A075039 -id:A075039 - cp's OEIS frontend

A052214 Numbers n with prime signature(n) = prime signature(n+1) = prime signature(n+2).

33, 85, 93, 141, 201, 213, 217, 301, 393, 445, 603, 633, 697, 921, 1041, 1137, 1261, 1274, 1309, 1345, 1401, 1641, 1761, 1837, 1885, 1893, 1924, 1941, 1981, 2013, 2101, 2181, 2217, 2305, 2361, 2433, 2461, 2517, 2523, 2641, 2665, 2721, 2733, 3097, 3385

Offset: 1

Erich Friedman, Jan 29 2000

			33 = 3^1 11^1, 34 = 2^1 17^1 and 35 = 5^1 7^1, so all have prime signature {1,1}.

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

A subsequence of A005238. - M. F. Hasler, Jan 05 2013

Cf. A075039.

pri[ n_ ] := Sort[ Transpose[ FactorInteger[ n ] ][ [ 2 ] ] ] Select[ Range[ 2,10000 ],pri[ # ]==pri[ #+1 ]==pri[ #+2 ]& ]

A052214(n, /*give optional 2nd arg =1 to print all terms*/ show_all=0, a=2*3)={until( !n-- || !a++, until(, vecsort(factor(a+=2)[, 2])!=vecsort(factor(a-1)[, 2]) && next; ((t=vecsort(factor(a-1)[, 2]))==vecsort(factor(a-2)[, 2]) || vecsort(factor(a++)[, 2])==t) && (a-=2) && break); show_all && print1(a", ")); a}  \\ - M. F. Hasler, Jan 06 2013

A086263 Smaller of two consecutive squarefree numbers having equal numbers of prime factors.

Original entry on oeis.org

2, 14, 21, 33, 34, 38, 57, 85, 86, 93, 94, 118, 122, 133, 141, 142, 145, 158, 177, 201, 202, 205, 213, 214, 217, 218, 230, 253, 285, 298, 301, 302, 326, 334, 381, 393, 394, 429, 434, 445, 446, 453, 481, 501, 514, 526, 537, 542, 553, 565, 609, 622, 633, 634

Offset: 1

Reinhard Zumkeller, Jul 14 2003

nonn

a(k) is a term of A075039 iff a(k)+1 = a(k+1).

If a prime divides a(n) then it does not divide a(n) + 1. If a prime divides a(n) + 1, then it does not divide a(n). The sets of prime divisors of a(n) and a(n) + 1 are disjoint. - Torlach Rush, Jan 13 2018

			230 = 2*5*23 and 230+1 = 3*7*11, therefore 230 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A005117, A263990 (2 prime factors), A215217 (3 prime factors), A318896 (4 prime factors), A318964 (5 prime factors), A001221, A001222, A075039.

Select[Range[2, 634], SquareFreeQ[#] && SquareFreeQ[# + 1] && Length[FactorInteger[#]] == Length[FactorInteger[# + 1]] &] (* T. D. Noe, Jun 26 2013 *)
#[[1,1]]&/@Select[Partition[Table[{n,If[SquareFreeQ[n],1,0], PrimeOmega[ n]},{n,700}],2,1],#[[1,2]]==#[[2,2]]==1&&#[[1,3]]==#[[2,3]]&] (* Harvey P. Dale, Dec 13 2014 *)

for(n=1,10^3, if ( issquarefree(n) && issquarefree(n+1) && (omega(n)==omega(n+1)) , print1(n,", "))); \\ Joerg Arndt, Jun 26 2013

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

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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A052214 Numbers n with prime signature(n) = prime signature(n+1) = prime signature(n+2).

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A086263 Smaller of two consecutive squarefree numbers having equal numbers of prime factors.

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

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