A358817 -id:A358817 - cp's OEIS frontend

A369020 Numbers k such that k and k+1 have the same maximal exponent in their prime factorization.

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, 99, 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, Jan 12 2024

Differs from A358817 by having the terms 99, 165, 166, ..., which are not in A358817, and not having the terms 1, 440, 1331, 1575, ..., which are in A358817.
Numbers k such that A051903(k) = A051903(k+1).
If k is a term then k*(k+1) is a term of A362605.
The asymptotic density of this sequence is d(2) + Sum_{k>=2} (d(k) + d(k+1) - 2 * d2(k)) = 0.36939178586283962461..., where d(k) = Product_{p prime} (1 - 2/p^k) and d2(k) = Product_{p prime} (1 - 1/p^k - 1/p^(k+1)).

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

Cf. A051903, A358817, A362605, A369022.

Subsequences: A007674, A071318, A369021.

emax[n_] := emax[n] = Max[FactorInteger[n][[;; , 2]]]; emax[1] = 0; Select[Range[200], emax[#] == emax[# + 1] &]

emax(n) = if(n == 1, 0, vecmax(factor(n)[, 2]));
lista(kmax) = {my(e1 = 0, e2); for(k = 2, kmax, e2 = emax(k); if(e1 == e2, print1(k-1, ", ")); e1 = e2);}

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

Original entry on oeis.org

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, 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).

A358818 a(n) is the least number k such that A046660(k) = A046660(k+1) = n.

Original entry on oeis.org

1, 44, 135, 80, 8991, 29888, 123200, 2316032, 1043199, 24151040, 217713663, 689278976, 11573190656, 76876660736, 311969153024, 2035980763136, 2741258240000, 215189482110975

Offset: 0

Amiram Eldar, Dec 02 2022

a(14) <= 314944159743.

a(18) > 10^14.5; a(19) = 275892612890624; a(20) > 10^14.5. - Martin Ehrenstein, Dec 11 2022

Cf. A046660.
Subsequence of A358817.
Similar sequences: A052215, A059737, A093548, A115186.

e[n_] := PrimeOmega[n] - PrimeNu[n]; a[n_] := Module[{k = 1}, While[e[k] != n || e[k + 1] != n, k++]; k]; Array[a, 10, 0]

e(n) = {my(f = factor(n)); bigomega(f) - omega(f)};
a(n) = {my(k=1); while(e(k) != n || e(k+1) !=n , k++); k};

a(14)-a(16) from Martin Ehrenstein, Dec 04 2022

a(17) from Martin Ehrenstein, Dec 09 2022

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

Original entry on oeis.org

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, 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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A369020 Numbers k such that k and k+1 have the same maximal exponent in their prime factorization.

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

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Formula

A358818 a(n) is the least number k such that A046660(k) = A046660(k+1) = n.

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

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