A245024 -id:A245024 - cp's OEIS frontend

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

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

A245272 Composite terms in sequence {A245024(n)-3}.

Original entry on oeis.org

25, 49, 55, 85, 91, 115, 121, 133, 143, 145, 169, 175, 187, 203, 205, 217, 235, 247, 253, 259, 265, 289, 295, 299, 301, 319, 323, 325, 341, 343, 355, 361, 385, 391, 403, 413, 415, 427, 445, 451, 469, 473, 475, 481, 493, 505, 511, 517, 529, 533, 535, 551, 553

Offset: 1

Vladimir Shevelev, Jul 16 2014

nonn

See comment in A245024.

The sequence contains all squares of primes more than 3. Indeed, since for such primes lpf(p^2+2)=3, then p^2+3 is in A245024.

Cf. A245024, A243937.

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

A243937 Even numbers n>=6 for which lpf(n-1) > lpf(n-3), where lpf = least prime factor.

Original entry on oeis.org

6, 8, 12, 14, 18, 20, 24, 30, 32, 36, 38, 42, 44, 48, 54, 60, 62, 66, 68, 72, 74, 78, 80, 84, 90, 96, 98, 102, 104, 108, 110, 114, 120, 122, 126, 128, 132, 138, 140, 144, 150, 152, 156, 158, 162, 164, 168, 174, 180, 182, 186, 188, 192, 194, 198, 200, 204, 210

Offset: 1

Vladimir Shevelev, Jul 10 2014

nonn

Complement of A245024 over even n >= 6.
Conjecture: All differences are 2, 4 or 6 such that there are 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 2, 4 appear 1, 2, 3 or 4 times. The conjecture is verified up to n = 2.5*10^7. - Vladimir Shevelev and Peter J. C. Moses, Jul 11 2014
Divisibility by 3 means 6m is in the sequence for all m > 0, and 6m + 4 never is, while 6m + 2 is undetermined. Divisibility by 5 means 30m + 8 is always in the sequence, and 30m + 26 never is. This proves the above conjecture. - Jens Kruse Andersen, Aug 19 2014
Note that,
1) Since numbers of the form 6*k evidently are in the sequence, then the counting function of the terms not exceeding x is not less than x/6.
2) Sequence {a(n)-1} contains all primes greater than 3 in the natural order. The subsequence of other terms of {a(n)-1} is 35, 65, 77, 95, ... - Vladimir Shevelev, Jul 15 2014

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

Cf. A242719, A242720, A245024.

select(n->factor(n-1)[1,1]>factor(n-3)[1,1], vector(200, x, 2*x+4)) \\ Jens Kruse Andersen, Aug 19 2014

More terms from Peter J. C. Moses, Jul 10 2014

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

Original entry on oeis.org

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

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

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

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A245272 Composite terms in sequence {A245024(n)-3}.

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A243937 Even numbers n>=6 for which lpf(n-1) > lpf(n-3), where lpf = least prime factor.

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A242034 a(n) = lpf(A243937(n)-3), 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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Programs

Mathematica

PARI

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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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Mathematica

Extensions

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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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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