A242034 -id:A242034 - cp's OEIS frontend

A242037 a(n) is the smallest k such that in the interval [1,k] of sequence A242034 all odd primes <= prime(n) are present.

1, 2, 23, 23, 63, 63, 120, 228, 228, 386, 460, 460, 602, 896, 1096, 1096, 1416, 1416, 1416, 3158, 3158, 3158, 3204, 3438, 3438, 3966, 3966, 3966, 8229, 8229, 8229, 8229, 8229, 8229, 8229, 8229, 8229, 8294, 8593, 8593, 11125, 11125, 11559, 11559, 12216, 13594

Offset: 2

Vladimir Shevelev, Aug 12 2014

nonn

Cf. A245024, A243937, A242033, A242034, A242036.

lpf[n_]:=lpf[n]=First[First[FactorInteger[n]]]; (* least prime factor *)
A242034=Map[lpf[#-3]&,Select[Range[6,100000,2],lpf[#-1]>lpf[#-3]&](*A243937*)];
pos={};NestWhile[#+1&,2,(AppendTo[pos,Position[A242034,Prime[#],1,1]];!Last[pos]=={})&];
A242037=Rest[FoldList[Max,-Infinity,Flatten[pos]]] (* Peter J. C. Moses, Aug 14 2014 *)

More terms from Peter J. C. Moses, Aug 12 2014

A242064 Smallest k such that the union of {A242033(i): 1 <= i <= k} and {A242034(i): 1 <= i <= k} includes all primes {3, ..., prime(n)}.

Original entry on oeis.org

1, 2, 9, 9, 36, 36, 81, 220, 220, 386, 386, 386, 434, 521, 896, 896, 896, 1167, 1167, 1695, 2065, 2096, 2096, 2968, 2968, 2968, 2968, 3341, 4561, 4561, 4561, 4561, 4672, 4672, 5964, 6203, 7158, 8294, 8294, 8294, 8740, 8740, 10452, 10452, 11075, 11075, 12092

Offset: 2

Vladimir Shevelev, Aug 13 2014

nonn

Cf. A242033, A242034, A242036, A242037.

lpf[n_]:=lpf[n]=First[First[FactorInteger[n]]];(*least prime factor*)
A242033=Map[lpf[#-1]&,Select[Range[6,100000,2],lpf[#-1]A245024*)];
A242034=Map[lpf[#-3]&,Select[Range[6,100000,2],lpf[#-1]>lpf[#-3]&](*A243937*)];
pos={};NestWhile[#+1&,2,(AppendTo[pos,Min[Position[A242033,Prime[#],1,1],Position[A242034,Prime[#],1,1]/.{}->0]];!Last[pos]==0)&];
A242064=Rest[FoldList[Max,-Infinity,Flatten[pos]]] (* Peter J. C. Moses, Aug 14 2014 *)

More terms from Peter J. C. Moses, Aug 14 2014

A242033 a(n) = lpf(A245024(n)-1), where lpf = least prime factor (A020639).

Original entry on oeis.org

3, 3, 3, 5, 3, 3, 3, 3, 7, 3, 5, 3, 3, 3, 3, 3, 5, 3, 7, 3, 3, 3, 3, 5, 3, 3, 3, 7, 3, 3, 5, 3, 3, 3, 3, 13, 3, 5, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 3, 5, 3, 3, 3, 11, 3, 7, 3, 5, 3, 3, 3, 3, 3, 5, 3, 7, 3, 3, 3, 11, 3, 5, 3, 3, 3, 7, 3, 3, 5, 3, 19, 3, 3, 3, 3, 5

Offset: 1

Vladimir Shevelev, Aug 12 2014

nonn

Conjecture. The sequence contains all odd primes.

The conjecture is true. Consider n-1 = p*q where p is an odd prime and q is a prime > p such that q == p^(-1) mod r for every odd prime r < p. Such primes q exist by Dirichlet's theorem on primes in arithmetic progressions. - Robert Israel, Aug 13 2014

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

Cf. A020639, A065091, A242034, A243937, A245024.

lpf:= n -> min(numtheory:-factorset(n)):
L:= [seq(lpf(2*i+1),i=1..1000)]:
L[select(i->L[i] < L[i-1], [$2..nops(L)])]; # Robert Israel, Aug 13 2014

lpf[n_]:=lpf[n]=First[First[FactorInteger[n]]]; (* least prime factor *)
A242033=Map[lpf[#-1]&,Select[Range[6,300,2],lpf[#-1]A245024*) ] (* Peter J. C. Moses, Aug 14 2014 *)

More terms from Peter J. C. Moses, Aug 12 2014

A242036 Smallest k such that in the interval [1,k] in A242033 all odd primes <= prime(n) are present.

Original entry on oeis.org

1, 4, 9, 54, 54, 88, 88, 220, 220, 444, 444, 570, 570, 570, 896, 1510, 1510, 1510, 1510, 1695, 2065, 2249, 2249, 2968, 2968, 2968, 2968, 3341, 4561, 4561, 4561, 4942, 4942, 6471, 6471, 6471, 7158, 9202, 9202, 10915, 10915, 10915, 10915, 12312, 12312, 12312

Offset: 2

Vladimir Shevelev, Aug 12 2014

nonn

Cf. A245024, A243937, A242033, A242034.

lpf[n_]:=lpf[n]=First[First[FactorInteger[n]]]; (* least prime factor *)
A242033=Map[lpf[#-1]&,Select[Range[6,100000,2],lpf[#-1]A245024*)];
pos={};NestWhile[#+1&,2,(AppendTo[pos,Position[A242033,Prime[#],1,1]];!Last[pos]=={})&];
A242036=Rest[FoldList[Max,-Infinity,Flatten[pos]]] (* Peter J. C. Moses, Aug 14 2014 *)

More terms from Peter J. C. Moses, Aug 12 2014

A242059 lpf_3(A242057(n)-1), where lpf_3(n) = lpf(n/3^t) (cf. A020639) such that 3^t (t>=0) is the maximal power of 3 which divides n.

Original entry on oeis.org

1, 5, 7, 5, 1, 11, 5, 13, 5, 7, 5, 7, 5, 23, 5, 1, 5, 7, 5, 11, 5, 37, 5, 7, 5, 43, 7, 5, 47, 11, 5, 17, 5, 53, 7, 5, 13, 5, 61, 5, 7, 5, 67, 7, 5, 11, 71, 5, 13, 5, 7, 5, 1, 11, 5, 7, 5, 7, 5, 31, 5, 5, 7, 5, 103, 5, 11, 17, 5, 7, 37, 5, 113, 11, 7, 5, 13, 5

Offset: 1

Vladimir Shevelev, Aug 13 2014

nonn

An analog of A242033.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A245024, A243937, A242057, A242058, A242033, A242034, A020639.

lpf[n_]:=lpf[n]=First[First[FactorInteger[n]]];
lpf3[n_]:=lpf3[n]=If[#==1,1,lpf[#]]&[n/3^IntegerExponent[n,3]]
Map[lpf3[#-1]&,Select[Range[4,300,2],lpf3[#-1]Peter J. C. Moses, Aug 13 2014 *)

lpf3(n)=m=n/3^valuation(n, 3); if(m>1, factor(m)[1,1], 1)
apply(n->lpf3(n-1), select(n->lpf3(n-1)Jens Kruse Andersen, Aug 19 2014

More terms from Peter J. C. Moses, Aug 13 2014

A242060 Lpf_3(A242058(n)-3), where lpf_3(n) = lpf(n/3^t) (cf. A020639) such that 3^t (t>=0) is the maximal power of 3 which divides n.

Original entry on oeis.org

1, 5, 1, 11, 5, 17, 7, 1, 29, 5, 13, 41, 5, 7, 17, 5, 19, 59, 5, 23, 71, 5, 7, 1, 5, 29, 7, 5, 11, 101, 5, 107, 37, 5, 7, 11, 5, 43, 5, 137, 5, 7, 149, 5, 7, 5, 13, 19, 5, 59, 179, 5, 7, 191, 5, 197, 5, 11, 5, 7, 13, 5, 227, 7, 5, 79, 239, 1, 5, 13, 83, 5, 7

Offset: 1

Vladimir Shevelev, Aug 13 2014

nonn

An analog of A242034. Records are lesser numbers of twin primes.

Cf. A245024, A243937, A242057, A242058, A242059, A242033, A242034.

lpf[n_]:=lpf[n]=First[First[FactorInteger[n]]];
lpf3[n_]:=lpf3[n]=If[#==1,1,lpf[#]]&[n/3^IntegerExponent[n,3]]
Map[lpf3[#-3]&,Select[Range[4,300,2],lpf3[#-1]>lpf3[#-3]&]](* Peter J. C. Moses, Aug 13 2014 *)

More terms from Peter J. C. Moses, Aug 13 2014

A242065 Smallest k such that the union of {A242059(i): 1 <= i <= k} and {A242060(i): 1 <= i <= k} includes all primes {5, ..., prime(n)}.

Original entry on oeis.org

2, 3, 4, 8, 8, 17, 17, 17, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 179, 179, 179, 179, 179, 179, 179, 179, 264, 264, 264, 319, 319, 319, 319, 365, 1112, 1112, 1112, 1112, 1112, 1112, 1112, 1112, 1112, 1112, 1112, 4372, 4372, 4372, 4372, 4372, 15504, 15504

Offset: 3

Vladimir Shevelev, Aug 13 2014

nonn

Cf. A242033, A242034, A242036, A242037, A242057, A242058, A242059, A242060, A242064.

lpf[n_]:=lpf[n]=First[First[FactorInteger[n]]];(*least prime factor*)
lpf3[n_]:=lpf3[n]=If[#==1,1,lpf[#]]&[n/3^IntegerExponent[n,3]];
A242059=Map[lpf3[#-1]&,Select[Range[4,100000,2],lpf3[#-1]A242057*)];
A242060=Map[lpf3[#-3]&,Select[Range[4,100000,2],lpf3[#-1]>lpf3[#-3]&](*A242058*)];
pos={};NestWhile[#+1&,3,(AppendTo[pos,Min[Position[A242059,Prime[#],1,1],Position[A242060,Prime[#],1,1]/.{}->0]];!Last[pos]==0)&];
A242065=Rest[FoldList[Max,-Infinity,Flatten[pos]]] (* Peter J. C. Moses, Aug 14 2014 *)

More terms from Peter J. C. Moses, Aug 14 2014

A242066 The smallest even k such that lpf_3(k-3) > lpf_3(k-1) >= p_n, where lpf_3(n) = lpf(n/3^t) (cf. A020639) such that 3^t (t>=0) is the maximal power of 3 which divides n.

Original entry on oeis.org

16, 22, 34, 40, 70, 70, 70, 112, 112, 112, 130, 130, 142, 160, 184, 184, 202, 214, 310, 310, 310, 310, 310, 310, 310, 340, 340, 340, 382, 412, 412, 490, 490, 490, 490, 490, 502, 544, 544, 544, 574, 580, 634, 634, 634, 754, 754, 754, 754, 754, 754, 754, 772

Offset: 3

Vladimir Shevelev, Aug 13 2014

nonn

a(n)-3 and (a(n)-1)/3 are primes.

Cf. A242057, A242058, A242059, A242060, A242064, A242065, A242033, A242034, A242036, A242037.

lpf[n_]:=lpf[n]=First[First[FactorInteger[n]]];(*least prime factor*)
lpf3[n_]:=lpf3[n]=If[#==1,1,lpf[#]]&[n/3^IntegerExponent[n,3]];
Table[NestWhile[#+2&,2,!(lpf3[#-3]>lpf3[#-1]>=Prime[n])&],{n,3,100}] (* Peter J. C. Moses, Aug 14 2014 *)

cp's OEIS Frontend

A242037 a(n) is the smallest k such that in the interval [1,k] of sequence A242034 all odd primes <= prime(n) are present.

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A242064 Smallest k such that the union of {A242033(i): 1 <= i <= k} and {A242034(i): 1 <= i <= k} includes all primes {3, ..., prime(n)}.

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A242033 a(n) = lpf(A245024(n)-1), where lpf = least prime factor (A020639).

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A242036 Smallest k such that in the interval [1,k] in A242033 all odd primes <= prime(n) are present.

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A242059 lpf_3(A242057(n)-1), where lpf_3(n) = lpf(n/3^t) (cf. A020639) such that 3^t (t>=0) is the maximal power of 3 which divides n.

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A242060 Lpf_3(A242058(n)-3), where lpf_3(n) = lpf(n/3^t) (cf. A020639) such that 3^t (t>=0) is the maximal power of 3 which divides n.

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A242065 Smallest k such that the union of {A242059(i): 1 <= i <= k} and {A242060(i): 1 <= i <= k} includes all primes {5, ..., prime(n)}.

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A242066 The smallest even k such that lpf_3(k-3) > lpf_3(k-1) >= p_n, where lpf_3(n) = lpf(n/3^t) (cf. A020639) such that 3^t (t>=0) is the maximal power of 3 which divides n.

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