cp's OEIS Frontend

A260143 Runs of consecutive integers with same prime signature.

%I A260143 #21 Feb 16 2025 08:33:26
%S A260143 2,3,14,15,21,22,33,34,35,38,39,44,45,57,58,75,76,85,86,87,93,94,95,
%T A260143 98,99,116,117,118,119,122,123,133,134,135,136,141,142,143,145,146,
%U A260143 147,148,158,159,171,172,177,178,201,202,203,205,206,213,214,215,217,218,219,230,231,244,245
%N A260143 Runs of consecutive integers with same prime signature.
%C A260143 This sequence is infinite, see A189982 and Theorem 4 in Goldston-Graham-Pintz-Yıldırım. - _Charles R Greathouse IV_, Jul 17 2015
%H A260143 Charles R Greathouse IV, <a href="/A260143/b260143.txt">Table of n, a(n) for n = 1..10000</a>
%H A260143 D. A. Goldston, S. W. Graham, J. Pintz, and C. Y. Yıldırım, <a href="http://arxiv.org/abs/0803.2636">Small gaps between almost primes, the parity problem, and some conjectures of Erdos on consecutive integers</a>, arXiv:0803.2636 [math.NT], 2008.
%H A260143 MathOverflow, <a href="http://mathoverflow.net/questions/32412">Question on consecutive integers with similar prime factorizations</a>
%H A260143 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeSignature.html">Prime Signature</a>
%H A260143 OEIS Wiki, <a href="http://oeis.org/wiki/Prime_signatures">Prime signatures</a>
%e A260143 Runs begin:
%e A260143 (terms)         (prime signature)
%e A260143 {2, 3},         [1]
%e A260143 {14, 15},       [1,1]
%e A260143 {21, 22},       [1,1]
%e A260143 {33, 34, 35},   [1,1]
%e A260143 {38, 39},       [1,1]
%e A260143 {44, 45},       [1,2]
%e A260143 {57, 58},       [1,1]
%e A260143 {75, 76},       [1,2]
%e A260143 {85, 86, 87},   [1,1]
%e A260143 {93, 94, 95},   [1,1]
%e A260143 {98, 99},       [1,2]
%e A260143 ...
%t A260143 Split[Range[2,250], Sort[FactorInteger[#1][[All, 2]]] === Sort[FactorInteger[#2][[All, 2]]]&] // Select[#, Length[#] > 1&]& // Flatten
%o A260143 (PARI) is(n)=my(f=vecsort(factor(n)[,2])); f==vecsort(factor(n-1)[,2]) || f==vecsort(factor(n+1)[,2]) \\ _Charles R Greathouse IV_, Jul 17 2015
%o A260143 (Python)
%o A260143 from sympy import factorint
%o A260143 def aupto(limit):
%o A260143     aset, prevsig = {2}, [1]
%o A260143     for k in range(3, limit+2):
%o A260143         sig = sorted(factorint(k).values())
%o A260143         if sig == prevsig: aset.update([k - 1, k])
%o A260143         prevsig = sig
%o A260143     return sorted(aset)
%o A260143 print(aupto(250)) # _Michael S. Branicky_, Sep 20 2021
%Y A260143 Main sequence is A052213.
%K A260143 nonn,easy,tabf
%O A260143 1,1
%A A260143 _Jean-François Alcover_, Jul 17 2015