A077657 -id:A077657 - cp's OEIS frontend

A045984 a(n) = smallest number m such that factorizations of n consecutive integers starting at m have same number of primes (counted with multiplicity).

1, 2, 33, 602, 602, 2522, 211673, 3405122, 3405122, 49799889, 202536181, 3195380868, 5208143601, 85843948321, 97524222465

Offset: 1

David W. Wilson

a(16) > 10^13. a(16) must have at least 5 prime factors (counted with multiplicity) because one of the 16 consecutive numbers is divisible by 2^4. - Donovan Johnson, Apr 01 2013

			a(4) = 602 as 602 = 2 * 7 * 43, 603 = 3 * 3 * 67, 604 = 2 * 2 * 151, 605 = 5 * 11 * 11 so four consecutive positive integers have the same number of prime factors starting at 602, the first such number. - _David A. Corneth_, Feb 24 2024

Cf. A001222, A045920, A045939, A045940, A045941, A045942, A077657, A117969, A356953.

More terms from Vladeta Jovovic, Aug 06 2002

More terms from Martin Fuller, Nov 21 2006

A077655 Number of consecutive successors of n having the same number of prime factors as n (counted with multiplicity).

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Nov 13 2002

nonn

If a(n) > 0 then a(n+1) = a(n)-1.

			33=3*11 has only two successors also with two factors: 34=2*17 and 35=5*7 (whereas 33+3=36=2*2*3*3), therefore a(33)=2.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A001222, A077656, A045920, A045939, A045940, A045941, A045942, A077657.

snpf[n_]:=Module[{f=PrimeOmega[n],k=0},While[f==PrimeOmega[n+k],k++];k]; Array[snpf,110]-1 (* Harvey P. Dale, Aug 01 2021 *)

A077655(n) = { my(k=n+1,w=bigomega(n)); while(bigomega(k)==w,k++); (k-n)-1; }; \\ Antti Karttunen, Jan 22 2020

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

A356953 Least nonzero starting number in the first run of exactly n consecutive numbers having the same number of prime factors counted with multiplicity, or -1 if no such number exists.

Original entry on oeis.org

1, 2, 33, 1083, 602, 2522, 211673, 6612470, 3405122, 49799889, 202536181, 3195380868, 5208143601, 85843948321, 97524222465, 361385490681003, 441826936079342

Offset: 1

Jean-Marc Rebert, Sep 06 2022

In the definition, "exactly" means the run is not part of a longer run.

a(18) > 2 * 10^15. - Toshitaka Suzuki, Aug 31 2025

			2 and 3 are 2 consecutive numbers and have the same number of prime factors, and 2 is the smallest such number, hence a(2) = 2.

Cf. A001222, A077657, A356855.

Cf. A045920, A115186, A113752, A356893.

card(m)=my(c=0,k=bigomega(m));if(bigomega(m-1)!=k,while(bigomega(m)==k,c++;m++));c
a(n)=if(n==1,return(1));for(m=2,+oo,if(card(m)==n,return(m)))

a(16)-a(17) from Toshitaka Suzuki, Aug 31 2025

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

A045984 a(n) = smallest number m such that factorizations of n consecutive integers starting at m have same number of primes (counted with multiplicity).

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A077655 Number of consecutive successors of n having the same number of prime factors as n (counted with multiplicity).

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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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A356953 Least nonzero starting number in the first run of exactly n consecutive numbers having the same number of prime factors counted with multiplicity, or -1 if no such number exists.

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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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