A071190 -id:A071190 - cp's OEIS frontend

A071189 Smallest prime factor of sum of divisors of n.

1, 3, 2, 7, 2, 2, 2, 3, 13, 2, 2, 2, 2, 2, 2, 31, 2, 3, 2, 2, 2, 2, 2, 2, 31, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 7, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 127, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 11, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Reinhard Zumkeller, May 15 2002

nonn

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

Cf. A000203, A020639, A028982, A028983, A071187, A071190.

Table[FactorInteger[DivisorSigma[1,n]][[1,1]],{n,90}] (* Harvey P. Dale, May 15 2011 *)

A071189(n) = if(1==n, n, my(f = factor(sigma(n))); vecmin(f[, 1])); \\ Antti Karttunen, Jul 24 2017

first(n) = {my(v = vector(n, i, 2), sq = List()); for(i=1, sqrtint(n), listput(sq, i^2); listput(sq, 2*i^2)); listsort(sq); v[1]=1; for(i=2, #sq, if(sq[i]>n,break); v[sq[i]] = factor(sigma(sq[i]))[, 1]~[1]);v} \\ David A. Corneth, Jul 24 2017

a(n) = A020639(A000203(n)).

a(n) = 2 if and only if n is in A028983. - Amiram Eldar, Mar 24 2024

A071188 Largest prime factor of number of divisors of n; a(1)=1.

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 5, 2, 3, 2, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 5, 3, 3, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 7, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 5, 5, 2, 2, 3, 2, 2, 2, 2, 2, 3

Offset: 1

Reinhard Zumkeller, May 15 2002

From Robert Israel, Dec 04 2016: (Start)
a(n)=2 if and only if every member of the prime signature of n is of the form 2^k-1.
a(m*k) = max(a(m),a(k)) if m and k are coprime. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000005, A006530, A071187, A071190, A078542, A078543, A078544, A124010.

Cf. A327839, A354181.

a071188 = a006530 . a000005  -- Reinhard Zumkeller, Sep 04 2013

f:= n -> max(1, numtheory:-factorset(numtheory:-tau(n))):
map(f, [$1..100]); # Robert Israel, Dec 04 2016

Max[Transpose[FactorInteger[#]][[1]]]&/@DivisorSigma[0,Range[100]] (* Harvey P. Dale, Aug 28 2013 *)

a(n) = if(n == 1, 1, vecmax(factor(numdiv(n))[, 1])); \\ Michel Marcus, Dec 05 2016

a(n) = A006530(A000005(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*d(1) + Sum_{k>=2} prime(k)*(d(k) - d(k-1)) = 2.4365518864..., where d(1) = A327839, and for k >= 2, d(k) is the asymptotic density of numbers whose number of divisors is a prime(k)-smooth number, i.e., d(k) = Product_{p prime} ((1 - 1/p) * Sum_{i, A006530(i) <= prime(k)} 1/p^(i-1)) (see A354181 for an example). - Amiram Eldar, Jan 15 2024

A071834 Numbers n > 1 such that n and sigma(n) have the same largest prime factor.

Original entry on oeis.org

6, 28, 40, 84, 117, 120, 135, 140, 224, 234, 270, 420, 468, 496, 585, 672, 756, 775, 819, 891, 931, 936, 1080, 1120, 1170, 1287, 1372, 1488, 1550, 1625, 1638, 1782, 1862, 2176, 2299, 2325, 2340, 2480, 2574, 2793, 3100, 3159, 3250, 3276, 3360, 3472

Offset: 1

Benoit Cloitre, Jun 08 2002

By pure convention, we could include a leading 1 to this sequence, as someone using the mathematically arguably value A006530(1) = 1 might search for this sequence with a leading 1. However, this was not done in view of the age of this sequence. - Rémy Sigrist, Jan 09 2018

			1550 = 2*5^2*31 and sigma(1550) = 2976 = 2^5*3*31 hence 1550 is in the sequence.

Michel Marcus, Table of n, a(n) for n = 1..1000

Cf. A000203 (sigma), A006530 (gpf), A071190.

A000396 (perfect numbers) is a subsequence.

fQ[n_] := FactorInteger[n][[-1, 1]] == FactorInteger[DivisorSigma[1, n]][[-1, 1]]; Rest@ Select[ Range@3500, fQ] (* Robert G. Wilson v, Jan 09 2018 *)

for(n=2,1000,if(component(component(factor(n),1),omega(n)) == component(component(factor(sigma(n)),1),omega(sigma(n))), print1(n,",")))

isok(n) = vecmax(factor(n)[,1]) == vecmax(factor(sigma(n))[,1]); \\ Michel Marcus, Sep 29 2017

n such that A006530(n) = A006530(sigma(n)).

n such that A006530(n) = A071190(n). - Michel Marcus, Oct 11 2017

A295078 Numbers n > 1 such that n and sigma(n) have the same smallest and simultaneously largest prime factors.

Original entry on oeis.org

6, 28, 40, 84, 120, 140, 224, 234, 270, 420, 468, 496, 672, 756, 936, 1080, 1120, 1170, 1372, 1488, 1550, 1638, 1782, 1862, 2176, 2340, 2480, 2574, 3100, 3250, 3276, 3360, 3472, 3564, 3724, 3744, 3780, 4116, 4464, 4598, 4650, 4680, 5148, 5456, 5586, 6048, 6200

Offset: 1

Jaroslav Krizek, Nov 13 2017

nonn

All even perfect numbers are terms.
Conjecture: A007691 (multiply-perfect numbers) is a subsequence.
Note that an odd perfect number (if it exists) would be a counterexample to the conjecture. - Robert Israel, Jan 08 2018
Intersection of A071834 and A295076.
Numbers n such that A020639(n) = A020639(sigma(n)) and simultaneously A006530(n) = A006530(sigma(n)).
Numbers n such that A020639(n) = A071189(n) and simultaneously A006530(n) = A071190(n).
Supersequence of A027598.

			40 = 2^3*5 and sigma(40) = 90 = 2*3^2*5 hence 40 is in the sequence.
The first odd term is 29713401 = 3^2 * 23^2 * 79^2; sigma(29713401) = 45441669 = 3*7^3*13*43*79.

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A000203, A006530, A007691, A020639, A027598, A071189, A071190, A295076.

[n: n in [2..10000] | Minimum(PrimeDivisors(n)) eq Minimum(PrimeDivisors(SumOfDivisors(n))) and Maximum(PrimeDivisors(n)) eq Maximum(PrimeDivisors(SumOfDivisors(n)))]

filter:= proc(n) local f, s; uses numtheory;
  f:= factorset(n);
  s:= factorset(sigma(n));
  min(f) = min(s) and max(f)=max(s)
end proc:
select(filter, [$2..10^4]); # Robert Israel, Jan 08 2018

Rest@ Select[Range@ 6200, SameQ @@ Map[{First@ #, Last@ #} &@ FactorInteger[#][[All, 1]] &, {#, DivisorSigma[1, #]}] &] (* Michael De Vlieger, Nov 13 2017 *)

isok(n) = if (n > 1, my(fn = factor(n)[,1], fs = factor(sigma(n))[,1]); (vecmin(fn) == vecmin(fs)) && (vecmax(fn) == vecmax(fs))); \\ Michel Marcus, Jan 08 2018

Added condition n>1 to definition. Corrected b-file. - N. J. A. Sloane, Feb 03 2018

A333646 Numbers k divisible by the largest prime factor of the sum of divisors of k; a(1) = 1.

Original entry on oeis.org

1, 6, 15, 28, 30, 33, 40, 42, 51, 66, 69, 84, 91, 95, 102, 105, 117, 120, 135, 138, 140, 141, 145, 159, 165, 182, 186, 190, 210, 213, 224, 231, 234, 255, 270, 273, 280, 282, 285, 287, 290, 295, 308, 318, 321, 330, 345, 357, 364, 395, 420, 426, 435, 440, 445, 455

Offset: 1

Amiram Eldar, Jun 05 2020

nonn

Pomerance (1973) proved that all the harmonic numbers (A001599) are in this sequence.
If m is a product of distinct Mersenne primes (A046528), m > 1 and 3 | m, then 2*m is a term.
If p is a term of A005105 then, 6*p is a term for p > 3, and 3*p is a term if p is not a Mersenne prime (A000668).

			15 is a term since sigma(15) = 24, 3 is the largest prime factor of 24, and 15 is divisible by 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Carl Pomerance, On a Problem of Ore: Harmonic Numbers, unpublished manuscript, 1973; abstract *709-A5, Notices of the American Mathematical Society, Vol. 20, 1973, page A-648, entire volume.

A001599 and A105402 are subsequences.

Cf. A000203, A000668, A005105, A006530, A046528, A071190.

Select[Range[500], Divisible[#, FactorInteger[DivisorSigma[1, #]][[-1, 1]]] &]

Numbers k such that A071190(k) | k.

A071965 Numbers k such that k = Gpf(k) * Gpf(sigma(k)) where Gpf(k) = A006530(k) is the greatest prime factor of k.

Original entry on oeis.org

15, 33, 51, 69, 91, 95, 141, 145, 159, 213, 287, 295, 321, 395, 445, 473, 573, 581, 679, 703, 745, 895, 973, 995, 1139, 1149, 1169, 1195, 1199, 1293, 1339, 1345, 1441, 1561, 1717, 1757, 1795, 1891, 1941, 2051, 2147, 2167, 2245, 2353, 2395, 2443, 2495, 2589

Offset: 1

Benoit Cloitre, Jun 16 2002

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

Cf. A000203, A006530, A071190.

Select[Range[2,3000],FactorInteger[#][[-1,1]]FactorInteger[ DivisorSigma[ 1,#]] [[-1,1]]==#&] (* Harvey P. Dale, Sep 15 2011 *)

for(n=1,3000,if(vecmax(component(factor(n),1))*vecmax(component(factor(sigma(n)),1))==n,print1(n,",")))

A078551 Largest prime dividing sigma(2,n).

Original entry on oeis.org

5, 5, 7, 13, 5, 5, 17, 13, 13, 61, 7, 17, 5, 13, 31, 29, 13, 181, 13, 5, 61, 53, 17, 31, 17, 41, 7, 421, 13, 37, 13, 61, 29, 13, 13, 137, 181, 17, 17, 29, 5, 37, 61, 13, 53, 17, 31, 43, 31, 29, 17, 281, 41, 61, 17, 181, 421, 1741, 13, 1861, 37, 13, 127, 17, 61, 449, 29, 53

Offset: 2

Labos Elemer, Dec 05 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 2..10000

Cf. A006530, A001157-A001160, A071190.

gpf[n_] := FactorInteger[n][[-1, 1]]; a[n_] := gpf[DivisorSigma[2, n]]; Array[a, 70, 2] (* Amiram Eldar, Aug 01 2019 *)

a(n) = A006530(A001157(n)).

A078552 Largest prime dividing sigma(3,n).

Original entry on oeis.org

3, 7, 73, 7, 7, 43, 13, 757, 7, 37, 73, 157, 43, 7, 151, 13, 757, 7, 73, 43, 37, 13, 13, 829, 157, 73, 73, 271, 7, 19, 73, 37, 13, 43, 757, 43, 7, 157, 13, 547, 43, 139, 73, 757, 13, 103, 151, 1063, 829, 13, 157, 919, 73, 37, 43, 7, 271, 163, 73, 523, 19, 757, 337, 157

Offset: 2

Labos Elemer, Dec 05 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 2..10000

Cf. A006530, A001157-A001160, A071190.

gpf[n_] := FactorInteger[n][[-1, 1]]; a[n_] := gpf[DivisorSigma[3, n]]; Array[a, 65, 2] (* Amiram Eldar, Aug 01 2019 *)

a(n) = A006530(A001158(n)).

A078553 Largest prime dividing sigma(4,n).

Original entry on oeis.org

17, 41, 13, 313, 41, 1201, 257, 73, 313, 7321, 41, 14281, 1201, 313, 41, 41761, 73, 3833, 313, 1201, 7321, 139921, 257, 601, 14281, 193, 1201, 353641, 313, 1129, 241, 7321, 41761, 1201, 73, 10529, 3833, 14281, 313, 10313, 1201, 521, 7321, 313

Offset: 2

Labos Elemer, Dec 05 2002

nonn

			Observe nontrivial frequent occurrence of several primes like 73,313,14281, etc.

Amiram Eldar, Table of n, a(n) for n = 2..10000

Cf. A006530, A001157-A001160, A071190.

gpf[n_] := FactorInteger[n][[-1, 1]]; a[n_] := gpf[DivisorSigma[4, n]]; Array[a, 50, 2] (* Amiram Eldar, Aug 01 2019 *)

a(n) = A006530(A001159(n)).

A078554 Largest prime dividing sigma(5,n).

Original entry on oeis.org

11, 61, 151, 521, 61, 191, 41, 4561, 521, 13421, 151, 2411, 191, 521, 1801, 101, 4561, 2251, 521, 191, 13421, 211, 61, 1741, 2411, 1181, 191, 401, 521, 21821, 331, 13421, 101, 521, 4561, 1824841, 2251, 2411, 521, 4111, 191, 3341101, 13421, 4561, 211

Offset: 2

Labos Elemer, Dec 05 2002

nonn

			Observe nontrivial frequent occurrence of several primes like 61,191,521,4561, etc.

Amiram Eldar, Table of n, a(n) for n = 2..10000

Cf. A006530, A001157-A001160, A071190.

gpf[n_] := FactorInteger[n][[-1, 1]]; a[n_] := gpf[DivisorSigma[5, n]]; Array[a, 50, 2] (* Amiram Eldar, Aug 01 2019 *)
FactorInteger[#][[-1,1]]&/@DivisorSigma[5,Range[2,50]] (* Harvey P. Dale, Feb 02 2025 *)

a(n) = A006530(A001160(n)).

cp's OEIS Frontend

A071189 Smallest prime factor of sum of divisors of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A071188 Largest prime factor of number of divisors of n; a(1)=1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

A071834 Numbers n > 1 such that n and sigma(n) have the same largest prime factor.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A295078 Numbers n > 1 such that n and sigma(n) have the same smallest and simultaneously largest prime factors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Extensions

A333646 Numbers k divisible by the largest prime factor of the sum of divisors of k; a(1) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A071965 Numbers k such that k = Gpf(k) * Gpf(sigma(k)) where Gpf(k) = A006530(k) is the greatest prime factor of k.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A078551 Largest prime dividing sigma(2,n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica