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A260143
Runs of consecutive integers with same prime signature.
3
2, 3, 14, 15, 21, 22, 33, 34, 35, 38, 39, 44, 45, 57, 58, 75, 76, 85, 86, 87, 93, 94, 95, 98, 99, 116, 117, 118, 119, 122, 123, 133, 134, 135, 136, 141, 142, 143, 145, 146, 147, 148, 158, 159, 171, 172, 177, 178, 201, 202, 203, 205, 206, 213, 214, 215, 217, 218, 219, 230, 231, 244, 245
OFFSET
1,1
COMMENTS
This sequence is infinite, see A189982 and Theorem 4 in Goldston-Graham-Pintz-Yıldırım. - Charles R Greathouse IV, Jul 17 2015
LINKS
Charles R Greathouse IV, 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, arXiv:0803.2636 [math.NT], 2008.
Eric Weisstein's World of Mathematics, Prime Signature
OEIS Wiki, Prime signatures
EXAMPLE
Runs begin:
(terms) (prime signature)
{2, 3}, [1]
{14, 15}, [1,1]
{21, 22}, [1,1]
{33, 34, 35}, [1,1]
{38, 39}, [1,1]
{44, 45}, [1,2]
{57, 58}, [1,1]
{75, 76}, [1,2]
{85, 86, 87}, [1,1]
{93, 94, 95}, [1,1]
{98, 99}, [1,2]
...
MATHEMATICA
Split[Range[2, 250], Sort[FactorInteger[#1][[All, 2]]] === Sort[FactorInteger[#2][[All, 2]]]&] // Select[#, Length[#] > 1&]& // Flatten
PROG
(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
(Python)
from sympy import factorint
def aupto(limit):
aset, prevsig = {2}, [1]
for k in range(3, limit+2):
sig = sorted(factorint(k).values())
if sig == prevsig: aset.update([k - 1, k])
prevsig = sig
return sorted(aset)
print(aupto(250)) # Michael S. Branicky, Sep 20 2021
CROSSREFS
Main sequence is A052213.
Sequence in context: A032806 A353870 A225756 * A353871 A275303 A281500
KEYWORD
nonn,easy,tabf
AUTHOR
STATUS
approved