A052214 -id:A052214 - cp's OEIS frontend

A034173 a(n) is minimal such that prime factorizations of a(n), ..., a(n)+n-1 have same exponents.

1, 2, 33, 19940, 204323, 380480345, 440738966073

Offset: 1

Dean Hickerson, Oct 01 1998

a(8) > 10^13. - Donovan Johnson, Oct 20 2009
Don Reble has shown that a(8) < 1.9*10^42, cf. link.
From David Wasserman, Jan 05 2019: (Start)
a(8) <= 108111092880293127811946663766147737122,
a(9) <= 6850672946809600696044301071559918192380244,
a(10) <= 96037988156124494415303285590850571857698741869620,
a(11) <= 9044737840075556371215937303485030235666252755947862558252154847122. (End)

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

Don Reble, A034173(8) exists, SeqFan list, Oct 23 2012.

Cf. A034174.
Cf. A006558, A083790.
Cf. A052213, A052214, A175590, A218448. This sequence is the first column of A083785 and first row of A113456. The latter generalizes to arithmetic progressions with step d>=1. - M. F. Hasler, Oct 28 2012

A034173(n)={my(f);for(k=1,oo,f=0;for(i=1,n, f==(f=vecsort(factor(k+n-i)[,2])) || i==1 || [k+=n-i; next(2)]);return(k))} \\ M. F. Hasler, Oct 23 2012

a(n) = A034174(n) - n + 1. - Max Alekseyev, Nov 10 2009

a(n) = A083785(n,1) = A113456(1,n); a(2) = A052213(1), a(3) = A052214(1), a(4) = A175590(1), a(5) = A218448(1), a(6) = A218448(62) = A218448(63)-1. - M. F. Hasler, Oct 28 2012

a(7) from Donovan Johnson, Oct 20 2009

Don Reble link repaired by N. J. A. Sloane, Oct 24 2024

A066509 a(n) is the first of a triple of consecutive integers, each of which is both the product of three distinct primes and also the product of three primes counted with multiplicity.

Original entry on oeis.org

1309, 1885, 2013, 2665, 3729, 5133, 6061, 6213, 6305, 6477, 6853, 6985, 7257, 7953, 8393, 8533, 8785, 9213, 9453, 9821, 9877, 10281, 10945, 11605, 12453, 12565, 12801, 12857, 12993, 13053, 14133, 14313, 14329, 14465, 14817, 15085, 15265, 15805, 16113, 16133

Offset: 1

Jason Earls, Jan 04 2002

nonn

A subsequence of A052214 and thus of A005238. - M. F. Hasler, Jan 05 2013
Also, the start of pairs of adjacent sphenic twins, i.e., a(n) = A215217(k) such that A215217(k+1) = A215217(k)+1. Therefore these triples might be called "sphenic triples". They form a subsequence of A242606. - M. F. Hasler, May 18 2014
Minimal difference is 4 which occurs at indices n = {316, 547, 566, 604, 666, 695, 821, 874, 979, ...}. - Zak Seidov, Jul 04 2020

			a(5) = 3729 because it along with 3730 and 3731 are all the product of three distinct primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
G. L. Honaker, Jr. and Chris K. Caldwell, Prime Curios! 1309.

Subsequence of A052214 and hence of A005238.

Subsequence of A215217, A007675, A242606 and A168626.

f[n_]:=Last/@FactorInteger[n]=={1,1,1};lst={};Do[If[f[n]&&f[n+1]&&f[n+2],AppendTo[lst,n]],{n,9!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 04 2010 *)
SequencePosition[Table[If[PrimeNu[n]==PrimeOmega[n]==3,1,0],{n,17000}],{1,1,1}][[;;,1]] (* Harvey P. Dale, Feb 28 2025 *)

Trip(n) = { local(f); f=factor(n); if (matsize(f)[1] != 3, return(0)); for(i=1, 3, if (f[i, 2] != 1, return(0))); return(1); } { n=0; for (m=1, 10^10, if (!Trip(m) || !Trip(m+1) || !Trip(m+2), next); write("b066509.txt", n++, " ", m); if (n==1000, return) ) } \\ Harry J. Smith, Feb 19 2010

A066509(n,show_all=0,a=2*3*5,s=[1,1,1]~)={until( !n-- || !a++, until(, factor(a+2)[,2]!=s && (a+=3) && next; factor(a+1)[,2]!=s && (a+=2) && next; factor(a)[,2]==s && break; factor(a+3)[,2]==s && a++ && break; a+=4);show_all && print1(a",")); a} \\ M. F. Hasler, Jan 05 2013

is3dp(n)=my(f=factor(n));matsize(f)==[3,2]&&vecmax(f[,2])==1
list(lim)=my(v=List(),t);forprime(p=17,lim\15, forprime(q=5,min(p-1,lim\3), forprime(r=3,min(q-1,lim\(p*q)), t=p*q*r; if(t%4==1 && is3dp(t+1) && is3dp(t+2), listput(v,t))))); Set(v) \\ Charles R Greathouse IV, Jan 05 2013; updated Jan 22 2025

list(lim)=my(v=List(),ct); forfactored(n=1309,lim\1+2, if(n[2][,2]==[1,1,1]~, if(ct++==3, listput(v,n[1]-2)), ct=0)); Vec(v) \\ Charles R Greathouse IV, Aug 30 2022

a(n) == 1 (mod 4). - Zak Seidov, Mar 31 2020

Definition clarified by Harvey P. Dale, Feb 28 2025

A175590 Numbers k with prime signature(k) = prime signature(k+1) = prime signature(k+2) = prime signature(k+3).

Original entry on oeis.org

19940, 49147, 54585, 118923, 136825, 183554, 204323, 204324, 262932, 304675, 361275, 361322, 476377, 486962, 506905, 619722, 668211, 734948, 854018, 937025, 938203, 999649, 1062025, 1118275, 1335572, 1336075, 1356324, 1466225, 1541491

Offset: 1

Charles R Greathouse IV, Jul 19 2010

nonn

			a(1) = 2^2 * 5 * 997; a(1)+1 = 3 * 17^2 * 23; a(1)+2 = 2 * 13^2 * 59; a(1)+3 = 7^2 * 11 * 37. All have prime signature {2, 1, 1}.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A052213, A052214, A218448. Subsequence of A070284.

SequencePosition[Table[Sort[FactorInteger[n][[All,2]]],{n,1542000}],{x_,x_,x_,x_}][[All,1]] (* Requires Mathematica version 10 or later *) (* The program will take a long time to run. *) (* Harvey P. Dale, Jun 09 2021 *)

sig(n)={vecsort(factor(n)[,2])}; s=sig(1);for(n=1,1e6,t=sig(n+1);if(s==t&t==sig(n+2)&t==sig(n+3),print1(n-1,","));s=t)

is_A175590(n)={my(f(n)=vecsort(factor(n)[,2]),t=f(n));!for(i=1,3,f(n+i)!=t & return)}  \\ M. F. Hasler, Nov 01 2012

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

A075039 Smallest of three consecutive squarefree numbers having equal numbers of prime factors.

Original entry on oeis.org

33, 85, 93, 141, 201, 213, 217, 301, 393, 445, 633, 697, 921, 1041, 1137, 1261, 1309, 1345, 1401, 1641, 1761, 1837, 1885, 1893, 1941, 1981, 2013, 2101, 2181, 2217, 2305, 2361, 2433, 2461, 2517, 2641, 2665, 2721, 2733, 3097, 3385, 3601, 3693, 3729, 3865

Offset: 1

Amarnath Murthy, Sep 03 2002

nonn

			33 is a member as 33, 34 and 35 are of the form p*q where p and q are primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A005117, A052214, A086263, A039833.

Select[Range[4000], AllTrue[# + Range[0, 2], SquareFreeQ] && Equal @@ PrimeNu[# + Range[0, 2]] &] (* Amiram Eldar, Feb 24 2021 *)

A001221(a(n)) = A001222(a(n)) = A001221(a(n)+1) = A001222(a(n)+1).

More terms from Matthew Conroy, Sep 08 2002
Edited by Reinhard Zumkeller, Jul 14 2003
Offset corrected by Amiram Eldar, Feb 24 2021

A113458 Least k such that k, k+n and k+2n have the same prime signature.

Original entry on oeis.org

33, 3, 155, 3, 77, 5, 51, 3, 77, 3, 35, 5, 50, 3, 187, 6, 21, 5, 39, 3, 145, 33, 39, 5, 69, 39, 91, 3, 33, 7, 15, 12, 221, 3, 28, 7, 21, 3, 55, 3, 33, 5, 91, 66, 209, 69, 35, 5, 50, 3, 115, 39, 141, 5, 51, 6, 145, 85, 15, 7, 21, 93, 95, 3, 57, 5, 51, 3, 65, 15, 35, 7, 69, 55, 287, 6

Offset: 1

David Wasserman, Jan 08 2006

Third row of A113456.

			a(4) = 3 because 3, 7 and 11 have the same prime signature.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A052214, A113456, A113467.

s:= n-> sort(map(i-> i[2], ifactors(n)[2])):
a:= proc(n) option remember; local k; for k
      while s(k)<>s(k+n) or s(k)<>s(k+2*n) do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 28 2018

s[n_] := FactorInteger[n][[All, 2]] // Sort;
a[n_] := Module[{k}, For[k = 2, True, k++, If[s[k] == s[k+n] == s[k+2n], Return[k]]]];
Array[a, 100] (* Jean-François Alcover, Nov 05 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

A279767 Numbers m such that m and m+2 have the same prime signature.

Original entry on oeis.org

3, 5, 11, 17, 18, 29, 33, 41, 50, 54, 55, 59, 71, 85, 91, 93, 101, 107, 137, 141, 143, 149, 159, 179, 183, 185, 191, 197, 201, 203, 213, 215, 217, 219, 227, 235, 239, 242, 247, 248, 265, 269, 281, 299, 301, 303, 306, 311, 319, 321, 327, 339, 340, 347, 348, 391, 393, 411, 413

Offset: 1

Altug Alkan, Dec 18 2016

The sequence contains some terms such that m and m + 2k (k > 1) have the same prime signature. For some terms where m and m + 2k share the same prime signature this means that every alternate element between, and including m and m + 2k have the same prime signature. The first such example is where a(41951) = 402677, a(41953) = 402679, and a(41955) = 402681, share the same prime signature {1, 1}. Also the remaining alternate terms excluding endpoints share the same prime signature. Using the above example, a(41952) = 402678 and a(41954) = 402680 share the prime signature {1,1,3}. - Torlach Rush, Feb 25 2018

			18 is a term because 18 = 2 * 3^2 and 18 + 2 = 20 = 2^2 * 5.
19 is not a term because it is prime and 21 is the product of two primes, so the prime signatures are different.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5585 from Michel Marcus)
Index to sequences related to prime signature.

Cf. A001359, A052213, A052214.

primeSignature[n_] := Sort[Transpose[FactorInteger[n]][[2]]]; Select[ Range[2, 1000], primeSignature[#] == primeSignature[# + 2] &] (* Adapted from A052213 *)

isok(n) = vecsort(factor(n)[,2]) == vecsort(factor(n+2)[,2]); \\ Michel Marcus, Feb 25 2018

A333056 Numbers k such that k, k+1 and k+2 have different prime signatures and d(k) = d(k+1) = d(k+2), where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

59318, 72063, 72224, 184190, 185192, 215648, 300320, 355454, 362624, 384128, 548936, 550016, 640790, 682624, 707966, 723896, 758888, 828872, 828873, 858494, 860030, 888704, 901503, 963486, 963710, 993375, 1039742, 1039743, 1081214, 1248776, 1261897, 1340630

Offset: 1

Amiram Eldar, Mar 06 2020

nonn

Apparently most of the numbers k such that d(k) = d(k+1) = d(k+2) (A005238) are terms of A052214, i.e., k, k+1 and k+2 have the same prime signature.

Of the first 10000 terms of A005238, 6406 are also in A052214, 3578 have a pair (k and k+1, k and k+2, or k+1 and k+2) with the same prime signature, and only 16 are in this sequence.

			59318 is a term since d(59318) = d(59319) = d(59320) = 16, and the prime signatures of these 3 numbers are different: 59318 = 2 * 7 * 19 * 223, 59319 = 3^3 * 13^3, and 59320 = 2^3 * 5 * 1483 have 3 different ordered prime signatures (A124010): [1, 1, 1, 1], [3, 3], and [1, 1, 3].

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A005238.

Cf. A000005, A052214, A124010, A333055.

psig[n_] := Sort @ FactorInteger[n][[;; , 2]]; d[sig_] := Times @@ (sig + 1); vsig = psig /@ Range[2, 4]; seqQ[v_] := Length@Union[v] == 3 && Length @ Union[d /@ v] == 1; seq = {}; Do[If[seqQ[vsig], AppendTo[seq, n - 3]]; vsig = Join[Rest[vsig], {psig[n]}], {n, 5, 10^6}]; seq

A359746 Numbers k such that k, k+1 and k+2 have the same ordered prime signature.

Original entry on oeis.org

33, 85, 93, 141, 201, 213, 217, 301, 393, 445, 633, 697, 921, 1041, 1137, 1261, 1309, 1345, 1401, 1641, 1761, 1837, 1885, 1893, 1941, 1981, 2013, 2101, 2181, 2217, 2305, 2361, 2433, 2461, 2517, 2641, 2665, 2721, 2733, 3097, 3385, 3601, 3693, 3729, 3865, 3901, 3957

Offset: 1

Amiram Eldar, Jan 13 2023

nonn

First differs from its subsequence A039833 at n = 17, and from its subsequence A075039 at n = 53.
The ordered prime signature of a number n is the list of exponents of the distinct prime factors in the prime factorization of n, in the order of the prime factors (A124010).
Can 4 consecutive integers have the same ordered prime signature? There are no such quadruples below 10^9.
The answer to the question above is no. Two out of every four consecutive numbers are even and their powers of 2 are different. - Ivan N. Ianakiev, Jan 13 2023

			33 is a term since 33 = 3^1 * 11^1, 34 = 2^1 * 17^1, and 35 = 5^1 * 7^1 have the same ordered prime signature, (1, 1).
4923 is a term since 4923 = 3^2 * 547^1, 4924 = 2^2 * 1231^1, and 4925 = 5^2 * 197^1 have the same ordered prime signature, (2, 1).
603 is a term of A052214 but not a term of this sequence, since 603 = 3^2 * 67^1, 604 = 2^2 * 151^1, and 605 = 5^1 * 11^2 have different ordered prime signatures, (2, 1) or (1, 2).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A052214 and A359745.
Subsequences: A039833, A075039.
Cf. A124010, A175590.

q[n_] := SameQ @@ (FactorInteger[#][[;; , 2]]& /@ (n + {0, 1, 2})); Select[Range[2, 4000], q]

lista(nmax) = {my(e1 = [], e2 = factor(2)[,2]); for(n = 3, nmax, e3 = factor(n)[,2]; if(e1 == e2 && e2 == e3, print1(n-2, ", ")); e1 = e2; e2 = e3); }

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A034173 a(n) is minimal such that prime factorizations of a(n), ..., a(n)+n-1 have same exponents.

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A066509 a(n) is the first of a triple of consecutive integers, each of which is both the product of three distinct primes and also the product of three primes counted with multiplicity.

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A175590 Numbers k with prime signature(k) = prime signature(k+1) = prime signature(k+2) = prime signature(k+3).

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

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A075039 Smallest of three consecutive squarefree numbers having equal numbers of prime factors.

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A113458 Least k such that k, k+n and k+2n have the same prime signature.

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

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