

A052213


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


15



2, 14, 21, 33, 34, 38, 44, 57, 75, 85, 86, 93, 94, 98, 116, 118, 122, 133, 135, 141, 142, 145, 147, 158, 171, 177, 201, 202, 205, 213, 214, 217, 218, 230, 244, 253, 285, 296, 298, 301, 302, 326, 332, 334, 375, 381, 387, 393, 394, 429, 434, 445, 446, 453, 481
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OFFSET

1,1


COMMENTS

This sequence is infinite, see A189982 and Theorem 4 in GoldstonGrahamPintzYıldırım.  Charles R Greathouse IV, Jul 17 2015
This is a subsequence of A005237, hence a(n) >> n sqrt(log log n) by the ErdősPomeranceSárközy result cited there.  Charles R Greathouse IV, Jul 17 2015
Sequence is not the same as A280074, first deviation is at a(212): a(212) = 2041, A280074(212) = 2024. Number 2024 is the smallest number n such that A007425(n) = A007425(n+1) with different prime signatures of numbers n and n+1 (2024 = 2^3 * 11 * 23, 2025 = 3^4 * 5^2; A007425(2024) = A007425(2025) = 90). Conjecture: also numbers n such that Product_{dn} tau(d) = Product_{dn+1} tau(d).  Jaroslav Krizek, Dec 25 2016


LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000
D. A. Goldston, S. W. Graham, J. Pintz, and C. Y. Yıldırım, Small gaps between almost primes, the parity problem, and some conjectures of Erdos on consecutive integers (2008)
MathOverflow, Question on consecutive integers with similar prime factorizations


EXAMPLE

14 = 2^1*7^1 and 15 = 3^1*5^1, so both have prime signature {1,1}.


MATHEMATICA

pri[n_] := Sort[ Transpose[ FactorInteger[n]] [[2]]]; Select[ Range[ 2, 1000], pri[#] == pri[#+1] &]
Rest[SequencePosition[Table[Sort[FactorInteger[n][[All, 2]]], {n, 500}], {x_, x_}][[All, 1]]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 28 2017 *)


PROG

(PARI) lista(nn) = for (n=1, nn1, if (vecsort(factor(n)[, 2]) == vecsort(factor(n+1)[, 2]), print1(n, ", ")); ); \\ Michel Marcus, Jun 10 2015


CROSSREFS

Cf. A005237, A189982.
Sequence in context: A138047 A005237 A140578 * A280074 A086263 A335071
Adjacent sequences: A052210 A052211 A052212 * A052214 A052215 A052216


KEYWORD

easy,nonn


AUTHOR

Erich Friedman, Jan 29 2000


STATUS

approved



