A034174 -id:A034174

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A034173

a(n) is minimal such that prime factorizations of a(n), ..., a(n)+n-1 have same exponents.

+10
13

1, 2, 33, 19940, 204323, 380480345, 440738966073

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

OFFSET

1,2

COMMENTS

a(8) > 10^13. - Donovan Johnson, Oct 20 2009

Don Reble has shown that a(8) < 1.9*10^42, cf. link.

From David Wasserman, Jan 05 2019: (Start)

a(8) <= 108111092880293127811946663766147737122,

a(9) <= 6850672946809600696044301071559918192380244,

a(10) <= 96037988156124494415303285590850571857698741869620,

a(11) <= 9044737840075556371215937303485030235666252755947862558252154847122. (End)

LINKS

Table of n, a(n) for n=1..7.

Don Reble, A034173(8) exists, SeqFan list, Oct 23 2012

FORMULA

a(n) = A034174(n) - n + 1. - Max Alekseyev, Nov 10 2009

a(n) = A083785(n,1) = A113456(1,n); a(2) = A052213(1), a(3) = A052214(1), a(4) = A175590(1), a(5) = A218448(1), a(6) = A218448(62) = A218448(63)-1. - M. F. Hasler, Oct 28 2012

EXAMPLE

a(4) = 19940 because 19940, ..., 19943 all have the form p^2 q r.

PROG

(PARI) A034173(n)={my(f); for(k=1, 9e9, f=0; for(i=1, n, f==(f=vecsort(factor(k+n-i)[, 2])) || i==1 || [k+=n-i; next(2)]); return(k))} \\ M. F. Hasler, Oct 23 2012

CROSSREFS

Cf. A034174.

Cf. A006558, A083790.

Cf. A052213, A052214, A175590, A218448. This sequence is the first column of A083785 and first row of A113456. The latter generalizes to arithmetic progressions with step d>=1. - M. F. Hasler, Oct 28 2012

KEYWORD

hard,nonn,more

AUTHOR

Dean Hickerson, Oct 01 1998

EXTENSIONS

a(7) from Donovan Johnson, Oct 20 2009

STATUS

approved

A218448

First of a run of 5 consecutive numbers with same prime signature.

+10
5

204323, 3252571, 5205074, 7201674, 20182921, 28387953, 36193650, 43216722, 51049537, 56155074, 57070850, 61961315, 62167075, 65425473, 76647074, 82507473, 92658049, 95943321, 100498849, 107236449, 109751473, 110899321, 112198075, 112477849, 116736323

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

OFFSET

1,1

COMMENTS

A number n is in this sequence iff n and n+1 is in A175590; also: iff n and n+2 are in A052214 (in which case n+1 is in A052214, too); and also: iff {n,n+1,n+2,n+3} are in A052213.

A034173(6) = A218448(62) = A218448(63)-1 is the least term n such that n+1 is also in the sequence.

LINKS

M. F. Hasler and Donovan Johnson, Table of n, a(n) for n = 1..1000 (first 140 terms from M. F. Hasler)

PROG

(PARI) is_A218448(n)={my(f); !for(i=0, 4, f!=(f=vecsort(factor(n+i)[, 2])) & i & return)}

(PARI) f(k)=vecsort(factor(k)[, 2]~, , 4)

t=f(n=2); while(n<1e8, for(i=n+1, n+4, tt=f(i); if(tt!=t, n=i; t=tt; next(2))); print1(n", "); n++) \\ Charles R Greathouse IV, Oct 28 2012

CROSSREFS

Cf. A034173, A034174, A006558, A083790, A052213, A052214, A175590.

KEYWORD

nonn

AUTHOR

M. F. Hasler, Oct 28 2012

EXTENSIONS

a(6)-a(8) from Charles R Greathouse IV, Oct 28 2012

a(9)-a(25) from Donovan Johnson, Oct 28 2012

Values up to a(140) computed using b175590.txt from Charles R Greathouse IV - M. F. Hasler, Oct 28 2012

STATUS

approved

A083785

Triangle read by rows in which the n-th row gives the smallest set of n consecutive numbers with the same prime signatures.

+10
3

1, 2, 3, 33, 34, 35, 19940, 19941, 19942, 19943, 204323, 204324, 204325, 204326, 204327, 380480345, 380480346, 380480347, 380480348, 380480349, 380480350, 440738966073, 440738966074, 440738966075, 440738966076, 440738966077, 440738966078, 440738966079

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

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..28.

Index entries for sequences related to primes in arithmetic progressions

EXAMPLE

2 3

33 34 35

19940 19941 19942 19943

204323 204324 204325 204326 204327

...

CROSSREFS

The first column is given by A034173, the final terms by A034174.

Cf. A034173, A034174, A083787.

KEYWORD

nonn,tabl,nice

AUTHOR

Amarnath Murthy, May 07 2003

EXTENSIONS

7th row added using A034173 by Michel Marcus, Dec 13 2015

STATUS

approved

A083787

Sequence contains the prime signature pertaining to the n-th row of A083785.

+10
2

1, 2, 6, 60, 60, 60, 420

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

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..7.

FORMULA

a(n) = A046523(A034173(n)). - Amiram Eldar, Jul 28 2024

EXAMPLE

a(4) = 2^2*3*5 = 60 because A083785(4, 1) = 19940 = 2^2*5*997, (p^2*q*r) etc.

CROSSREFS

Cf. A034173, A034174, A046523, A083785.

KEYWORD

nonn,more

AUTHOR

Amarnath Murthy, May 07 2003

EXTENSIONS

a(6)-a(7) from Jinyuan Wang, May 24 2020

STATUS

approved

A218455

First of a run of 6 consecutive numbers with same prime signature.

+10
2

380480345, 2713001274, 6282718946, 7209536449, 9809067073, 10684724346, 12008728850, 14824913049, 17231547073, 17552118546, 17659180314, 18036555273, 20473171322, 21507097001, 23676804346, 24742649321, 25401767522, 25694056449, 27656894273, 28259097818

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

OFFSET

1,1

COMMENTS

A number n is in this sequence iff n and n+1 is in A218448; see the comment there for other characterizations in terms of membership in A175590 or A052214 or A052213.

LINKS

Donovan Johnson, Table of n, a(n) for n = 1..1000

PROG

(PARI) is_A218455(n)={my(s(n)=vecsort(factor(n)[, 2]), t=s(n)); !for(m=n+1, n+5, t!=s(m) & return)}

CROSSREFS

Cf. A034173, A034174, A006558, A083790, A052213, A052214, A175590, A218448, A218865.

KEYWORD

nonn

AUTHOR

M. F. Hasler, Oct 29 2012

EXTENSIONS

a(2)-a(20) from Donovan Johnson, Oct 29 2012

STATUS

approved

A218865

First of a run of 7 consecutive numbers with same prime signature.

+10
1

440738966073, 464881210073, 645462662449, 914549532721, 932537185321, 936083045673, 1043710445721, 1091100709673, 1225660528209, 1349165568945, 1602377996921, 1682159750473, 1714275593649, 1730241245545, 1756186304521, 1872671302049, 1956516794721, 1987087485225

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

OFFSET

1,1

COMMENTS

a(1) = A034173(7) = A218455(443) = A218455(444)-1.

a(19) > 2*10^12. - Donovan Johnson, May 11 2013

LINKS

Table of n, a(n) for n=1..18.

EXAMPLE

The 7 consecutive numbers from 645462662449 to 645462662455 all have a prime signature of 2,1,1,1.

CROSSREFS

Cf. A034173, A034174, A052213, A052214, A175590, A218448, A218455.

KEYWORD

nonn

AUTHOR

Donovan Johnson, Nov 08 2012

EXTENSIONS

a(7)-a(18) from Donovan Johnson, May 11 2013

STATUS

approved

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