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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 (list; graph; refs; listen; history; text; internal format)
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.

MathOverflow, Question on consecutive integers with similar prime factorizations

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: A244295 A032806 A225756 * A275303 A281500 A041009

Adjacent sequences:  A260140 A260141 A260142 * A260144 A260145 A260146

KEYWORD

nonn,easy,tabf

AUTHOR

Jean-François Alcover, Jul 17 2015

STATUS

approved

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Last modified January 20 03:25 EST 2022. Contains 350467 sequences. (Running on oeis4.)