A260143 Runs of consecutive integers with same prime signature.

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

Author

Jean-François Alcover, Jul 17 2015

Keywords

• nonn
• easy
• tabf

Comments

This sequence is infinite, see A189982 and Theorem 4 in Goldston-Graham-Pintz-Yıldırım. - Charles R Greathouse IV, Jul 17 2015

Examples

			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]
...
		

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

Crossrefs

Main sequence is A052213.

Programs

Mathematica

Split[Range[2,250], Sort[FactorInteger[#1][[All, 2]]] === Sort[FactorInteger[#2][[All, 2]]]&] // Select[#, Length[#] > 1&]& // Flatten

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