A073607 -id:A073607 - cp's OEIS frontend

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

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

Offset: 1

Robert G. Wilson v, Aug 06 2002

nonn

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

			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

Robert Israel, Table of n, a(n) for n = 1..320

Cf. A073606, A073607, A072555, A069561.

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

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 *)

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

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

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

Amarnath Murthy, Aug 04 2002

nonn

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

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

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 *)

Edited by Robert G. Wilson v, Aug 07 2002

A072555 Smallest of four consecutive integers divisible by four consecutive primes respectively.

Original entry on oeis.org

158, 368, 578, 788, 789, 790, 998, 1208, 1418, 1628, 1838, 1944, 2048, 2258, 2468, 2678, 2888, 3098, 3099, 3308, 3518, 3728, 3938, 4148, 4254, 4358, 4367, 4568, 4778, 4988, 5198, 5408, 5409, 5618, 5795, 5828, 6038, 6248, 6458, 6564, 6668, 6797, 6878

Offset: 1

Robert G. Wilson v, Aug 06 2002

nonn

			158 is a term as 158, 159, 160 and 161 are divisible by 2, 3, 5 and 7 respectively.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A073606, A073607, A073754, A073755, A073756, A072562.

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, 3} ]; k = 1; While[ k < l + 1 && Union[ Mod[ t, NestList[ NextPrim, p[ [ k ] ], 3 ] ] ] != {0}, k++ ]; If[ k < l + 1, Print[ n ] ], {n, 2, 7297} ]

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

Robert G. Wilson v, Aug 07 2002

nonn

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

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

Robert G. Wilson v, Aug 07 2002

nonn

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

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

Robert G. Wilson v, Aug 08 2002

nonn

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

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} ]

A073608 a(1) = 1, a(n) = smallest number such that a(n) - a(n-k) is a prime power > 1 for all k.

Original entry on oeis.org

1, 3, 5, 8, 10, 12

Offset: 1

Amarnath Murthy, Aug 04 2002

Differences |a(i)-a(j)| are prime powers for all i,j. Conjecture: sequence is bounded.
Proof that sequence is complete: Assume there is some k after the term 12. Then {k-1, k-3, k-5} must contain a multiple of 3. Also {k-8,k-10,k-12} also contains a multiple of 3. No prime > 12 is a multiple of 3, so the multiples of 3 are both prime powers. This implies there must be two powers of 3 that have a difference at most 11, but no such pair exists > 12 (only 1,3 and 3,9 qualify.) - Jim Nastos, Aug 09 2002
There is an elementary proof that no set of seven integers of this kind exists. - Don Reble, Aug 10 2002

			a(5) = 10 as 10-8, 10-5, 10-3, 10-1 or 2, 5, 7, 9 are prime powers.

Cf. A073607.

Sixth term from Jim Nastos, Aug 09 2002

cp's OEIS Frontend

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

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

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A072555 Smallest of four consecutive integers divisible by four consecutive primes respectively.

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

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

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

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A073608 a(1) = 1, a(n) = smallest number such that a(n) - a(n-k) is a prime power > 1 for all k.

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