A020639 -id:A020639 - 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

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

A242489 Smallest even k such that lpf(k-1) = prime(n), while lpf(k-3) > prime(n), where lpf=least prime factor (A020639).

Original entry on oeis.org

10, 26, 50, 254, 170, 392, 362, 944, 842, 1892, 1370, 2420, 1850, 2210, 3764, 6314, 3722, 4892, 5042, 7082, 8612, 9380, 7922, 12320, 11414, 10610, 11450, 13844, 18872, 16130, 17162, 20414, 19322, 26672, 24614, 25592, 29504, 37910, 29930, 44930, 36020, 36482

Offset: 2

Vladimir Shevelev, May 16 2014

nonn

This sequence is connected with a sufficient condition for the infinitude of twin primes.
Almost all numbers of the form a(n)-3 are primes. For composite numbers of such a form, see A242716.
Primes p for which a(p) = p^2+1 form sequence A062326 for p >= 3. - Vladimir Shevelev, May 21 2014

			Let n=2, prime(2)=3. Then lpf(10-1)=3, but lpf(10-3)=7>3.
Since k=10 is the smallest such k, then a(2)=10.

Peter J. C. Moses, Table of n, a(n) for n = 2..2501

Cf. A001359, A006512.

lpf[n_]:=lpf[n]=First[Select[Divisors[n],PrimeQ[#]&]];
Table[test=Prime[n];NestWhile[#+2&,test^2+1,!((lpf[#-1]==test)&&(lpf[#-3]>test))&],{n,2,60}] (* Peter J. C. Moses, May 21 2014 *)

a(n) = {k = 6; p = prime(n); while ((factor(k-1)[1, 1] != p) || (factor(k-3)[1, 1] <= p), k+= 2); k;} \\ Michel Marcus, May 16 2014

a(n) >= prime(n)^2+1. - Vladimir Shevelev, May 21 2014

More terms from Michel Marcus, May 16 2014

A242490 Smallest even number k such that lpf(k-3) = prime(n) while lpf(k-1) > lpf(k-3), where lpf=least prime factor (A020639).

Original entry on oeis.org

6, 8, 80, 14, 224, 20, 440, 854, 32, 1460, 1742, 44, 2282, 3434, 4190, 62, 5432, 4760, 74, 12194, 8930, 8054, 12374, 13292, 104, 15350, 110, 14282, 31982, 17402, 18212, 140, 24050, 152, 25220, 29990, 28202, 32234, 33392, 182, 43262, 194, 44972, 200, 47564

Offset: 2

Vladimir Shevelev, May 16 2014

nonn

Note that the "small terms" {6,8,14,20,32,44,...} correspond to a(n) for which {a(n)-3, a(n)-1} is a twin pair such that the corresponding positions form sequence A029707.

If we change the definition to consider k for which {k-3, k-1} is not a twin pair, we obtain a closely related sequence 12,38,80,212,224,530,440,854,1250,1460,1742,... which shows a "model behavior" of A242490, if there are only a finite number of twin primes. - Vladimir Shevelev, May 19 2014

			Let n=2, prime(2)=3. Then lpf(6-3)=3, but lpf(6-1)=5>3. Since k=6 is the smallest such k, a(2)=6.

Peter J. C. Moses, Table of n, a(n) for n = 2..1001

Cf. A001359, A006512, A242489.

a(n)=my(p=prime(n),k=p+3); while(factor(k-3)[1,1]Charles R Greathouse IV, May 30 2014

Correction and more terms from Peter J. C. Moses, May 19 2014

A252375 a(n) = smallest r such that r^k <= spf(n) and gpf(n) < r^(k+1), for some k >= 0, where spf and gpf (smallest and greatest prime factor of n) are given by A020639(n) and A006530(n).

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 8, 3, 2, 2, 2, 2, 6, 3, 12, 2, 2, 2, 14, 2, 8, 2, 6, 2, 2, 12, 18, 2, 2, 2, 20, 14, 6, 2, 8, 2, 12, 3, 24, 2, 2, 2, 6, 18, 14, 2, 2, 4, 8, 20, 30, 2, 6, 2, 32, 3, 2, 4, 12, 2, 18, 24, 8, 2, 2, 2, 38, 3, 20, 4, 14, 2, 6, 2, 42, 2, 8, 5, 44, 30, 12, 2, 6, 4, 24, 32

Offset: 1

Antti Karttunen, Dec 17 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

A252374 gives the corresponding exponents.
Cf. A251726 (those n for which a(n) <= A006530(n)).
Cf. A251727 (those n > 1 for which a(n) = A006530(n)+1).
Cf. A006530, A020639, A066048, A138510, A251725.

(define (A252375 n) (let ((spf (A020639 n)) (gpf (A006530 n))) (let outerloop ((r 2)) (let innerloop ((rx 1)) (cond ((and (<= rx spf) (< gpf (* r rx))) r) ((<= rx spf) (innerloop (* r rx))) (else (outerloop (+ 1 r))))))))
(define (A252375 n) (let ((x (A251725 n))) (if (= 1 x) 2 x))) ;; Alternatively, using the implementation of A251725.

If A251725(n) = 1, a(n) = 2, otherwise a(n) = A251725(n).
Other identities. For all n >= 1:
a(n) = a(A066048(n)). [The result depends only on the smallest and the largest prime factor of n.]

A284260 Greatest prime dividing n which is less than A020639(n)^2, where A020639(n) is the smallest prime dividing n, a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 2, 3, 2, 11, 3, 13, 2, 5, 2, 17, 3, 19, 2, 7, 2, 23, 3, 5, 2, 3, 2, 29, 3, 31, 2, 3, 2, 7, 3, 37, 2, 3, 2, 41, 3, 43, 2, 5, 2, 47, 3, 7, 2, 3, 2, 53, 3, 11, 2, 3, 2, 59, 3, 61, 2, 7, 2, 13, 3, 67, 2, 3, 2, 71, 3, 73, 2, 5, 2, 11, 3, 79, 2, 3, 2, 83, 3, 17, 2, 3, 2, 89, 3, 13, 2, 3, 2, 19, 3, 97, 2, 3, 2, 101, 3, 103, 2, 7, 2, 107, 3, 109

Offset: 1

Antti Karttunen, Mar 24 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001

Cf. A006530, A020639, A284252, A284253, A284254, A284255, A284256, A284257, A284258, A284259.

Cf. A251726 (gives n > 1 such that a(n) = A006530(n)).

Table[Last[Function[s, Select[s, # < First[s]^2 &]]@ FactorInteger[n][[All, 1]] /. {} -> {1}], {n, 109}] (* Michael De Vlieger, Mar 24 2017 *)

A(n) = if(n<2, return(1), my(f=factor(n)[, 1]); for(i=2, #f, if(f[i]>f[1]^2, return(f[i]))); return(1));
a(n) = if(A(n)==1, 1, A(n)*a(n/A(n)));
gpf(n) = if(n>1, vecmax(factor(n)[,1]),1);
for(n=1, 150, print1(gpf(n/a(n)),", ")) \\ Indranil Ghosh, Mar 24 2017, after David A. Corneth

from sympy import primefactors
def A(n):
    for i in primefactors(n):
        if i>min(primefactors(n))**2: return i
    return 1
def a(n): return 1 if A(n)==1 else A(n)*a(n//A(n))
def gpf(n): return 1 if n<2 else max(primefactors(n))
print([gpf(n//a(n)) for n in range(1, 151)]) # Indranil Ghosh, Mar 24 2017

(define (A284260 n) (A006530 (A284255 n)))

a(n) = A006530(A284255(n)).

A325965 a(n) is the least k >= A020639(n) such that n-k and n-(sigma(n)-k) are relatively prime.

Original entry on oeis.org

1, 2, 4, 2, 6, 5, 8, 2, 3, 3, 12, 3, 14, 3, 4, 2, 18, 2, 20, 3, 4, 3, 24, 5, 5, 3, 4, 27, 30, 5, 32, 2, 4, 3, 6, 2, 38, 3, 4, 3, 42, 5, 44, 3, 4, 3, 48, 3, 7, 2, 4, 3, 54, 5, 6, 3, 4, 3, 60, 5, 62, 3, 4, 2, 6, 5, 68, 5, 4, 3, 72, 2, 74, 3, 4, 3, 8, 5, 80, 3, 3, 3, 84, 3, 6, 3, 4, 3, 90, 5, 8, 3, 4, 3, 6, 5, 98, 2, 4, 2

Offset: 1

Antti Karttunen, May 29 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A000396, A020639, A325817, A325966.

A020639(n) = if(1==n, n, factor(n)[1, 1]);
A325965(n) = { my(s=sigma(n)); for(i=A020639(n), s, if(1==gcd(n-i, n-(s-i)), return(i))); };

a(n) = A000203(n) - A325966(n).
For all n:
a(A000396(n)) = A000396(n)-1.
a(n) >= A325817(n).

A325966 a(n) is the largest i <= sigma(n)-A020639(n) such that n-i and n-(sigma(n)-i) are relatively prime.

Original entry on oeis.org

0, 1, 0, 5, 0, 7, 0, 13, 10, 15, 0, 25, 0, 21, 20, 29, 0, 37, 0, 39, 28, 33, 0, 55, 26, 39, 36, 29, 0, 67, 0, 61, 44, 51, 42, 89, 0, 57, 52, 87, 0, 91, 0, 81, 74, 69, 0, 121, 50, 91, 68, 95, 0, 115, 66, 117, 76, 87, 0, 163, 0, 93, 100, 125, 78, 139, 0, 121, 92, 141, 0, 193, 0, 111, 120, 137, 88, 163, 0, 183, 118, 123, 0

Offset: 1

Antti Karttunen, May 29 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A000396, A020639, A324213, A325818, A325961, A325962, A325965.

A020639(n) = if(1==n, n, factor(n)[1, 1]);
A325966(n) = { my(s=sigma(n)); forstep(i=s-A020639(n), 0, -1, if(1==gcd(n-i, n-(s-i)), return(i))); };

a(n) = A000203(n) - A325965(n).
For all n:
a(A000396(n)) = A000396(n)+1.
a(n) <= A325818(n).

A366825 Numbers of the form p^2 * m, squarefree m > 1, prime p < lpf(m), where lpf(m) = A020639(m).

Original entry on oeis.org

12, 20, 28, 44, 45, 52, 60, 63, 68, 76, 84, 92, 99, 116, 117, 124, 132, 140, 148, 153, 156, 164, 171, 172, 175, 188, 204, 207, 212, 220, 228, 236, 244, 260, 261, 268, 275, 276, 279, 284, 292, 308, 315, 316, 325, 332, 333, 340, 348, 356, 364, 369, 372, 380, 387

Offset: 1

Michael De Vlieger, Dec 15 2023

Proper subset of A126706. Proper subset of A364996.
Prime signature of a(n) is 2 followed by at least one 1.
Numbers of the form A065642(A120944(k)) for some k.
The asymptotic density of this sequence is (6/Pi^2) * Sum_{p prime} (1/p^2) * (Product_{primes q <= p} (q/(q+1))) = 0.155068688392... . - Amiram Eldar, Dec 18 2023

			a(1) = 12 = 4*3 = p^2 * m, squarefree m > 1; sqrt(4) < lpf(3), i.e., 2 < 3.
a(5) = 45 = 9*5 = p^2 * m, squarefree m > 1; sqrt(9) < lpf(5), i.e., 3 < 5.
Prime powers p^k, k > 2, are not in the sequence since m = p^(k-2) is not squarefree and p = lpf(m).

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007947, A020639, A065642, A120944, A126706, A364996.

Select[Select[Range[500], PrimeOmega[#] > PrimeNu[#] > 1 &], First[#1] == 2 && Union[#2] == {1} & @@ TakeDrop[FactorInteger[#][[All, -1]], 1] &]

is(n) = {my(e = factor(n)[, 2]); #e > 1 && e[1] == 2 && vecmax(e[2..#e]) == 1;} \\ Amiram Eldar, Dec 18 2023

A117358 a(n) = A032742(A032742(A032742(n))) = ((n/lpf(n))/lpf(n/lpf(n)))/lpf((n/lpf(n))/lpf(n/lpf(n))), where lpf=A020639, least prime factor.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 3, 1, 7, 1, 1, 1, 5, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 10, 3, 1, 1, 7, 1, 1, 1, 11, 1, 5, 1, 1, 1, 1, 1, 12, 1, 1, 1, 5, 1, 1, 1

Offset: 1

Reinhard Zumkeller, Mar 10 2006

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Eric Weisstein's World of Mathematics, Least Prime Factor

Cf. A014673, A032742, A037144, A033987, A054576, A115561.

f[n_] := n/FactorInteger[n][[1, 1]]; (* f is A032742 *)
a[n_] := f@ f@ f@ n;
Array[a, 100] (* Jean-François Alcover, Dec 09 2021 *)
Table[Nest[#/FactorInteger[#][[1,1]]&,n,3],{n,110}] (* Harvey P. Dale, Oct 10 2024 *)

(define (A117358 n) (A032742 (A032742 (A032742 n)))) ;; Antti Karttunen, Dec 07 2017

a(n) = A032742(A032742(A032742(n))) = A032742(A054576(n)) = A054576(n)/A115561(n).

a(A037144(n)) = 1, a(A033987(n)) > 1.

cp's OEIS Frontend

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

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

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A242489 Smallest even k such that lpf(k-1) = prime(n), while lpf(k-3) > prime(n), where lpf=least prime factor (A020639).

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A242490 Smallest even number k such that lpf(k-3) = prime(n) while lpf(k-1) > lpf(k-3), where lpf=least prime factor (A020639).

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A252375 a(n) = smallest r such that r^k <= spf(n) and gpf(n) < r^(k+1), for some k >= 0, where spf and gpf (smallest and greatest prime factor of n) are given by A020639(n) and A006530(n).

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A284260 Greatest prime dividing n which is less than A020639(n)^2, where A020639(n) is the smallest prime dividing n, a(1) = 1.

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A325965 a(n) is the least k >= A020639(n) such that n-k and n-(sigma(n)-k) are relatively prime.

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A325966 a(n) is the largest i <= sigma(n)-A020639(n) such that n-i and n-(sigma(n)-i) are relatively prime.

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