A080569
a(n) is the first number in the first run of at least n successive numbers, all having exactly 3 distinct prime factors.
Original entry on oeis.org
30, 230, 644, 1308, 2664, 6850, 10280, 39693, 44360, 48919, 218972, 526095, 526095, 526095, 17233173, 127890362, 29138958036, 146216247221, 23671413563491, 36966736685739
Offset: 1
a(3) = 644 because 644 = 2^2 * 7 * 23, so omega(644) = 3, 645 = 3*5*43, so omega(645) = 3 and 646 = 2*17*19, so omega(646) = 3 and no other number n < 644 has omega(n)=omega(n+1)=omega(n+2)=3.
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k = 1; Do[ While[ Union[ Table[ Length[ FactorInteger[i]], {i, k, k + n - 1}]] != {3}, k++ ]; Print[k], {n, 1, 16}]
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k=1; for(i=1,600000,s=1; for(j=1,k,if(omega(i+j-1)!=3,s=0,)); if(s==1,print1(i,", "); k++; i--,) )
A048971
Let d(n) = number of distinct primes dividing n (A001221); sequence gives t such that d(t)=d(t+1)=...=d(t+n-1) is a run of record length.
Original entry on oeis.org
1, 14, 54, 91, 323, 44360, 48919, 218972, 534078, 2699915, 3355906, 17233173
Offset: 1
54, 55, 56, 57, 58 is a run of length 5 in A001221 all with d = 2.
More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Sep 04 2006
A048972
Length of record run mentioned in A048971.
Original entry on oeis.org
1, 2, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16
Offset: 1
54, 55, 56, 57, 58 is a run of length 5 in A001221 all with d = 2.
More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Sep 04 2006
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
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.
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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++)
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.
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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.
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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 1-6 of 6 results.
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