A034173 -id:A034173 - cp's OEIS frontend

A006558 Start of first run of n consecutive integers with same number of divisors.

1, 2, 33, 242, 11605, 28374, 171893, 1043710445721, 2197379769820, 2642166652554075

Offset: 1

The entry 40311 given by Guy and by Wells is incorrect. - Jud McCranie, Jan 20 2002
a(10) <= 2642166652554075, a(11) <= 17707503256664346, a(12) <= 9827470582657267545. - David Wasserman, Feb 22 2008
a(10) > 10^13. - Giovanni Resta, Jul 13 2015
a(12) <= 3842083249515874843. - Hugo van der Sanden, Sep 20 2022
a(13) <= 34169215324203592637988571. - Hugo van der Sanden, Apr 13 2022
a(14) <= 9721439902882994590514319997146. - Hugo van der Sanden, Jun 14 2022
a(15) <= 80215613469168729088982885848674841. - Natalia Makarova, Sep 18 2022
a(16) <= 37981337212463143311694743672867136611416. - Vladimir Letsko, Mar 17 2017
a(17) <= 768369049267672356024049141254832375543516. - Vladimir Letsko, Sep 12 2017
a(18) <= 488900003598703704335810037459507226590256411. - Vladimir Letsko, Jun 03 2022
a(19) <= 5908388043825578351730345292813071711296723319324. - Vladimir Letsko, Apr 09 2022
a(20) <= 17668887847524548413038893976018715843277693308027547. Vladimir Letsko, May 30 2022
Spătaru proves that the longest such run up to N is at most exp(C*sqrt(log N log log N)) for some constant C, hence a(n) >> exp(exp(W((log^2 n)/C))) which is approximately exp(log^2 n/(2 log log n)). - Charles R Greathouse IV, Feb 06 2023

			33 has four divisors (1, 3, 11, and 33), 34 has four divisors (1, 2, 17, and 34), 35 has four divisors (1, 5, 7, and 35).  These are the first three consecutive numbers with the same number of divisors, so a(3)=33.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 33, pp 12, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems in Number Theory, section B18.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 87.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, NY, 1986, pages 147 and 176.

Pentti Haukkanen, Some computational results concerning the divisor functions d(n) and sigma(n), The Mathematics Student, Vol. 62 Nos. 1-4 (1993) pp. 166-168. See p. 167.
Vladimir A. Letsko, Some new results on consecutive equidivisible integers, arXiv:1510.07081 [math.NT], 2015.
Vladimir A. Letsko and Vasilii Dziubenko On consecutive equidivisible integers (in Russian), Boundaries of knowledge, 2 (45) 2016.
Carlos Rivera, Problem 20: k consecutive numbers with the same number of divisors, The Prime Puzzles and Problems Connection.
Carlos Rivera, Problem 61: problem 20 revisited, The Prime Puzzles and Problems Connection.
Vlad-Titus Spătaru, Runs of consecutive integers having the same number of divisors, arXiv preprint (2023). arXiv:2301.04464 [math.NT]

Cf. A000005, A005237, A005238, A006601, A049051, A019273, A039665.

Cf. A034173, A115158, A119479.

tau = DivisorSigma[0, #]&;
A006558[q_, w_] := Module[{a, k, j, ok, n}, For[j = 0, j <= w, j++, For[n = 1, n <= q, n++, ok = 1; a = tau[n]; For[k = 1, k <= j, k++, If[a != tau[n + k], ok = 0; Break[]]]; If [ok == 1, Print[n]; Break[]]]]];
A006558[2*10^5, 7] (* Jean-François Alcover, Dec 10 2017 *)

isok(n, k)=nb = numdiv(k); for (j=k+1, k+n-1, if (numdiv(j) != nb, return(0));); 1;
a(n) = {k=1; while (!isok(n, k), k++); k;} \\ Michel Marcus, Feb 17 2016

a(8) from Jud McCranie, Jan 20 2002
a(9) conjectured by David Wasserman, Jan 08 2006
a(9) confirmed by Jud McCranie, Jan 14 2006
a(10) by Jud McCranie, Nov 27 2018

A083790 Smallest start of n consecutive numbers with distinct prime signatures.

Original entry on oeis.org

1, 1, 4, 6, 8, 23, 24, 1804, 1968, 8375, 53750, 243864, 639324, 41381620, 102621343, 7802708736, 7802708736, 178874468615, 1133934949368

Offset: 1

Amarnath Murthy, May 07 2003

a(16) > 750000000. - Ray Chandler, Aug 21 2003
a(16) > 6.646*10^9. - David Wasserman, Jan 08 2006
a(18) > 10^11. - Donovan Johnson, Oct 24 2009
a(20) > 10^13. - Giovanni Resta, Jul 13 2015

A124058 gives another version of the same sequence.
First column of A083788.
Cf. A083789, A083791, A034173.

Corrected and extended by Ray Chandler, Aug 17 2003
More terms from Ray Chandler, Aug 21 2003
a(16)-a(17) from Donovan Johnson, Oct 24 2009
a(18)-a(19) from Giovanni Resta, Jul 13 2015
New name from Jon E. Schoenfield, Nov 17 2022

A034174 a(n) is minimal such that prime factorizations of a(n)-n+1, ..., a(n) have same exponents.

Original entry on oeis.org

1, 3, 35, 19943, 204327, 380480350, 440738966079

Offset: 1

Dean Hickerson, Oct 01 1998

The final terms of the arithmetic progressions defined in A083785. - N. J. A. Sloane, Oct 18 2007
a(8) > 10^13. - Donovan Johnson, Oct 20 2009. [See Reble link for an upper limit.]
The main entry is A034173, which should be updated whenever something relevant is added here. - M. F. Hasler, Oct 28 2012

			a(4)=19943 because 19940, ..., 19943 all have the form p^2 q r.

Don Reble, A034173(8) exists, SeqFan list, Oct 23 2012
Index entries for sequences related to primes in arithmetic progressions

Diagonal of A083785. Cf. A034173, A083785, A083787. See A034173 for more.

A034174(n)={my(f); for(k=n, oo, f=0; for(i=0, n-1, f==(f=vecsort(factor(k-i)[, 2])) || !i || [k+=n-i-1; next(2)]); return(k))} \\ For illustrative purpose; not useful for n>=6. - M. F. Hasler, Oct 28 2012

a(n) = A034173(n) + n - 1. - Max Alekseyev, Nov 10 2009

a(7) from Donovan Johnson, Oct 20 2009

A218448 First of a run of 5 consecutive numbers with same prime signature.

Original entry on oeis.org

204323, 3252571, 5205074, 7201674, 20182921, 28387953, 36193650, 43216722, 51049537, 56155074, 57070850, 61961315, 62167075, 65425473, 76647074, 82507473, 92658049, 95943321, 100498849, 107236449, 109751473, 110899321, 112198075, 112477849, 116736323

Offset: 1

M. F. Hasler, Oct 28 2012

nonn

A number n is in this sequence iff n and n+1 is in A175590; also: iff n and n+2 are in A052214 (in which case n+1 is in A052214, too); and also: iff {n,n+1,n+2,n+3} are in A052213.

A034173(6) = A218448(62) = A218448(63)-1 is the least term n such that n+1 is also in the sequence.

M. F. Hasler and Donovan Johnson, Table of n, a(n) for n = 1..1000 (first 140 terms from _M. F. Hasler_)

Cf. A034173, A034174, A006558, A083790, A052213, A052214, A175590.

is_A218448(n)={my(f);!for(i=0,4,f!=(f=vecsort(factor(n+i)[,2])) & i & return)}

f(k)=vecsort(factor(k)[,2]~,,4)
t=f(n=2);while(n<1e8, for(i=n+1, n+4, tt=f(i); if(tt!=t, n=i; t=tt; next(2))); print1(n", "); n++) \\ Charles R Greathouse IV, Oct 28 2012

a(6)-a(8) from Charles R Greathouse IV, Oct 28 2012
a(9)-a(25) from Donovan Johnson, Oct 28 2012
Values up to a(140) computed using b175590.txt from Charles R Greathouse IV  - M. F. Hasler, Oct 28 2012

A113456 Square array read by antidiagonals: a(n, d) is the smallest number that begins an arithmetic progression with common difference d of n numbers with the same prime signature.

Original entry on oeis.org

1, 2, 1, 33, 3, 1, 19940, 3, 2, 1, 204323, 213, 155, 3, 1, 380480345, 213, 7572, 3, 2, 1

Offset: 1

David Wasserman, Jan 08 2006; corrected Jan 08 2006

First two columns are A034173 and A113457. First three rows are A000012, A086489 and A113458.

			a(3, 2) = 3 because 3, 5 and 7 have the same prime signature.

A083785 Triangle read by rows in which the n-th row gives the smallest set of n consecutive numbers with the same prime signatures.

Original entry on oeis.org

1, 2, 3, 33, 34, 35, 19940, 19941, 19942, 19943, 204323, 204324, 204325, 204326, 204327, 380480345, 380480346, 380480347, 380480348, 380480349, 380480350, 440738966073, 440738966074, 440738966075, 440738966076, 440738966077, 440738966078, 440738966079

Offset: 1

Amarnath Murthy, May 07 2003

			       1
       2      3
      33     34     35
   19940  19941  19942  19943
  204323 204324 204325 204326 204327
  ...

Index entries for sequences related to primes in arithmetic progressions

The first column is given by A034173, the final terms by A034174.

Cf. A034173, A034174, A083787.

7th row added using A034173 by Michel Marcus, Dec 13 2015

A083787 Sequence contains the prime signature pertaining to the n-th row of A083785.

Original entry on oeis.org

1, 2, 6, 60, 60, 60, 420

Offset: 1

Amarnath Murthy, May 07 2003

			a(4) = 2^2*3*5 = 60 because A083785(4, 1) = 19940 = 2^2*5*997, (p^2*q*r) etc.

Cf. A034173, A034174, A046523, A083785.

a(n) = A046523(A034173(n)). - Amiram Eldar, Jul 28 2024

a(6)-a(7) from Jinyuan Wang, May 24 2020

A218455 First of a run of 6 consecutive numbers with same prime signature.

Original entry on oeis.org

380480345, 2713001274, 6282718946, 7209536449, 9809067073, 10684724346, 12008728850, 14824913049, 17231547073, 17552118546, 17659180314, 18036555273, 20473171322, 21507097001, 23676804346, 24742649321, 25401767522, 25694056449, 27656894273, 28259097818

Offset: 1

M. F. Hasler, Oct 29 2012

nonn

A number n is in this sequence iff n and n+1 is in A218448; see the comment there for other characterizations in terms of membership in A175590 or A052214 or A052213.

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

Cf. A034173, A034174, A006558, A083790, A052213, A052214, A175590, A218448, A218865.

is_A218455(n)={my(s(n)=vecsort(factor(n)[,2]),t=s(n));!for(m=n+1,n+5, t!=s(m) & return)}

a(2)-a(20) from Donovan Johnson, Oct 29 2012

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

Original entry on oeis.org

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

A124058 Smallest of the first occurrence of n consecutive integers with all different prime signatures.

Original entry on oeis.org

2, 3, 4, 6, 8, 23, 24, 1804, 1968, 8375, 53750, 243864, 639324, 41381620, 102621343, 7802708736, 7802708736, 178874468615, 1133934949368

Offset: 1

Joe McCauley (mccauleyj(AT)insightbb.com), Nov 03 2006

Essentially the same as A083790. - T. D. Noe, Nov 10 2006
a(18) > 10^11. - Donovan Johnson, Oct 24 2009
a(20) > 10^13. - Giovanni Resta, Jul 13 2015

			a(6) = 23 because it begins the first occurrence of 6 consecutive integers that have all different prime signatures ({1}, {1,3}, {2}, {1,1}, {3}, {1,2} respectively for 23 through 28).

Eric Weisstein's World of Mathematics, Prime Signature.

Cf. A034173, A025487, A083790, A083788.

a(12)-a(15) added from A083790, Nov 10 2006
a(16)-a(17) from Donovan Johnson, Oct 24 2009
a(18)-a(19) from Giovanni Resta, Jul 13 2015

cp's OEIS Frontend

A006558 Start of first run of n consecutive integers with same number of divisors.

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A083790 Smallest start of n consecutive numbers with distinct prime signatures.

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A034174 a(n) is minimal such that prime factorizations of a(n)-n+1, ..., a(n) have same exponents.

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A218448 First of a run of 5 consecutive numbers with same prime signature.

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A113456 Square array read by antidiagonals: a(n, d) is the smallest number that begins an arithmetic progression with common difference d of n numbers with the same prime signature.

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A083785 Triangle read by rows in which the n-th row gives the smallest set of n consecutive numbers with the same prime signatures.

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A083787 Sequence contains the prime signature pertaining to the n-th row of A083785.

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A218455 First of a run of 6 consecutive numbers with same prime signature.

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

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