A243937 -id:A243937 - cp's OEIS frontend

A242034 a(n) = lpf(A243937(n)-3), where lpf = least prime factor (A020639).

3, 5, 3, 11, 3, 17, 3, 3, 29, 3, 5, 3, 41, 3, 3, 3, 59, 3, 5, 3, 71, 3, 7, 3, 3, 3, 5, 3, 101, 3, 107, 3, 3, 7, 3, 5, 3, 3, 137, 3, 3, 149, 3, 5, 3, 7, 3, 3, 3, 179, 3, 5, 3, 191, 3, 197, 3, 3, 11, 3, 5, 3, 13, 3, 227, 3, 3, 239, 3, 5, 3, 3, 3, 3, 269, 3, 5, 3

Offset: 1

Vladimir Shevelev, Aug 12 2014

nonn

The records of the sequence form sequence of lesser numbers of twin primes.

The sequence contains all odd primes. Cf. comment by Robert Israel in A242033. - Vladimir Shevelev, Aug 16 2014

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

Cf. A001359, A020639, A065091, A242033, A243937, A245024.

lpf[n_]:=lpf[n]=First[First[FactorInteger[n]]]; (* least prime factor *)
A242034=Map[lpf[#-3]&,Select[Range[6,300,2],lpf[#-1]>lpf[#-3]&](*A243937*)] (* Peter J. C. Moses, Aug 14 2014 *)

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

A245274 Composite terms in sequence {A243937(n)-1}.

Original entry on oeis.org

35, 65, 77, 95, 119, 121, 125, 143, 155, 161, 185, 187, 203, 209, 215, 217, 221, 245, 247, 275, 287, 289, 299, 305, 323, 329, 335, 341, 365, 371, 377, 395, 407, 413, 425, 427, 437, 455, 473, 485, 497, 515, 517, 527, 529, 533, 539, 545, 551, 575, 581, 583, 605

Offset: 1

Vladimir Shevelev, Jul 16 2014

nonn

See comment in A243937.
If prime p is not in A062326, then p^2 is in the sequence.
If p>3 and p,p+2 are twin primes, then p*(p+2) is in the sequence. Indeed, it can be shown that in this case (p+1)^2 is in A243937. lpf((p+1)^2-1)=p while lpf((p+1)^2-3)=3, since for lesser p>3 of twin prime p+1==0(mod 6).

Cf. A245272, A243937, A245024.

More terms from Peter J. C. Moses, Jul 16 2014

A245024 Even numbers n for which lpf(n-1) < lpf(n-3), where lpf = least prime factor.

Original entry on oeis.org

10, 16, 22, 26, 28, 34, 40, 46, 50, 52, 56, 58, 64, 70, 76, 82, 86, 88, 92, 94, 100, 106, 112, 116, 118, 124, 130, 134, 136, 142, 146, 148, 154, 160, 166, 170, 172, 176, 178, 184, 190, 196, 202, 206, 208, 214, 220, 226, 232, 236, 238, 244, 250, 254, 256, 260

Offset: 1

Vladimir Shevelev, Jul 10 2014

nonn

By the definition, either a(n)==1 (mod 3) or, for every pair of primes (p,q), p>q>=3, a(n)==1 (mod p) and a(n) not==3 (mod q).
Conjecture: All differences are 2,4 or 6 such that no two consecutive terms 2 (...,2,2,...), no two consecutive terms 4, while consecutive terms 6 occur 1,2,3 or 4 times; also consecutive pairs of terms 4,2 appear 1,2,3 or 4 times.
Conjecture is verified up to n = 2.5*10^7. - Vladimir Shevelev and Peter J. C. Moses, Jul 11 2014
The first comment is wrong as stated. This would fix it: for every pair of primes (p,q), p>q>=3, if a(n)==1 (mod p) then a(n) not==3 (mod q). Divisibility by 3 means 6m+4 is in the sequence for all m>0, and 6m never is, while 6m+2 is undetermined. Divisibility by 5 means 30m+26 is always in the sequence, and 30m+8 never is. This proves the above conjecture. - Jens Kruse Andersen, Jul 13 2014
Note that the sequence {a(n)-3} contains all odd primes, except for lesser primes in twin primes pairs (A001359). Other terms of {a(n)-3} are 25,49,55,85,91,... - Vladimir Shevelev,  Jul 15 2014

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

Cf. A020639, A242719, A242720, A243937.

lpf:= n -> min(numtheory:-factorset(n)):
select(n -> lpf(n-1) < lpf(n-3),[seq(2*k,k=3..1000)]); # Robert Israel, Jul 15 2014

lpf[n_] := FactorInteger[n][[1, 1]];
Reap[For[n = 6, n <= 300, n += 2, If[lpf[n-1] < lpf[n-3], Sow[n]]]][[2, 1]] (* Jean-François Alcover, Feb 25 2019 *)

More terms from Peter J. C. Moses, Jul 10 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

A242057 Even numbers n for which lpf_3(n-1) < lpf_3(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

10, 16, 22, 26, 28, 34, 36, 40, 46, 50, 56, 64, 66, 70, 76, 82, 86, 92, 96, 100, 106, 112, 116, 120, 126, 130, 134, 136, 142, 144, 146, 154, 156, 160, 162, 166, 170, 176, 184, 186, 190, 196, 202, 204, 206, 210, 214, 216, 222, 226, 232, 236, 244, 254, 256, 260

Offset: 1

Vladimir Shevelev, Aug 13 2014

nonn

An analog of A245024.

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

Cf. A245024, A243937, A020639.

lpf[n_]:=lpf[n]=First[First[FactorInteger[n]]];
lpf3[n_]:=lpf3[n]=If[#==1,1,lpf[#]]&[n/3^IntegerExponent[n,3]]
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)
select(n->lpf3(n-1)Jens Kruse Andersen, Aug 19 2014

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

A242058 Even numbers n for which lpf_3(n-1) > lpf_3(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

6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 52, 54, 58, 60, 62, 68, 72, 74, 78, 80, 84, 88, 90, 94, 98, 102, 104, 108, 110, 114, 118, 122, 124, 128, 132, 138, 140, 148, 150, 152, 158, 164, 168, 172, 174, 178, 180, 182, 188, 192, 194, 198, 200, 208, 212

Offset: 1

Vladimir Shevelev, Aug 13 2014

nonn

An analog of A243937.

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

Cf. A245024, A243937, A242057, A020639.

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

lpf3(n)=m=n/3^valuation(n, 3); if(m>1, factor(m)[1,1], 1)
select(n->lpf3(n-1)>lpf3(n-3), vector(200, x, 2*x)) \\ Jens Kruse Andersen, Aug 19 2014

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

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

Original entry on oeis.org

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

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

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

cp's OEIS Frontend

A242034 a(n) = lpf(A243937(n)-3), where lpf = least prime factor (A020639).

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A245274 Composite terms in sequence {A243937(n)-1}.

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A245024 Even numbers n for which lpf(n-1) < lpf(n-3), where lpf = least prime factor.

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

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A242057 Even numbers n for which lpf_3(n-1) < lpf_3(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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A242036 Smallest k such that in the interval [1,k] in A242033 all odd primes <= prime(n) are present.

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A242058 Even numbers n for which lpf_3(n-1) > lpf_3(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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Mathematica

PARI

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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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Mathematica

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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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