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.

A072562 Smallest of exactly n consecutive integers divisible respectively by the first n primes.

Original entry on oeis.org

4, 2, 8, 158, 3098, 788, 210998, 5316098, 34415168, 703693778, 194794490678, 5208806743928, 138782093170508, 5006786309605868, 253579251611336438, 12551374903381164638, 142908008812141343558, 77053322014980646906358
Offset: 1

Views

Author

Robert G. Wilson v, Aug 06 2002

Keywords

Comments

a(n) often equals A069561(n).
For n>2, a(n) == 8 (mod 30). - Robert G. Wilson v, Oct 30 2014
For n<1000 a(n)>a(n+1) when n = 1, 5, 90. - Robert G. Wilson v, Oct 30 2014

Examples

			a(4)=158 because 158 is the least number such that 158, 159, 160 and 161 are divisible by 4 consecutive primes, namely 2, 3, 5 and 7 respectively.
a(5) does not equal A069561(5)=788 because 788 is the smallest integer in a run of 6 (not 5) consecutive integers that are divisible respectively by the first 6 consecutive primes. - _Geoffrey Critzer_, Oct 29 2014

Links

Crossrefs

Cf. A073606, A073607, A072555, A069561.

Programs

  • Maple
    A:= proc(n)
local r;
if n = 1 then return 4 fi;
r:= chrem([seq(-i,i=0..n-1)],[seq(ithprime(i),i=1..n)]);
if r + n mod ithprime(n+1) = 0 then  r + mul(ithprime(i),i=1..n) else r fi
end proc:
seq(A(n),n=1..50); # Robert Israel, Oct 29 2014
  • Mathematica
    f[n_] := Block[{p = Prime@ Range@ n}, r = ChineseRemainder[-Range@ n + 1, p]; If[ Mod[r + n, Prime[n + 1]] == 0, r + Times @@ p, r]]; f[1] = 4; Array[f, 20] (* Robert G. Wilson v, Oct 30 2014 *)
  • PARI
    a(n)=if(n==1,return(4)); my(m=chinese(vector(n, k, Mod(1-k, prime(k)))), p=prime(n+1), t=lift(m)); if((t+n)%p, t, t+m.mod) \\ Charles R Greathouse IV, Jun 20 2015

Formula

If A069561(n+1) = A069561(n), then a(n) = A069561(n) + A002110(n). Otherwise, then a(n) = A069561(n). - David Wasserman, Oct 21 2004
a(n) = A069561(n) or A069561(n) + A002110(n), hence log a(n) << n log n. - Charles R Greathouse IV, Jun 20 2015

Extensions

More terms from David Wasserman, Oct 21 2004

A073606 Smaller of two consecutive integers divisible respectively by two consecutive primes.

Original entry on oeis.org

2, 8, 9, 14, 20, 21, 24, 26, 32, 38, 39, 44, 50, 54, 55, 56, 62, 68, 69, 74, 77, 80, 84, 86, 90, 92, 98, 99, 104, 110, 114, 115, 116, 122, 125, 128, 129, 134, 140, 144, 146, 152, 158, 159, 160, 164, 169, 170, 174, 175, 176, 182, 188, 189, 194, 195, 200, 204, 206
Offset: 1

Views

Author

Amarnath Murthy, Aug 04 2002

Keywords

Comments

There are arbitrarily long strings of consecutive integers in this sequence; for example, A072562(k+1) is followed by at least k-1 more consecutive members. - David Wasserman, Oct 21 2004

Examples

			54 is a term as 54 and 55 are divisible by 3 and 5 respectively. 55 is also a term as 55 and 56 are divisible by 5 and 7. 56 is also a term as 56 and 57 are divisible by 2 and 3.

Links

Crossrefs

Cf. A073755, A073607, A072555, A073754, A073756, A072562.

Programs

  • Mathematica
    f[n_Integer] := Flatten[ Table[ #1] & @@@ FactorInteger[n]]; NextPrim[n_] := Block[ {k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; Do[ p = f[ n ]; l = Length[ p ]; t = Table[n + i, {i, 0, 1} ]; k = 1; While[ k < l + 1 && Union[ Mod[ t, NestList[ NextPrim, p[[ k ]], 1 ]]] != {0}, k++ ]; If[ k < l + 1, Print[ n ]], {n, 2, 220} ]
npQ[n_] := Or @@ Divisible[n + 1, NextPrime[First /@ FactorInteger[n]]]; Select[Range[2, 210], npQ[#] &] (* Jayanta Basu, Jul 03 2013 *)

Extensions

Edited by Robert G. Wilson v, Aug 07 2002

A073607 Smallest of three consecutive integers divisible by three consecutive primes respectively.

Original entry on oeis.org

8, 20, 38, 54, 68, 98, 114, 128, 158, 159, 169, 188, 218, 248, 264, 278, 308, 338, 368, 369, 398, 405, 428, 458, 474, 488, 518, 548, 578, 579, 608, 638, 668, 684, 698, 728, 758, 788, 789, 790, 791, 818, 848, 878, 894, 908, 938, 968, 998, 999, 1028, 1058
Offset: 1

Views

Author

Amarnath Murthy, Aug 04 2002

Keywords

Comments

The sequence is infinite as 30*k + 8 is a member for all k. What is the longest string of consecutive integers?

Examples

			20 is a term as 20,21 and 22 are divisible by 5,7 and 11 respectively.
114 is a term as 114, 115 and 116 are divisible by 19, 23 and 29 respectively.

Links

Crossrefs

Cf. A073606, A072555, A073754, A073755, A073756, A072562.

Programs

  • Mathematica
    f[ n_Integer ] := Flatten[ Table[ #1 ] & @@@ FactorInteger[ n ]]; NextPrim[ n_ ] := Block[ {k = n + 1}, While[ !PrimeQ[ k ], k++ ]; k ]; Do[ p = f[ n ]; l = Length[ p ]; t = Table[ n + i, {i, 0, 2} ]; k = 1; While[ k < l + 1 && Union[ Mod[ t, NestList[ NextPrim, p[[ k ]], 2 ] ]] != {0}, k++ ]; If[ k < l + 1, Print[ n ]], {n, 2, 1117} ]

Extensions

Edited, corrected and extended by Robert G. Wilson v, Aug 06 2002

A072722 Smallest of 6 consecutive integers divisible respectively by 6 consecutive primes.

Original entry on oeis.org

788, 30818, 60848, 90878, 120908, 150938, 180968, 210998, 210999, 241028, 271058, 301088, 331118, 361148, 391178, 421208, 451238, 466254, 466255, 481268, 511298, 541328, 571358, 601388, 631418, 661448, 691478, 721508, 721509, 751538
Offset: 1

Views

Author

Robert G. Wilson v, Aug 07 2002

Keywords

Examples

			30818 is a term as 30818, 30818, 30819, 30820, 30821 and 30822 are divisible by 2, 3, 5, 7 and 11 respectively.

Links

Crossrefs

Cf. A073606, A073607, A072555, A073754, A073756, A072562.

Programs

  • Mathematica
    f[n_Integer] := Flatten[ Table[ #1] & @@@ FactorInteger[n]]; NextPrim[n_] := Block[ {k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; Do[ p = f[ n ]; l = Length[ p ]; t = Table[n + i, {i, 0, 5} ]; k = 1; While[ k < l + 1 && Union[ Mod[ t, NestList[ NextPrim, p[[ k ]], 5 ]]] != {0}, k++ ]; If[ k < l + 1, Print[ n ]], {n, 2, 811597} ]

A072730 Smallest of 5 consecutive integers divisible respectively by 5 consecutive primes.

Original entry on oeis.org

788, 789, 3098, 5408, 7718, 10028, 12338, 14648, 15804, 16958, 19268, 21578, 23888, 26198, 28508, 30818, 30819, 33128, 35438, 37748, 40058, 40830, 42368, 44678, 45834, 46988, 49298, 51608, 53918, 56228, 58538, 60848, 60849, 63158
Offset: 1

Views

Author

Robert G. Wilson v, Aug 07 2002

Keywords

Examples

			3098 is a term as 3098, 3099, 3100, 3101 and 3102 are divisible by 2, 3, 5, 7 and 11 respectively.

Links

Crossrefs

Cf. A073606, A073607, A072555, A073755, A073756, A072562.

Programs

  • Mathematica
    f[n_Integer] := Flatten[ Table[ #1] & @@@ FactorInteger[n]]; NextPrim[n_] := Block[ {k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; Do[ p = f[ n ]; l = Length[ p ]; t = Table[n + i, {i, 0, 4} ]; k = 1; While[ k < l + 1 && Union[ Mod[ t, NestList[ NextPrim, p[[ k ]], 4 ]]] != {0}, k++ ]; If[ k < l + 1, Print[ n ]], {n, 2, 72397} ]
cicpQ[n_]:=Module[{num=Range[n,n+4],pr=PrimePi[n+4]-4},Total [Boole[ AllTrue[ #, IntegerQ]&/@Table[num/Prime[Range[k,k+4]],{k,pr}]]]>0]; Select[ Range[ 64000],cicpQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 11 2019 *)

A072760 Smallest of 7 consecutive integers divisible respectively by 7 consecutive primes.

Original entry on oeis.org

210998, 466254, 721508, 1232018, 1742528, 2253038, 2763548, 3274058, 3784568, 4295078, 4805588, 5316098, 5316099, 5826608, 6337118, 6847628, 7358138, 7868648, 8379158, 8889668, 9400178, 9910688, 10165944, 10421198, 10931708
Offset: 1

Views

Author

Robert G. Wilson v, Aug 08 2002

Keywords

Examples

			210998 is a term as 210998, 210999, 211000, 211001, 211002, 211003 and 211004 are divisible by 2, 3, 5, 7, 11, 13 and 17 respectively.

Links

Crossrefs

Cf. A073606, A073607, A072555, A073754, A073755, A072562.

Programs

  • Mathematica
    f[n_Integer] := Flatten[ Table[ #1] & @@@ FactorInteger[n]]; NextPrim[n_] := Block[ {k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; Do[ p = f[ n ]; l = Length[ p ]; t = Table[n + i, {i, 0, 6} ]; k = 1; While[ k < l + 1 && Union[ Mod[ t, NestList[ NextPrim, p[[ k ]], 6 ]]] != {0}, k++ ]; If[ k < l + 1, Print[ n ]], {n, 2, 811597} ]
Showing 1-6 of 6 results.

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