cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A034173 a(n) is minimal such that prime factorizations of a(n), ..., a(n)+n-1 have same exponents.

Original entry on oeis.org

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

Views

Author

Dean Hickerson, Oct 01 1998

Keywords

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)

Examples

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

Links

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

Programs

  • PARI
    A034173(n)={my(f);for(k=1,oo,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

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

Extensions

a(7) from Donovan Johnson, Oct 20 2009
Don Reble link repaired by N. J. A. Sloane, Oct 24 2024

A218448 First of a run of 5 consecutive numbers with same prime signature.

Original entry on oeis.org

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
Offset: 1

Views

Author

M. F. Hasler, Oct 28 2012

Keywords

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

Crossrefs

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

Programs

  • 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

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

A083785 Triangle read by rows in which the n-th row gives the smallest set of n consecutive numbers with the same prime signatures.

Original entry on oeis.org

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
Offset: 1

Views

Author

Amarnath Murthy, May 07 2003

Keywords

Examples

			       1
       2      3
      33     34     35
   19940  19941  19942  19943
  204323 204324 204325 204326 204327
  ...

Links

Crossrefs

The first column is given by A034173, the final terms by A034174.
Cf. A034173, A034174, A083787.

Extensions

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

A083787 Sequence contains the prime signature pertaining to the n-th row of A083785.

Original entry on oeis.org

1, 2, 6, 60, 60, 60, 420
Offset: 1

Views

Author

Amarnath Murthy, May 07 2003

Keywords

Examples

			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.

Formula

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

Extensions

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

A218455 First of a run of 6 consecutive numbers with same prime signature.

Original entry on oeis.org

380480345, 2713001274, 6282718946, 7209536449, 9809067073, 10684724346, 12008728850, 14824913049, 17231547073, 17552118546, 17659180314, 18036555273, 20473171322, 21507097001, 23676804346, 24742649321, 25401767522, 25694056449, 27656894273, 28259097818
Offset: 1

Views

Author

M. F. Hasler, Oct 29 2012

Keywords

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

Crossrefs

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

Programs

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

Extensions

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

A218865 First of a run of 7 consecutive numbers with same prime signature.

Original entry on oeis.org

440738966073, 464881210073, 645462662449, 914549532721, 932537185321, 936083045673, 1043710445721, 1091100709673, 1225660528209, 1349165568945, 1602377996921, 1682159750473, 1714275593649, 1730241245545, 1756186304521, 1872671302049, 1956516794721, 1987087485225
Offset: 1

Views

Author

Donovan Johnson, Nov 08 2012

Keywords

Comments

a(1) = A034173(7) = A218455(443) = A218455(444)-1.
a(19) > 2*10^12. - Donovan Johnson, May 11 2013

Examples

			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.

Extensions

a(7)-a(18) from Donovan Johnson, May 11 2013
Showing 1-6 of 6 results.

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