A052213 -id:A052213 - cp's OEIS frontend

A280074 Numbers k such that Sum_{d|k} tau(d) = Sum_{d|k+1} tau(d).

2, 14, 21, 33, 34, 38, 44, 57, 75, 85, 86, 93, 94, 98, 116, 118, 122, 133, 135, 141, 142, 145, 147, 158, 171, 177, 201, 202, 205, 213, 214, 217, 218, 230, 244, 253, 285, 296, 298, 301, 302, 326, 332, 334, 375, 381, 387, 393, 394, 429, 434, 445, 446, 453, 481

Offset: 1

Jaroslav Krizek, Dec 25 2016

nonn

tau(n) is the number of positive divisors of n (A000005).
Numbers k such that A007425(k) = A007425(k+1).
Subsequence of A052213 and A005237.
Sequence is not the same as A052213, first deviation is at a(212): A052213(212) = 2041, a(212) = 2024. Number 2024 is the smallest number n such that A007425(n) = A007425(n+1) with different prime signatures of numbers n and n+1 (2024 = 2^3 * 11 * 23, 2025 = 3^4 * 5^2; A007425(2024) = A007425(2025) = 90).
Sequence of the smallest numbers k such that Sum_{d|k} tau(d) = Sum_{d|k+1} tau(d) = ... = Sum_{d|k+n-1} tau(d) for n>=1: 1, 2, 33, 19940, 204323, 380480345, 440738966073, ...; conjecture: this sequence is different from A034173.

			2 is a term because Sum_{d|2} tau(d) = Sum_{d|3} tau(d) = 1 + 2 = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Jaroslav Krizek)

Cf. A000005, A005237, A007425, A052213.

[n: n in [1..10000] | &+[NumberOfDivisors(d): d in Divisors(n)]  eq &+[NumberOfDivisors(d): d in Divisors(n+1)]]

Select[Range@ 500, Total@ Map[DivisorSigma[0, #] &, Divisors@ #] == Total@ Map[DivisorSigma[0, #] &, Divisors[# + 1]] &] (* Michael De Vlieger, Dec 25 2016 *)

sd(n) = sumdiv(n, d, numdiv(d)); \\ A007425
isok(m) = sd(m) == sd(m+1); \\ Michel Marcus, Apr 28 2020

A280684 a(n) = number of divisors of the product of the divisors of n.

Original entry on oeis.org

1, 2, 2, 4, 2, 9, 2, 7, 4, 9, 2, 28, 2, 9, 9, 11, 2, 28, 2, 28, 9, 9, 2, 65, 4, 9, 7, 28, 2, 125, 2, 16, 9, 9, 9, 100, 2, 9, 9, 65, 2, 125, 2, 28, 28, 9, 2, 126, 4, 28, 9, 28, 2, 65, 9, 65, 9, 9, 2, 637, 2, 9, 28, 22, 9, 125, 2, 28, 9, 125, 2, 247, 2, 9, 28

Offset: 1

Jaroslav Krizek, Jan 07 2017

nonn

			For n = 4; a(n) = tau (1*2*4) = tau(8) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A000040, A001248, A007955, A008578, A052213, A280685, A280582.

[#[d: d in Divisors(&*[d: d in Divisors(n)])]: n in [1..100]]

A007955(n) = if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2)); \\ This function from Charles R Greathouse IV, Feb 11 2011
A280684(n) = numdiv(A007955(n)); \\ Antti Karttunen, May 19 2017

a(n) = A000005(A007955(n)).
a(p) = 2 for p = primes (A000040).
a(n) = 4 for squares of primes (A001248).
a(n) = n for numbers 1, 2, 4, 28, 100, ...
a(n) = tau(n) for noncomposites (A008578).
a(n) = a(n+1) for numbers in A052213.

A333057 Numbers k such that k and k+1 have different (ordered) prime signatures and d_3(k) = d_3(k+1), where d_3 is A007425.

Original entry on oeis.org

2024, 5624, 13688, 15375, 21608, 50300, 62775, 69375, 70784, 108927, 110888, 116864, 118016, 130815, 149768, 152703, 164024, 213759, 221823, 224720, 238975, 242432, 255231, 257175, 283904, 297135, 324224, 341887, 346544, 365295, 366848, 366975, 379647, 455552

Offset: 1

Amiram Eldar, Mar 06 2020

nonn

Apparently most of the numbers k such that k and k+1 have the same value of d_3 also have the same prime signature. a(1) = 2024 is the 212th number k such that d_3(k) = d_3(k+1), and up to 10^8 there are 8026247 such numbers k of them only 6414 are not in A052213.

			2024 is a term since d_3(2024) = d_3(2025) = 90, and the prime signatures of 2024 = 2^3 * 11 * 23 and 2025 = 3^4 * 5^2 are different ([1, 1, 3] and [2, 4]).

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

Cf. A007425, A052213, A124010, A333055.

f[p_, e_] := (e+1)*(e+2)/2;  d3[1] = 1; d3[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[10^5], d3[#] == d3[#+1] && Sort[FactorInteger[#][[;;,2]]] != Sort[FactorInteger[#+1][[;;,2]]] &]

A359745 Numbers k such that k and k+1 have the same ordered prime signature.

Original entry on oeis.org

2, 14, 21, 33, 34, 38, 44, 57, 85, 86, 93, 94, 116, 118, 122, 133, 135, 141, 142, 145, 158, 171, 177, 201, 202, 205, 213, 214, 217, 218, 230, 253, 285, 296, 298, 301, 302, 326, 332, 334, 381, 387, 393, 394, 429, 434, 445, 446, 453, 481, 501, 514, 526, 537, 542

Offset: 1

Amiram Eldar, Jan 13 2023

nonn

The ordered prime signature of a number n is the list of exponents of the distinct prime factors in the prime factorization of n, in the order of the prime factors (A124010).

			14 is a term since 14 = 2^1 * 7^1 and 15 = 3^1 * 5^1 have the same ordered prime signature, (1, 1).
44 is a term since 44 = 2^2 * 11^1 and 45 = 3^2 * 5^1 have the same ordered prime signature, (2, 1).
75 is a term of A052213 but not a term of this sequence, since 75 = 3^1 * 5^2 and 76 = 2^2 * 19^1 have different ordered prime signatures, (1, 2) and (2, 1).

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

Subsequence of A052213.
A359746 is a subsequence.
Cf. A124010.

q[n_] := SameQ @@ (FactorInteger[#][[;; , 2]]& /@ (n + {0, 1})); Select[Range[2, 600], q]

lista(nmax) = {my(e1 = [], e2); for(n = 2, nmax, e2 = factor(n)[,2]; if(e1 == e2, print1(n-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).

A178800 Smaller of two consecutive numbers with prime signature {1,2,3}.

Original entry on oeis.org

41624, 177624, 336500, 515096, 625975, 686375, 767124, 789399, 816776, 829575, 985959, 1035800, 1059624, 1173500, 1184183, 1276856, 1340199, 1365875, 1369575, 1614500, 1784871, 1790199, 1890567, 2078199, 2175875, 2219624, 2364200, 2379351, 2540600

Offset: 1

Will Nicholes, Jun 15 2010

nonn

The numbers in this sequence are described by both A052213 (consecutive numbers with the same prime signature) and A163569 (numbers with prime signature {1,2,3}).

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A163569, A052213.

e=vector(3); for(n=360, 2540600, if(omega(n)==3, if(bigomega(n)==6, if(omega(n+1)==3, if(bigomega(n+1)==6, for(m=n, n+1, f=factorint(m); for(j=1, 3, e[j]=f[j,2]); e=vecsort(e); for(j=1, 3, if(e[j]<>j, next(3)))); print1(n ", ")))))) \\ Donovan Johnson, Dec 17 2013

Extended by Ray Chandler, Jul 30 2010

A218865 First of a run of 7 consecutive numbers with same prime signature.

Original entry on oeis.org

440738966073, 464881210073, 645462662449, 914549532721, 932537185321, 936083045673, 1043710445721, 1091100709673, 1225660528209, 1349165568945, 1602377996921, 1682159750473, 1714275593649, 1730241245545, 1756186304521, 1872671302049, 1956516794721, 1987087485225

Offset: 1

Donovan Johnson, Nov 08 2012

nonn

a(1) = A034173(7) = A218455(443) = A218455(444)-1.

a(19) > 2*10^12. - Donovan Johnson, May 11 2013

			The 7 consecutive numbers from 645462662449 to 645462662455 all have a prime signature of 2,1,1,1.

Cf. A034173, A034174, A052213, A052214, A175590, A218448, A218455.

a(7)-a(18) from Donovan Johnson, May 11 2013

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

A280074 Numbers k such that Sum_{d|k} tau(d) = Sum_{d|k+1} tau(d).

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A280684 a(n) = number of divisors of the product of the divisors of n.

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A333057 Numbers k such that k and k+1 have different (ordered) prime signatures and d_3(k) = d_3(k+1), where d_3 is A007425.

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A359745 Numbers k such that k and k+1 have the same ordered prime signature.

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

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A178800 Smaller of two consecutive numbers with prime signature {1,2,3}.

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A218865 First of a run of 7 consecutive numbers with same prime signature.

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

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