A331751 -id:A331751 - cp's OEIS frontend

A097248 a(n) is the eventual stable point reached when iterating k -> A097246(k), starting from k = n.

1, 2, 3, 3, 5, 6, 7, 6, 5, 10, 11, 5, 13, 14, 15, 5, 17, 10, 19, 15, 21, 22, 23, 10, 7, 26, 15, 21, 29, 30, 31, 10, 33, 34, 35, 15, 37, 38, 39, 30, 41, 42, 43, 33, 7, 46, 47, 15, 11, 14, 51, 39, 53, 30, 55, 42, 57, 58, 59, 7, 61, 62, 35, 15, 65, 66, 67, 51, 69, 70, 71, 30, 73, 74, 21

Offset: 1

Reinhard Zumkeller, Aug 03 2004

nonn

a(n) = r(n,m) with m such that r(n,m)=r(n,m+1), where r(n,k) = A097246(r(n,k-1)), r(n,0)=n. (The original definition.)
A097248(n) = r(n,a(n)).
From Antti Karttunen, Nov 15 2016: (Start)
The above remark could be interpreted to mean that A097249(n) <= a(n).
All terms are squarefree, and the squarefree numbers are the fixed points.
These are also fixed points eventually reached when iterating A277886.
(End)

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

Range of values is A005117.
Cf. A008683, A019565, A048675, A097246, A097247, A097249, A277905, A332824.
A003961, A225546, A277885, A277886, A331590 are used to express relationship between terms of this sequence.
The formula section also details how the sequence maps the terms of A007913, A260443, A329050, A329332.
See comments/formulas in A283475, A283478, A331751 giving their relationship to this sequence.

Table[FixedPoint[Times @@ Map[#1^#2 & @@ # &, Partition[#, 2, 2] &@ Flatten[FactorInteger[#] /. {p_, e_} /; e >= 2 :> {If[OddQ@ e, {p, 1}, {1, 1}], {NextPrime@ p, Floor[e/2]}}]] &, n], {n, 75}] (* Michael De Vlieger, Mar 18 2017 *)

A097246(n) = { my(f=factor(n)); prod(i=1, #f~, (nextprime(f[i,1]+1)^(f[i,2]\2))*((f[i,1])^(f[i,2]%2))); };
A097248(n) = { my(k=A097246(n)); while(k<>n, n = k; k = A097246(k)); k; };
\\ Antti Karttunen, Mar 18 2017

from sympy import factorint, nextprime
from operator import mul
def a097246(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [(nextprime(i)**int(f[i]/2))*(i**(f[i]%2)) for i in f])
def a(n):
    k=a097246(n)
    while k!=n:
        n=k
        k=a097246(k)
    return k # Indranil Ghosh, May 15 2017

;; with memoization-macro definec
;; Two implementations:
(definec (A097248 n) (if (not (zero? (A008683 n))) n (A097248 (A097246 n))))
(definec (A097248 n) (if (zero? (A277885 n)) n (A097248 (A277886 n))))
;; Antti Karttunen, Nov 15 2016

a(A005117(n)) = A005117(n).
From Antti Karttunen, Nov 15 2016: (Start)
If A008683(n) <> 0 [when n is squarefree], a(n) = n, otherwise a(n) = a(A097246(n)).
If A277885(n) = 0, a(n) = n, otherwise a(n) = a(A277886(n)).
A007913(a(n)) = a(n).
a(A007913(n)) = A007913(n).
A048675(a(n)) = A048675(n).
a(A260443(n)) = A019565(n).
(End)
From Peter Munn, Feb 06 2020: (Start)
a(1) = 1; a(p) = p, for prime p; a(m*k) = A331590(a(m), a(k)).
a(A331590(m,k)) = A331590(a(m), a(k)).
a(n^2) = a(A003961(n)) = A003961(a(n)).
a(A225546(n)) = a(n).
a(n) = A225546(2^A048675(n)) = A019565(A048675(n)).
a(A329050(n,k)) = prime(n+k-1) = A000040(n+k-1).
a(A329332(n,k)) = A019565(n * k).
Equivalently, a(A019565(n)^k) = A019565(n * k).
(End)
From Antti Karttunen, Feb 22-25 & Mar 01 2020: (Start)
a(A019565(x)*A019565(y)) = A019565(x+y).
a(A332461(n)) = A332462(n).
a(A332824(n)) = A019565(n).
a(A277905(n,k)) = A277905(n,1) = A019565(n), for all n >= 1, and 1 <= k <= A018819(n).
(End)

Name changed and the original definition moved to the Comments section by Antti Karttunen, Nov 15 2016

A331750 a(n) = A048675(sigma(n)).

Original entry on oeis.org

0, 2, 2, 8, 3, 4, 3, 6, 32, 5, 4, 10, 9, 5, 5, 1024, 5, 34, 6, 11, 5, 6, 5, 8, 1024, 11, 7, 11, 7, 7, 5, 12, 6, 7, 6, 40, 129, 8, 11, 9, 11, 7, 18, 12, 35, 7, 6, 1026, 130, 1026, 7, 17, 7, 9, 7, 9, 8, 9, 8, 13, 1025, 7, 35, 1073741824, 12, 8, 66, 13, 7, 8, 7, 38, 2049, 131, 1026, 14, 7, 13, 8, 1027, 32, 13, 12, 13, 8, 20, 9

Offset: 1

Antti Karttunen, Feb 05 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n)

Cf. A000203, A048675, A324573, A331751.

{0}~Join~Array[Total@ Map[#[[-1]]*2^(PrimePi[#[[1]]] - 1) &, FactorInteger@ DivisorSigma[1, #]] &, 86, 2] (* Michael De Vlieger, Feb 08 2020, after Jean-François Alcover at A048675 *)

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A331750(n) = A048675(sigma(n));

a(n) = A048675(A000203(n)).

Additive with a(p^e) = A048675(sigma(p^e)) = A048675(1 + p + p^2 + ... + p^e).

A331752 Numbers k such that squarefree part of sigma(k) is equal to squarefree part of 2*k.

Original entry on oeis.org

6, 28, 468, 496, 775, 2268, 3780, 4655, 7448, 8128, 9000, 10880, 10976, 25137, 40131, 40176, 58752, 62775, 66960, 91000, 137541, 137940, 140800, 160930, 167400, 173600, 195938, 224450, 307125, 377055, 399360, 406224, 417477, 494832, 569184, 603288, 634725, 639158, 658368, 773175, 869022, 881280, 889056, 1005480

Offset: 1

Antti Karttunen, Feb 06 2020

nonn

Numbers k such that A007913(sigma(k)) is equal to A007913(2*k), thus numbers for which sigma(k) has the same set of distinct prime factors with an odd exponent as 2*k.
Among the first 257 terms, these four are also in A228058:
  46277101  = 61 * 13^2 * 67^2,
  49889853  = 13 * 3^2 * 653^2,
  106706925 = 13 * 3^2 * 5^2 * 191^2,
  676830973 = 37 * 7^2 * 13^2 * 47^2.

			For n = 46277101 = 61 * 13^2 * 67^2, sigma(46277101) = 51703722 = 2 * 3^2 * 7^2 * 31^2 * 61, with A007913(sigma(46277101)) = 2*61 = A007913(2*46277101), thus 46277101 is included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..257
Index entries for sequences where any odd perfect numbers must occur

Cf. A000203, A006532, A007913, A228058, A331751, A332208.

Cf. A000396 (a subsequence).

Select[Range[10^6], SameQ @@ Map[Sqrt[#] /. (c_: 1)*a_^(b_: 0) :> (c*a^b)^2 &, {DivisorSigma[1, #], 2 #}] &] (* Michael De Vlieger, Feb 08 2020, after Bill Gosper at A007913 *)

isA331752(n) = (core(2*n)==core(sigma(n)));

A332208 Numbers k such that the squarefree kernel of sigma(k) is equal to the squarefree kernel of 2*k.

Original entry on oeis.org

6, 28, 120, 135, 270, 496, 672, 891, 1080, 1638, 1782, 3780, 8128, 18600, 20580, 24948, 26208, 30240, 32640, 32760, 35640, 41850, 44226, 55860, 66960, 164640, 167400, 185220, 199584, 200655, 273000, 293760, 307125, 401310, 441936, 446880, 502740, 523776, 544635, 614250, 707616, 802620, 819000, 884520

Offset: 1

Antti Karttunen, Feb 07 2020

nonn

Numbers k such that sigma(k) has the same set of distinct prime factors as 2*k.
Numbers k such that A007947(sigma(k)) is equal to A007947(2*k), or equally, that A087207(sigma(k)) is equal to A087207(2*k).
Of the first 256 terms 44 are odd, and none occurs in A228058. Compare also to A331752.

Antti Karttunen, Table of n, a(n) for n = 1..256
Index entries for sequences where any odd perfect numbers must occur

Cf. A000203, A007947, A080398, A087207, A228058, A331751, A331752.

Subsequences: A000396, A027687.

Select[Range[10^6], SameQ @@ Map[Times @@ FactorInteger[#][[All, 1]] &, {DivisorSigma[1, #], 2 #}] &] (* Michael De Vlieger, Feb 08 2020 *)

A007947(n) = factorback(factorint(n)[, 1]);
isA332208(n) = (A007947(sigma(n)) == A007947(2*n));

{n: A080398(n) == A007947(2n)}.

A332446 Numbers k for which A087808(sigma(k)) is equal to A087808(2*k).

Original entry on oeis.org

3, 6, 11, 19, 28, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 216, 227, 251, 267, 283, 286, 307, 331, 347, 379, 419, 443, 467, 491, 496, 499, 523, 547, 563, 571, 587, 598, 619, 643, 659, 683, 691, 726, 739, 787, 811, 827, 859, 883, 907, 947, 971, 1019, 1051, 1091, 1123, 1163, 1171, 1187, 1259, 1283, 1291, 1307, 1427, 1451

Offset: 1

Antti Karttunen, Feb 14 2020

nonn

Conjecture: includes all terms of A007520. - Bill McEachen, Dec 10 2023

Cf. A000203, A007520, A087808.
Subsequences: A000396, A332445.
Cf. A331751, A331752, A332208 for similar sequences.

A087808(n) = if(n<1, 0, if(n%2==0, 2*A087808(n/2), A087808((n-1)/2)+1));
isA332446(n) = (A087808(2*n)==A087808(sigma(n)));

A335913 a(n) = A225546(sigma(n)).

Original entry on oeis.org

1, 4, 3, 256, 8, 12, 6, 64, 4294967296, 18, 12, 768, 512, 24, 24

Offset: 1

Antti Karttunen, Jul 07 2020

nonn

Term a(16) has 309 digits (1025 bits).

This is not multiplicative. Even though a(2*3) = a(2)*a(3), a(2*7) = a(2)*a(7), and a(3*5) = a(3)*a(5), we still have a(2*5) = 18 <> a(2)*a(5) = 4*8 = 32.

Index entries for sequences related to sigma(n)

Cf. A000203, A048675, A225546, A331733, A331750, A331751.

Cf. also A326042.

Array[If[# == 1, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]] &@ DivisorSigma[1, #] &, 15] (* Michael De Vlieger, Jul 08 2020 *)

PARI
```
A335913(n) = A225546(sigma(n));
```

a(n) = A225546(A000203(n)).

A048675(a(n)) = A331750(n).

A347875 Numbers k such that A323905(sigma(k)) is equal to A323905(2*k).

Original entry on oeis.org

1, 6, 21, 28, 496, 8128

Offset: 1

Antti Karttunen, Sep 19 2021

Numbers k such that A323905(sigma(k)) = A332221(k) - A331750(k) is equal to 2*A156552(k) - A048675(k) = A156552(k) + A323905(k).

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Cf. A000203, A048675, A156552, A323905, A347876.

Cf. also A000396 (subsequence), A331751, A347392.

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; }; \\ From A048675
A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A323905(n) = (A156552(n) - A048675(n));
isA347875(n) = (A323905(sigma(n))==A323905(2*n));

cp's OEIS Frontend

A097248 a(n) is the eventual stable point reached when iterating k -> A097246(k), starting from k = n.

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A331750 a(n) = A048675(sigma(n)).

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A331752 Numbers k such that squarefree part of sigma(k) is equal to squarefree part of 2*k.

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A332208 Numbers k such that the squarefree kernel of sigma(k) is equal to the squarefree kernel of 2*k.

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A332446 Numbers k for which A087808(sigma(k)) is equal to A087808(2*k).

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A335913 a(n) = A225546(sigma(n)).

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A347875 Numbers k such that A323905(sigma(k)) is equal to A323905(2*k).

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