIS frontend

A358817 Numbers k such that A046660(k) = A046660(k+1).

1, 2, 5, 6, 10, 13, 14, 21, 22, 29, 30, 33, 34, 37, 38, 41, 42, 44, 46, 49, 57, 58, 61, 65, 66, 69, 70, 73, 75, 77, 78, 80, 82, 85, 86, 93, 94, 98, 101, 102, 105, 106, 109, 110, 113, 114, 116, 118, 122, 129, 130, 133, 135, 137, 138, 141, 142, 145, 147, 154, 157

Offset: 1

Amiram Eldar, Dec 02 2022

nonn

First differs from its subsequence A007674 at n=18.

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 5, 38, 369, 3655, 36477, 364482, 3644923, 36449447, 364494215, 3644931537, ... . Apparently, the asymptotic density of this sequence exists and equals 0.36449... .

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

Cf. A046660.
Subsequences: A007674, A052213, A085651, A358818.
Similar sequences: A002961, A005237, A006049, A045920.

seq[kmax_] := Module[{s = {}, e1 = 0, e2}, Do[e2 = PrimeOmega[k] - PrimeNu[k]; If[e1 == e2, AppendTo[s, k - 1]]; e1 = e2, {k, 2, kmax}]; s]; seq[160]

e(n) = {my(f = factor(n)); bigomega(f) - omega(f)};
lista(nmax) = {my(e1 = e(1), e2); for(n=2, nmax, e2=e(n); if(e1 == e2, print1(n-1,", ")); e1 = e2);}

A369166 Numbers k such that A000688(k) = A000688(k+1).

Original entry on oeis.org

Offset: 1

Amiram Eldar, Jan 15 2024

First differs from A358817 at n = 165.

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 5, 38, 368, 3632, 36266, 362468, 3624664, 36246863, 362468411, 3624675258, ... . From these values the asymptotic density of this sequence, whose existence was proven by Erdős and Ivić (1987) (the constant c in the Formula section), can be empirically evaluated by 0.36246... .

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter XIII, pp. 475-476.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Erdős and Aleksandar Ivić, The distribution of values of a certain class of arithmetic functions at consecutive integers, Colloq. Math. Soc. János Bolyai, 51, Number Theory, Budapest, 1987, pp. 45-91. See p. 60.

Cf. A000688, A358817.

Subsequences: A007674, A052213, A085651, A335328.

Select[Range[300], FiniteAbelianGroupCount[#] == FiniteAbelianGroupCount[#+1] &]

lista(kmax) = {my(c1 = 1, c2); for(k = 2, kmax, c2 = vecprod(apply(numbpart, factor(k)[, 2])); if(c1 == c2, print1(k-1, ", ")); c1 = c2);}

The number of terms not exceeding x, N(x) = c * x + O(x^(3/4) * log(x)^4), where c > 0 is a constant (Erdős and Ivić, 1987).

A369211 Numbers k such that A005361(k) = A005361(k+1).

Original entry on oeis.org

Offset: 1

Amiram Eldar, Jan 16 2024

First differs from A358817 at n = 165.
First differs from A369166 at n = 558. a(558) = 1520 is the least term that is not in A369166. A369166(144273) = 397952 is the least term of A369166 that is not a term of this sequence.
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 5, 38, 368, 3638, 36337, 363163, 3631569, 36315800, 363156839, 3631559150, ... . Apparently, the asymptotic density of this sequence exists and equals 0.36315... .

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

Cf. A005361, A219452, A358817, A369166.

Subsequences: A007674, A052213, A085651, A335328.

s[n_] := s[n] = Times @@ FactorInteger[n][[;; , 2]]; Select[Range[300], s[#] == s[# + 1] &]

lista(kmax) = {my(c1 = 1, c2); for(k = 2, kmax, c2 = vecprod(factor(k)[, 2]); if(c1 == c2, print1(k-1, ", ")); c1 = c2);}

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A358817 Numbers k such that A046660(k) = A046660(k+1).

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A369166 Numbers k such that A000688(k) = A000688(k+1).

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A369211 Numbers k such that A005361(k) = A005361(k+1).

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