A045983 -id:A045983 - cp's OEIS frontend

A358044 a(n) is the smallest number k such that n consecutive integers starting at k have the same number of triangular divisors (A007862).

1, 1, 55, 5402, 2515069

Offset: 1

Ilya Gutkovskiy, Oct 26 2022

Any subsequent terms are > 10^10. - Lucas A. Brown, Jan 06 2023

			55 has 2 triangular divisors {1, 55}, 56 has 2 triangular divisors {1, 28} and 57 has 2 triangular divisors {1, 3}. These are the first 3 consecutive numbers with the same number of triangular divisors, so a(3) = 55.

Index entries for sequences related to divisors of numbers
Lucas A. Brown, Python program.

Cf. A000217, A006558, A007862, A045983, A045984, A324593, A324594, A338628.

A305235 Smallest positive number k such that there are exactly n successive equal values of A001221 starting at k, i.e., such that A305234(k) = n.

Original entry on oeis.org

1, 4, 3, 2, 54, 91, 142, 141, 44360, 48919, 218972, 526097, 526096, 526095, 17233173, 127890362, 29138958036, 118968284929, 118968284928, 585927201065, 585927201064, 585927201063, 585927201062, 313978488186061, 453918847597185, 453918847597184, 455626105596320

Offset: 0

Felix Fröhlich, May 28 2018

nonn

a(27) > 2 * 10^15. - Toshitaka Suzuki, Jun 22 2025

			For n = 5: A001221(91+k) = 2 for k = 0..5 and 91 is the smallest number x with exactly 5 successors that have the same value of A001221 as x, so a(5) = 91.

Cf. A001221, A001274, A045983, A048932, A048971, A077657, A305234.

a305234(n) = my(k=n+1, i=0); while(omega(k)==omega(n), i++; k++); i
a(n) = my(k=1); while(1, if(a305234(k)==n, return(k)); k++)

a(16)-a(22) from Toshitaka Suzuki, Apr 01 2025

a(23)-a(26) from Toshitaka Suzuki, Jun 22 2025

A075041 Erroneous version of A006049.

Original entry on oeis.org

2, 14, 16, 20, 21, 31, 33, 34, 35, 38, 39, 44, 45, 50, 51, 54, 55, 56, 57, 62, 68, 74, 75, 76, 85, 86, 87

Offset: 1

Amarnath Murthy, Sep 03 2002

dead

This appears to be either an erroneous version of A045920 or an erroneous version of A006049. - R. J. Mathar, Aug 13 2012

Cf. A045983.

A084296 Triangle: number of distinct prime factors in n-th primorial numbers when n prime factors first appears and in n-1 subsequent integers after.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 4, 1, 2, 2, 5, 1, 2, 2, 3, 6, 2, 2, 3, 2, 2, 7, 3, 2, 3, 3, 2, 4, 8, 2, 3, 2, 4, 2, 3, 2, 9, 2, 3, 3, 3, 2, 4, 3, 4, 10, 3, 3, 2, 2, 2, 4, 3, 3, 2, 11, 1, 4, 3, 2, 4, 5, 4, 3, 3, 4, 12, 3, 3, 4, 2, 3, 6, 2, 3, 5, 4, 3, 13, 3, 4, 2, 3, 3, 3, 3, 3, 3, 6, 2, 4, 14, 2, 3, 2, 4, 5, 4, 5, 3, 3, 6, 4

Offset: 1

Labos Elemer, May 27 2003

Omega-values(=A001221) in the subsequent neighborhood of radius n, for primorial numbers are usually neither all distinct or all equal items as it is required in A068069, A045983 sequences.

			n-th row of table consists of n numbers A001221[A002110(n+j)], j=0...n-1:
1,
2,1,
3,1,1,
4,1,2,2,
5,1,2,2,3,
6,2,2,3,2,2,
7,3,2,3,3,2,4,
Rows starts with n at indices which are central polygonal numbers:a[A000124(n)]=n; rows ends at a[A000217(n)] terms, at triangular number indices.

Cf. A001221, A002110, A068069, A045983, A000217, A000124.

lf[x_] := Length[FactorInteger[x]] q[x_] := Apply[Times, Table[Prime[w], {w, 1, x}]] Flatten[Table[Table[lf[q[n]+j], {j, 0, n-1}], {n, 1, 20}], 1]

A318529 a(n) begins the first run of at least n consecutive numbers with same number of exponential divisors.

Original entry on oeis.org

1, 1, 1, 242, 3624, 22020, 671346, 8870024, 49250144, 463239475, 1407472722, 82462576220, 82462576220, 5907695879319

Offset: 1

Amiram Eldar, Aug 28 2018

From David A. Corneth, Aug 28 2018: (Start)
For 4 <= n <= 10, a(n) has two exponential divisors. Most numbers have 1 or 2 exponential divisors.
For n > 3, a(n) isn't squarefree. (End)
For n >= 2^(k+1), A049419(a(n)) must be divisible by A051548(k), because for 1 <= j <= k at least one of a(n),...,a(n)+n-1 has 2-adic order j. - Robert Israel, Sep 07 2018

			a(4) = 242 since the number of exponential divisors of 242, 243, 244, and 245 is 2, and this is the first run of 4 consecutive numbers.

Eric Weisstein's World of Mathematics, e-Divisor

Cf. A005117, A006558, A025487, A045983, A049419, A051548, A304463, A318166.

edivnum[1] = 1; edivnum [p_?PrimeQ] = 1; edivnum [p_?PrimeQ, e_] := DivisorSigma[ 0, e ]; edivnum [n_] := Times @@ (edivnum [#[[1]], #[[2]]] & ) /@ FactorInteger[ n ]; Seq[n_,q_] := Map[edivnum, Range[n, n + q - 1]]; findConsec[q_, nmin_, nmax_] := Module[{}, s = Seq[1, q]; n = q + 1; found = False; Do[ If[ CountDistinct[s] == 1, found = True; Break[] ]; s = Rest[AppendTo[s, edivnum[n] ]]; n++, {k, nmin, nmax}]; If[found, n - q, 0]]; seq = {1}; nmax = 100000000; Do[n1 = Last[seq]; s1 = findConsec[m, n1, nmax]; If[s1 == 0, Break[]]; AppendTo[ seq, s1 ], {m, 2, 7}]; seq (* after Jean-François Alcover in A049419 *)

a(11)-a(13) from Giovanni Resta, Aug 28 2018

a(14) from Giovanni Resta, Sep 07 2018

A349262 a(n) is the start of the least run of exactly n consecutive numbers with the same value of A349258.

Original entry on oeis.org

1, 14, 20, 2, 91, 6850, 2302, 141, 56014, 184171, 2800171, 27805034, 35297611, 8313366182, 1791416073, 3618621410

Offset: 1

Amiram Eldar, Nov 12 2021

a(17) > 10^11, if it exists.

			a(2) = 14 since A349258(14) = A349258(15) = 2, but A349258(13) != 2 and A349258(16) != 2.

Cf. A349258.

Similar sequences: A006558, A045983, A048932, A067813, A077657, A318166.

f[p_, e_] := 2^DigitCount[e, 2, 1] - 1; d[1] = 0; d[n_] := Plus @@ f @@@ FactorInteger[n]; seq[len_, nmax_] := Module[{s = Table[0, {len}], dprev = 0, n = 2, c = 1, k = 1}, s[[1]] = 1; While[k < len && n < nmax, d1 = d[n]; If[d1 == dprev, c++, If[c > 0 && c <= len && s[[c]] == 0, k++; s[[c]] = n - c]; c = 1]; n++; dprev = d1]; TakeWhile[s, # > 0 &]]; seq[8, 10^4]

A349305 a(n) is the start of the least run of exactly n consecutive numbers with the same number of nonunitary divisors.

Original entry on oeis.org

4, 10, 1, 19940, 54584, 204323, 2789143044, 27092041443

Offset: 1

Amiram Eldar, Nov 14 2021

a(9) > 10^11, if it exists.

			a(2) = 10 since A048105(10) = A048105(11) = 0, and A048105(9) != 0 and A048105(12) != 0.

Cf. A048105, A344315.

Similar sequences: A006558, A045983, A048932, A067813, A077657, A318166.

d[n_] := DivisorSigma[0, n] - 2^PrimeNu[n]; seq[len_, nmax_] := Module[{s = Table[0, {len}], dprev = -1, n = 1, c = 0, k = 0}, While[k < len && n < nmax, d1 = d[n]; If[d1 == dprev, c++, If[c > 0 && c <= len && s[[c]] == 0, k++; s[[c]] = n - c]; c = 1]; n++; dprev = d1]; TakeWhile[s, # > 0 &]]; seq[6, 10^6]

cp's OEIS Frontend

A358044 a(n) is the smallest number k such that n consecutive integers starting at k have the same number of triangular divisors (A007862).

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A305235 Smallest positive number k such that there are exactly n successive equal values of A001221 starting at k, i.e., such that A305234(k) = n.

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A075041 Erroneous version of A006049.

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A084296 Triangle: number of distinct prime factors in n-th primorial numbers when n prime factors first appears and in n-1 subsequent integers after.

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Mathematica

A318529 a(n) begins the first run of at least n consecutive numbers with same number of exponential divisors.

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Mathematica

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A349262 a(n) is the start of the least run of exactly n consecutive numbers with the same value of A349258.

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A349305 a(n) is the start of the least run of exactly n consecutive numbers with the same number of nonunitary divisors.

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