A335328 -id:A335328 - cp's OEIS frontend

A344314 Number k such that k and k+1 have the same number of nonunitary divisors (A048105).

1, 2, 5, 6, 10, 13, 14, 21, 22, 27, 29, 30, 33, 34, 37, 38, 41, 42, 44, 46, 57, 58, 61, 65, 66, 69, 70, 73, 75, 77, 78, 82, 85, 86, 93, 94, 98, 101, 102, 105, 106, 109, 110, 113, 114, 116, 118, 122, 124, 129, 130, 133, 135, 137, 138, 141, 142, 145, 147, 154, 157

Offset: 1

Amiram Eldar, May 14 2021

nonn

			1 is a term since A048105(1) = A048105(2) = 0.
27 is a term since A048105(27) = A048105(28) = 2.

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

Cf. A048105, A064115, A335328, A344315.

Similar sequences: A005237, A006049, A343819, A344312, A344313.

nd[n_] := DivisorSigma[0, n] - 2^PrimeNu[n]; Select[Range[200], nd[#] == nd[# + 1] &]

A335397 Starts of runs of 3 consecutive numbers that have an equal number of unitary and nonunitary divisors (A048109).

Original entry on oeis.org

22625, 28375, 40472, 48248, 49624, 58374, 59750, 102248, 103624, 107702, 112374, 129623, 136214, 136375, 164295, 187623, 190375, 197910, 199624, 211624, 221750, 246616, 264248, 275750, 280231, 298375, 300806, 312471, 346086, 349623, 352375, 356375, 372248, 382374

Offset: 1

Zak Seidov and Amiram Eldar, Jun 06 2020

nonn

			22625 is a term since 22625, 22626 and 22627 each have an equal number of unitary and nonunitary divisors. 22625 has 4 unitary divisors (1, 125, 181 and 22625) and 4 nonunitary divisors (5, 25, 905 and 4525), 22626 has 8 unitary divisors and 8 nonunitary divisors, and 22627 has 4 unitary divisors and 4 nonunitary divisors.

Zak Seidov, Table of n, a(n) for n = 1..10000

Subsequence of A048109 and A335328.

Cf. A000005, A034444, A048105.

q[n_] := DivisorSigma[0, n] == 2^(PrimeNu[n] + 1); v = q /@ Range[3]; seq = {}; Do[v = Append[Drop[v, 1], q[k]]; If[And @@ v, AppendTo[seq, k - 2]], {k, 4, 2 * 10^5}]; seq

A348098 Number k such that k and k+1 both have an equal number of unitary and nonunitary prime divisors (A348097).

Original entry on oeis.org

44, 75, 98, 116, 135, 147, 152, 171, 175, 188, 207, 244, 296, 332, 351, 368, 375, 387, 404, 423, 424, 507, 548, 567, 603, 604, 639, 656, 711, 724, 775, 832, 844, 847, 872, 891, 908, 927, 931, 963, 1016, 1028, 1052, 1075, 1083, 1107, 1183, 1215, 1250, 1251, 1268

Offset: 1

Amiram Eldar, Sep 30 2021

nonn

			44 is a term since 44 = 2^2 * 11 and 44 + 1 = 45 = 3^2 * 5 both have one unitary prime divisor and one nonunitary prime divisor.

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

Subsequence of A348097.

Cf. A335328.

q[n_] := n == 1 || Count[(e = FactorInteger[n][[;; , 2]]), 1] == Length[e]/2; Select[Range[10^3], q[#] && q[# + 1] &]

A335398 Starts of runs of 4 consecutive numbers that have an equal number of unitary and nonunitary divisors (A048109).

Original entry on oeis.org

2068373, 2948373, 3571749, 3916374, 4730373, 4757750, 6755750, 8109125, 11290872, 12248872, 13071750, 13648311, 13903623, 15278247, 15448374, 15449749, 16793622, 17446374, 17991125, 19407624, 20080248, 20250375, 21594248, 22577750, 24190758, 25297622, 26140373

Offset: 1

Zak Seidov and Amiram Eldar, Jun 06 2020

nonn

			2068373 is a term since 2068373, 2068374, 2068375 and 2068376 each have an equal number of unitary and nonunitary divisors. 2068373 and 2068375 each have 4 unitary divisors and 4 nonunitary divisors, 2068374 has 32 unitary divisors and 32 nonunitary divisors, and 2068376 has 8 unitary divisors and 8 nonunitary divisors.

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

Subsequence of A048109, A335328 and A335397.

Cf. A000005, A034444, A048105.

q[n_] := DivisorSigma[0, n] == 2^(PrimeNu[n] + 1); v = q /@ Range[4]; seq = {}; Do[v = Append[Drop[v, 1], q[k]]; If[And @@ v, AppendTo[seq, k - 3]], {k, 5, 10^7}]; seq

A335399 Starts of runs of 5 consecutive numbers that have an equal number of unitary and nonunitary divisors (A048109).

Original entry on oeis.org

146447622, 2259799749, 2559357269, 2647718871, 3660580374, 4262858871, 4708102374, 5188831623, 5341658373, 5494129749, 5728055749, 5876715750, 6127708374, 6455588247, 6608437623, 6612840374, 6617111750, 6689113623, 6722600373, 7456747623, 7923798375, 8272111445

Offset: 1

Zak Seidov and Amiram Eldar, Jun 06 2020

nonn

Do longer runs of consecutive numbers with an equal number of unitary and nonunitary divisors exist for any length of run?

Starts of runs of 6 consecutive numbers that have an equal number of unitary and nonunitary divisors, from Giovanni Resta's bfile, 80566783622, 117243671750, 390773539750, 573122731621, 636972066374. - Zak Seidov, Jun 07 2020

			146447622 is a term since 146447622, 146447623, 146447624, 146447625 and 146447626 each have an equal number of unitary and nonunitary divisors. 146447622 has 32 unitary divisors and 32 nonunitary divisors, 146447623, 146447625 and 146447626 each have 8 and 8, and 146447624 has 16 and 16.

Giovanni Resta, Table of n, a(n) for n = 1..3000

Subsequence of A048109, A335328, A335397 and A335398.

Cf. A000005, A034444, A048105.

q[n_] := DivisorSigma[0, n] == 2^(PrimeNu[n] + 1); v = q /@ Range[5]; seq = {}; Do[v = Append[Drop[v, 1], q[k]]; If[And @@ v, AppendTo[seq, k - 4]], {k, 6, 3*10^8}]; seq

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).

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);}

cp's OEIS Frontend

A344314 Number k such that k and k+1 have the same number of nonunitary divisors (A048105).

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A335397 Starts of runs of 3 consecutive numbers that have an equal number of unitary and nonunitary divisors (A048109).

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A348098 Number k such that k and k+1 both have an equal number of unitary and nonunitary prime divisors (A348097).

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A335398 Starts of runs of 4 consecutive numbers that have an equal number of unitary and nonunitary divisors (A048109).

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A335399 Starts of runs of 5 consecutive numbers that have an equal number of unitary and nonunitary divisors (A048109).

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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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