A046665 -id:A046665 - cp's OEIS frontend

A135170 Primes equal to a sum c1+c2 of two consecutive composite numbers such that lpf(c1)-spf(c1)+lpf(c2)-spf(c2) from their largest and smallest prime factors is prime.

19, 29, 31, 41, 43, 53, 67, 71, 79, 89, 101, 109, 131, 149, 151, 173, 197, 199, 233, 239, 241, 251, 269, 271, 283, 307, 311, 317, 331, 337, 349, 367, 401, 419, 439, 449, 461, 487, 491, 499, 509, 521, 593, 599, 617, 641, 647, 683, 691, 727, 739, 751, 769, 809

Offset: 1

Giovanni Teofilatto, Feb 14 2008

nonn

Cf. A111426.

A002808 := proc(n) option remember ; local a ; if n = 1 then 4; else for a from A002808(n-1)+1 do if not isprime(a) then RETURN(a) ; fi ; od: fi ; end:
isA060254 := proc(n) local i,sComp ; if isprime(n) then for i from 1 do sComp := A002808(i)+A002808(i+1) ; if sComp = n then RETURN(i); elif sComp > n then RETURN(-1) ; fi ; od: else -1 ; fi ; end:
A046665 := proc(n) local a,ifs ; a := 0 ; ifs := seq(op(1, i),i=ifactors(n)[2]) ; max(ifs)-min(ifs) ; end:
A111426 := proc(n) A046665(A002808(n)) ; end:
isA135170 := proc(p) local i ; i := isA060254(p) ; if i > 0 then A111426(i) + A111426(i+1) ; isprime(%) ; else false ; fi ; end:
for n from 1 to 300 do p := ithprime(n) ; if isA135170(p) then printf("%d,",p) ; fi ; od: # R. J. Mathar, Feb 19 2008

{A060254(j): A002808(i)+A002808(i+1)=A060254(j) and A111426(i)+A111426(i+1) in A000040}. Subsequence of A060254. - R. J. Mathar, Feb 19 2008

Corrected and extended by R. J. Mathar, Feb 19 2008

More precise definition by R. J. Mathar, Sep 17 2009

A381214 a(n) is the difference between the largest and smallest element of the set of bases and exponents (including exponents = 1) in the prime factorization of n.

Original entry on oeis.org

1, 2, 0, 4, 2, 6, 1, 1, 4, 10, 2, 12, 6, 4, 2, 16, 2, 18, 4, 6, 10, 22, 2, 3, 12, 0, 6, 28, 4, 30, 3, 10, 16, 6, 1, 36, 18, 12, 4, 40, 6, 42, 10, 4, 22, 46, 3, 5, 4, 16, 12, 52, 2, 10, 6, 18, 28, 58, 4, 60, 30, 6, 4, 12, 10, 66, 16, 22, 6, 70, 1, 72, 36, 4, 18

Offset: 2

Paolo Xausa, Feb 19 2025

			a(36) = 1 because 36 = 2^2*3^2, the set of these bases and exponents is {2, 3} and 3 - 2 = 1.
a(31500) = 6 because 31500 = 2^2*3^2*5^3*7^1, the set of these bases and exponents is {1, 2, 3, 5, 7} and 7 - 1 = 6.

Paolo Xausa, Table of n, a(n) for n = 2..10000

Cf. A046665, A081812, A381212.

Cf. A051674 (positions of zeros), A381215 (positions of ones).

A381214[n_] := Max[#] - Min[#] & [Flatten[FactorInteger[n]]];
Array[A381214, 100, 2]

a(n) = my(f=factor(n), s=setunion(Set(f[,1]), Set(f[,2]))); vecmax(s) - vecmin(s); \\ Michel Marcus, Feb 20 2025

a(n) = A081812(n) - A381212(n).

A089341 Numbers k with lpf(k) < gpf(k) < 2*lpf(k), where lpf = A020639, gpf = A006530.

Original entry on oeis.org

6, 12, 15, 18, 24, 35, 36, 45, 48, 54, 72, 75, 77, 91, 96, 108, 135, 143, 144, 162, 175, 187, 192, 209, 216, 221, 225, 245, 247, 288, 299, 323, 324, 375, 384, 391, 405, 432, 437, 486, 493, 527, 539, 551, 576, 589, 637, 648, 667, 675, 703, 713, 768, 847, 851

Offset: 1

Reinhard Zumkeller, Jan 03 2004

nonn

A081306 without prime powers.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Smooth Number.

Cf. A005279, A006530, A020639, A046665, A081306.

Subsequence of A024619.

a089341 n = a089341_list !! (n-1)
a089341_list = filter (\x -> a006530 x < 2 * a020639 x) a024619_list
-- Reinhard Zumkeller, Sep 29 2014

q[k_] := Module[{p = FactorInteger[k][[;;, 1]]}, Length[p] > 1 && p[[-1]] < 2*p[[1]]]; Select[Range[1000], q] (* Amiram Eldar, May 16 2025 *)

isok(k) = if(k == 1, 0, my(p = factor(k)[,1], np = #p);  np > 1 && p[np] < 2*p[1]); \\ Amiram Eldar, May 16 2025

A046665(a(n)) < A020639(a(n)).

A100573 Smallest difference between distinct prime divisors of n, or 0 if n is a prime power.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 5, 2, 0, 0, 1, 0, 3, 4, 9, 0, 1, 0, 11, 0, 5, 0, 1, 0, 0, 8, 15, 2, 1, 0, 17, 10, 3, 0, 1, 0, 9, 2, 21, 0, 1, 0, 3, 14, 11, 0, 1, 6, 5, 16, 27, 0, 1, 0, 29, 4, 0, 8, 1, 0, 15, 20, 2, 0, 1, 0, 35, 2, 17, 4, 1, 0, 3, 0, 39, 0, 1, 12, 41, 26, 9, 0, 1, 6, 21, 28, 45

Offset: 1

Leroy Quet, Jan 02 2005

nonn

			For 30 = 2*3*5, 2 and 3 are separated by only 1, so a(30) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. also A046665, A297173.

<Labos Elemer, Jan 05 2005 *)
Table[Min[Differences[Transpose[FactorInteger[n]][[1]]]],{n,100}]/.\[Infinity]->0 (* Harvey P. Dale, Jul 27 2015 *)

A100573(n) = if(omega(n)<=1,0,my(ps=factor(n)[,1]); vecmin(vector((#ps)-1,i,ps[i+1]-ps[i]))); \\ Antti Karttunen, Mar 03 2018

More terms from Labos Elemer, Jan 05 2005

A120454 a(n) = ceiling(GPF(n)/LPF(n)) where GPF is greatest prime factor, LPF is least prime factor.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 4, 2, 1, 1, 2, 1, 3, 3, 6, 1, 2, 1, 7, 1, 4, 1, 3, 1, 1, 4, 9, 2, 2, 1, 10, 5, 3, 1, 4, 1, 6, 2, 12, 1, 2, 1, 3, 6, 7, 1, 2, 3, 4, 7, 15, 1, 3, 1, 16, 3, 1, 3, 6, 1, 9, 8, 4, 1, 2, 1, 19, 2, 10, 2, 7, 1, 3, 1, 21, 1, 4, 4, 22, 10, 6, 1, 3, 2, 12, 11, 24, 4, 2, 1, 4, 4

Offset: 1

Jonathan Vos Post, Aug 16 2006

Given GPF(n) and LPF(n), the sum is A074320, the difference is A046665 and the product is A066048. a(n) = 1 iff n is p^k iff n is in A000961.

			a(26) = ceiling(GPF(26)/LPF(26)) = ceiling(13/2) = 7.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000961, A006530, A020639, A024619, A046665, A066048, A069897, A074320.

A120454 := proc(n) local ifs ; if n = 1 then RETURN(1) ; else ifs := ifactors(n)[2] ; RETURN( ceil(op(1,op(-1,ifs))/op(1,op(1,ifs))) ) ; fi ; end ; for n from 1 to 100 do printf("%d, ",A120454(n)) ; od ; # R. J. Mathar, Dec 16 2006

a[n_] := Module[{p = FactorInteger[n][[;;, 1]]}, Ceiling[p[[-1]] / p[[1]]]]; Array[a, 100] (* Amiram Eldar, Oct 24 2024 *)

A120454(n) = if(1==n,1, my(f = factor(n), lpf = f[1, 1], gpf = f[#f~, 1]); ceil(gpf/lpf)); \\ Antti Karttunen, Sep 06 2018

a(n) = ceiling(A006530(n)/A020639(n)).

a(n) = A069897(n) + 1 if n is not a power of a prime (A024619), and 1 otherwise. - Amiram Eldar, Oct 24 2024

Corrected and extended by R. J. Mathar, Dec 16 2006

A086770 Numbers k such that the difference between the largest and the smallest prime divisor of k equals the number of prime divisors of k (counted with multiplicity).

Original entry on oeis.org

1, 15, 20, 30, 35, 50, 112, 143, 168, 189, 252, 280, 315, 323, 378, 392, 420, 441, 525, 588, 630, 700, 735, 882, 899, 980, 1029, 1050, 1372, 1470, 1750, 1763, 2058, 2450, 2816, 3430, 3599, 3773, 4224, 4802, 5183, 5929, 6336, 7040, 9317, 9504, 9856, 10403

Offset: 1

Jason Earls, Aug 02 2003

nonn

			112 is a term because 112 = 2^4*7 with 5 primes dividing it and 7-2=5.

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

Cf. A001222, A006530, A046665.

f:=func; [1] cat [k:k in [2..10000]| Max(PrimeDivisors(k))-Min(PrimeDivisors(k)) eq f(k)]; // Marius A. Burtea, Dec 16 2019

seqQ[1] = True; seqQ[n_] := Plus @@ Last /@ (f = FactorInteger[n]) == f[[-1, 1]] - f[[1, 1]]; Select[Range[10^4], seqQ] (* Amiram Eldar, Dec 16 2019 *)

print1("1, "); for(k=2,10500,my(f=factor(k));if(bigomega(k)==vecmax(f[, 1])-f[1,1],print1(k,", "))) \\ Hugo Pfoertner, Dec 16 2019

Name edited by Hugo Pfoertner, Dec 16 2019

A212332 The difference between the largest and smallest prime factor of n as n runs through the numbers with at least two distinct prime factors.

Original entry on oeis.org

1, 3, 1, 5, 2, 1, 3, 4, 9, 1, 11, 5, 3, 8, 15, 2, 1, 17, 10, 3, 5, 9, 2, 21, 1, 3, 14, 11, 1, 6, 5, 16, 27, 3, 29, 4, 8, 9, 15, 20, 5, 1, 35, 2, 17, 4, 11, 3, 39, 5, 12, 41, 26, 9, 3, 6, 21, 28, 45, 14, 1, 5, 8, 3, 15, 11, 4, 51, 1, 9, 34, 5, 17, 18, 27, 10

Offset: 1

Michel Lagneau, Aug 07 2012

nonn

a(n) = A100576(n) if A168638(n) contains only two prime distinct divisors.

Nonzero entries of A046665 in the order of appearance. R. J. Mathar, Aug 07 2012

			a(13) = 3 because A168638(13) = 30 = 2*3*5 and 5-2 = 3.

Michel Lagneau, Table of n, a(n) for n = 1..10000

Cf. A100576, A168638.

for n from 2 to 200 do
    gpf := A006530(n) ;
    spf := A020639(n) ;
    if gpf <> spf then
        printf("%d,",gpf-spf) ;
    end if;
end do: # R. J. Mathar, Aug 07 2012

A243981 Minimum range of sets of natural numbers with a product of n.

Original entry on oeis.org

1, 2, 0, 4, 1, 6, 0, 0, 3, 10, 1, 12, 5, 2, 0, 16, 1, 18, 1, 4, 9, 22, 1, 0, 11, 0, 3, 28, 1, 30, 0, 8, 15, 2, 0, 36, 17, 10, 3, 40, 1, 42, 7, 2, 21, 46, 1, 0, 3, 14, 9, 52, 1, 6, 1, 16, 27, 58, 2, 60, 29, 2, 0, 8, 5, 66, 13, 20, 3, 70, 1, 72, 35, 2, 15, 4, 7, 78, 1, 0, 39, 82, 4, 12, 41, 26, 3, 88, 1, 6, 19, 28, 45, 14, 1, 96, 5, 2, 0

Offset: 2

Paul Richards, Nov 11 2014

The minimum difference between the largest and smallest values in the sets of positive integers with a product of n, excluding the singleton set {n}.

			For 45 the sets are {1,45}, {3,15}, {5,9}, {3,3,5} with differences of 44, 12, 4 and 2 respectively.  2 is the minimum and so a(45) = 2.

Alois P. Heinz, Table of n, a(n) for n = 2..10000

Cf. A056737, A060794, A076388, A082119.

a(n) <= A046665(n) for all composite n, a(p) = p - 1 for primes p. - Charlie Neder, Jan 13 2019

cp's OEIS Frontend

A135170 Primes equal to a sum c1+c2 of two consecutive composite numbers such that lpf(c1)-spf(c1)+lpf(c2)-spf(c2) from their largest and smallest prime factors is prime.

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A381214 a(n) is the difference between the largest and smallest element of the set of bases and exponents (including exponents = 1) in the prime factorization of n.

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A089341 Numbers k with lpf(k) < gpf(k) < 2*lpf(k), where lpf = A020639, gpf = A006530.

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A100573 Smallest difference between distinct prime divisors of n, or 0 if n is a prime power.

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A120454 a(n) = ceiling(GPF(n)/LPF(n)) where GPF is greatest prime factor, LPF is least prime factor.

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A086770 Numbers k such that the difference between the largest and the smallest prime divisor of k equals the number of prime divisors of k (counted with multiplicity).

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Magma

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A212332 The difference between the largest and smallest prime factor of n as n runs through the numbers with at least two distinct prime factors.

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Maple