A071189 -id:A071189 - cp's OEIS frontend

A071190 Greatest prime factor of sum of divisors of n, for n >= 2; a(1) = 1.

1, 3, 2, 7, 3, 3, 2, 5, 13, 3, 3, 7, 7, 3, 3, 31, 3, 13, 5, 7, 2, 3, 3, 5, 31, 7, 5, 7, 5, 3, 2, 7, 3, 3, 3, 13, 19, 5, 7, 5, 7, 3, 11, 7, 13, 3, 3, 31, 19, 31, 3, 7, 3, 5, 3, 5, 5, 5, 5, 7, 31, 3, 13, 127, 7, 3, 17, 7, 3, 3, 3, 13, 37, 19, 31, 7, 3, 7, 5, 31

Offset: 1

Reinhard Zumkeller, May 15 2002

nonn

a(n) is the greatest noncomposite factor of the sum of divisors of n. (Cf. A008578.) - Omar E. Pol, Feb 11 2018

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

Cf. A000203, A006530, A071188, A071189.

Table[FactorInteger[DivisorSigma[1,n]][[-1,1]],{n,80}] (* Harvey P. Dale, Nov 18 2014 *)

a(n) = if (n==1, 1, my(f = factor(sigma(n))); f[#f~,1]); \\ Michel Marcus, Sep 21 2014

a(n) = A006530(A000203(n)).

Name clarified by David A. Corneth and Omar E. Pol, Feb 11 2018

A071187 Smallest 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, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2

Offset: 1

Reinhard Zumkeller, May 15 2002

a(n) = 2 for nonsquare n. - David A. Corneth, Jul 24 2017

			324 = 18^2 = 2^2 * 3^4 has (2+1)*(4+1) = 3 * 5 = 15 divisors, thus a(324) = A020639(15) = 3. - _Antti Karttunen_, Nov 18 2019

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences computed from exponents in factorization of n.

Cf. A000005, A020639, A071188, A071189, A108951.

Differs from A329614 for the first time at n=324, where a(324) = 3, while A329614(324) = 5. A329613 gives the positions of differences.

a[n_] := FactorInteger[DivisorSigma[0, n]][[1, 1]]; Array[a, 90] (* Jean-François Alcover, Oct 01 2016 *)

A071187(n) = if(1==n, n, my(f = factor(numdiv(n))); vecmin(f[, 1])); \\ Antti Karttunen, Jul 24 2017

first(n) = my(v = vector(n, i, 2)); for(i=1,sqrtint(n), v[i^2] = numdiv(i^2)); v

a(n) = A020639(A000005(n)).
a(A108951(n)) = A329614(n). - Antti Karttunen, Nov 17 2019
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2. - Amiram Eldar, Jan 15 2024

Data section extended up to term a(105) by Antti Karttunen, Nov 17 2019

A371418 The largest aliquot divisor of the sum of divisors of n; a(1) = 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 6, 4, 5, 1, 9, 6, 14, 7, 12, 12, 1, 9, 13, 10, 21, 16, 18, 12, 30, 1, 21, 20, 28, 15, 36, 16, 21, 24, 27, 24, 13, 19, 30, 28, 45, 21, 48, 22, 42, 39, 36, 24, 62, 19, 31, 36, 49, 27, 60, 36, 60, 40, 45, 30, 84, 31, 48, 52, 1, 42, 72, 34, 63, 48, 72

Offset: 1

Amiram Eldar, Mar 23 2024

Carmichael (1921) defined this arithmetic function for the purpose of studying periodic chains that are formed by repeatedly applying the mapping x -> a(x) starting at a given positive integer. This results in a sequence that is analogous to an aliquot sequence.
Periodic chains of cycle 1 are the fixed points of this sequence. 1 and the even perfect numbers (the even terms of A000396) are fixed points. Are there any other numbers k such that a(k) = k?
If a(k) = k and k is even, then a(k) is even and so is sigma(k), and therefore sigma(k) = 2*k and k is an even perfect number. If k is odd, then it is an odd multiperfect number, and no odd multiperfect number above 1 is known.
More specifically, if a(k) = k and k is odd, then k must be a square, and an m-multiperfect number (number k such that sigma(k) = m * k), with m being an odd prime number that is the least prime factor of sigma(k). For example, if there is an odd triperfect number (A005820) then it is a fixed point of this sequence.
Periodic chains of cycle 2 are amicable pairs (A371419 and A371420). Are there any longer cycles?

			The sum of the divisors of 3 is 1 + 3 = 4. The divisors of 4 are 1, 2, 4. 2 is the largest aliquot divisor of 4. Therefore a(3) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Robert D. Carmichael, Empirical Results in the Theory of Numbers, The Mathematics Teacher, Vol. 14, No. 6 (1921), pp. 305-310; alternative link. See p. 309.
Eric Weisstein's World of Mathematics, Aliquot Sequence.
Eric Weisstein's World of Mathematics, Multiperfect Number.
Wikipedia, Aliquot sequence.

Cf. A000203, A000396, A005820, A023194, A028982, A028983, A032742, A071189, A371419, A371420, A371421, A371422, A371423.

r[n_] := n/FactorInteger[n][[1, 1]]; a[n_] := r[DivisorSigma[1, n]]; Array[a, 100]

a(n) = {my(s = sigma(n)); if(s == 1, 1, s/factor(s)[1, 1]);}

a(n) = A032742(A000203(n)).
a(n) = A000203(n)/A071189(n).
a(n) = A000203(n)/2 if n is in A028983 (i.e., n is not in A028982).
a(k) = 1 if and only if k = 1 or k is in A023194.

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

A336924 a(n) = spf(1+sigma(n)), where spf is the smallest prime factor and sigma is the sum of divisors function.

Original entry on oeis.org

2, 2, 5, 2, 7, 13, 3, 2, 2, 19, 13, 29, 3, 5, 5, 2, 19, 2, 3, 43, 3, 37, 5, 61, 2, 43, 41, 3, 31, 73, 3, 2, 7, 5, 7, 2, 3, 61, 3, 7, 43, 97, 3, 5, 79, 73, 7, 5, 2, 2, 73, 3, 5, 11, 73, 11, 3, 7, 61, 13, 3, 97, 3, 2, 5, 5, 3, 127, 97, 5, 73, 2, 3, 5, 5, 3, 97, 13, 3, 11, 2, 127, 5, 3, 109, 7, 11, 181, 7, 5, 113, 13

Offset: 1

Antti Karttunen, Aug 09 2020

nonn

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

Cf. A000203, A020639, A071189, A088580.

Cf. also A336691, A336692, A336693, A336694, A336695, A336696.

A336924(n) = (factor((1+sigma(n)))[1, 1]);

a(n) = A020639(1+A000203(n)) = A020639(A088580(n)).

A280348 Irregular triangle read by rows in which row n lists the divisors of the sum of the divisors of n.

Original entry on oeis.org

1, 1, 3, 1, 2, 4, 1, 7, 1, 2, 3, 6, 1, 2, 3, 4, 6, 12, 1, 2, 4, 8, 1, 3, 5, 15, 1, 13, 1, 2, 3, 6, 9, 18, 1, 2, 3, 4, 6, 12, 1, 2, 4, 7, 14, 28, 1, 2, 7, 14, 1, 2, 3, 4, 6, 8, 12, 24, 1, 2, 3, 4, 6, 8, 12, 24, 1, 31, 1, 2, 3, 6, 9, 18, 1, 3, 13, 39, 1, 2, 4, 5, 10, 20, 1, 2, 3, 6, 7, 14, 21, 42, 1, 2, 4, 8, 16, 32, 1, 2, 3, 4, 6, 9, 12, 18, 36

Offset: 1

Omar E. Pol, Jan 09 2017

			Triangle begins:
1;
1, 3;
1, 2, 4;
1, 7;
1, 2, 3, 6;
1, 2, 3, 4, 6, 12;
1, 2, 4, 8;
1, 3, 5, 15;
1, 13;
1, 2, 3, 6, 9, 18;
1, 2, 3, 4, 6, 12;
1, 2, 4, 7, 14, 28;
1, 2, 7, 14;
1, 2, 3, 4, 6, 8, 12, 24;
1, 2, 3, 4, 6, 8, 12, 24;
1, 31;
1, 2, 3, 6, 9, 18;
1, 3, 13, 39;
1, 2, 4, 5, 10, 20;
1, 2, 3, 6, 7, 14, 21, 42;
1, 2, 4, 8, 16, 32;
1, 2, 3, 4, 6, 9, 12, 18, 36;
1, 2, 3, 4, 6, 8, 12, 24;
1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60;
...
For n = 10 the sum of the divisors of 10 is 1 + 2 + 5 + 10 = 18, so row 10 of the triangle lists the divisors of 18: 1, 2, 3, 6, 9, 18.

Paolo Xausa, Table of n, a(n) for n = 1..11428 (rows 1..700 of triangle, flattened).

Right border gives A000203.
Row sums give A051027.
Row lengths give A062068.
Row products give A280581.
Column 1 is A000012.
Column 2 gives A071189, n >= 2.
Cf. A000005, A027750.

Divisors[DivisorSigma[1, Range[25]]] (* Paolo Xausa, Dec 11 2024 *)

T(n,k) = A027750(A000203(n),k).

A295076 Numbers n > 1 such that n and sigma(n) have the same smallest prime factor.

Original entry on oeis.org

6, 10, 12, 14, 20, 22, 24, 26, 28, 30, 34, 38, 40, 42, 44, 46, 48, 52, 54, 56, 58, 60, 62, 66, 68, 70, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 130, 132, 134, 136, 138, 140, 142, 146, 148

Offset: 1

Jaroslav Krizek, Nov 13 2017

nonn

Supersequence of A088829; this sequence contains also odd numbers: 441, 1521, 3249, 3969, 8649, 11025, ...
Even terms of A000396 (perfect numbers) are a subsequence.
Subsequence of A295078.
Numbers n such that A020639(n) = A020639(sigma(n)).
Numbers n such that A020639(n) = A071189(n).

			30 = 2*3*5 and sigma(30) = 72 = 2^3*3^2 hence 30 is in the sequence.

Cf. A000203, A020639, A071189, A088829, A295078.

Cf. A071834 (numbers n such that n and sigma(n) have the same largest prime factor).

[n: n in [2..1000000] | Minimum(PrimeDivisors(SumOfDivisors(n))) eq Minimum(PrimeDivisors(n))]

select(t -> min(numtheory:-factorset(t))=min(numtheory:-factorset(numtheory:-sigma(t))), [$2..1000]); # Robert Israel, Nov 14 2017

Rest@ Select[Range@ 150, SameQ @@ Map[FactorInteger[#][[1, 1]] &, {#, DivisorSigma[1, #]}] &] (* Michael De Vlieger, Nov 13 2017 *)

isok(n) = factor(n)[1,1] == factor(sigma(n))[1,1]; \\ Michel Marcus, Nov 14 2017

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

cp's OEIS Frontend

A071190 Greatest prime factor of sum of divisors of n, for n >= 2; a(1) = 1.

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A071187 Smallest prime factor of number of divisors of n; a(1) = 1.

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A371418 The largest aliquot divisor of the sum of divisors of n; a(1) = 1.

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

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A336924 a(n) = spf(1+sigma(n)), where spf is the smallest prime factor and sigma is the sum of divisors function.

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A280348 Irregular triangle read by rows in which row n lists the divisors of the sum of the divisors of n.

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A295076 Numbers n > 1 such that n and sigma(n) have the same smallest prime factor.

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