A374673
a(n) is the start of the least run of exactly n consecutive positive numbers with an equal value of A177329, or -1 if no such run exists.
Original entry on oeis.org
2, 8, 44, 83, 4475, 75093, 164903, 59480, 1342805
Offset: 1
n | a(n) | A177329(k), k = a(n), a(n)+1, ..., a(n)+n-1
--|--------|------------------------------------------------
1 | 2 | A177329(2) = 1
2 | 8 | A177329(8) = A177329(9) = 6
3 | 44 | A177329(44) = A177329(45) = A177329(46) = 21
4 | 83 | A177329(83) = ... = A177329(86) = 35
5 | 4475 | A177329(4475) = ... A177329(4479) = 923
6 | 75093 | A177329(75093) = ... = A177329(75098) = 10857
7 | 164903 | A177329(164903) = ... = A177329(164909) = 22038
8 | 59480 | A177329(59480) = ... = A177329(59487) = 8814
-
s[n_] := Module[{e = FactorInteger[n!][[;; , 2]]}, Sum[DigitCount[e[[k]], 2, 1], {k, 1, Length[e]}]]; seq[len_] := Module[{v = Table[0, {len}], w = {0}, c = 0, k = 3, m, s1}, While[c < len, s1 = s[k]; m = Length[w]; If[s1 == w[[m]], AppendTo[w, s1], If[m <= len && v[[m]] == 0, v[[m]] = k-m; c++]; w = {s1}]; k++]; v]; seq[5]
-
s(n) = {my(e = factor(n!)[, 2]); sum(k=1, #e, hammingweight(e[k]));}
lista(len) = {my(v = vector(len), w = [0], c = 0, k = 3, m, s1); while(c < len, s1 = s(k); m = #w; if(s1 == w[m], w = concat(w, s1), if(m < = len && v[m] == 0, v[m] = k-m; c++); w = [s1]); k++); v;}
-
from itertools import count
from collections import Counter
from sympy import factorint
def A374673(n):
if n==1: return 2
c, a, l = Counter(), 0, 0
for m in count(2):
c += Counter(factorint(m))
b = sum(map(int.bit_count,c.values()))
if b==a:
l += 1
else:
if l==n-1:
return m-n
l = 0
a = b # Chai Wah Wu, Jul 18 2024
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
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
-
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 *)
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
a(2) = 14 since A349258(14) = A349258(15) = 2, but A349258(13) != 2 and A349258(16) != 2.
-
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
a(2) = 10 since A048105(10) = A048105(11) = 0, and A048105(9) != 0 and A048105(12) != 0.
-
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]
Showing 1-4 of 4 results.
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