A159886 -id:A159886 - cp's OEIS frontend

A254035 Sequence A255412 sorted into ascending order, with duplicates removed.

4800, 28800, 57600, 67200, 86400, 96000, 115200, 142800, 144000, 148800, 153600, 182400, 201600, 211200, 230400, 259200, 288000, 297600, 326400, 345600, 355200, 384000, 403200, 432000, 470400, 489600, 499200, 518400, 528000, 547200, 576000, 614400, 633600, 638400, 662400, 672000, 691200, 720000, 729600

Offset: 1

Naohiro Nomoto, Jan 23 2015

nonn

Numbers n such that n = A000203(j) = A000203(k) and A007947(j) = A007947(k), where j != k.
In other words, numbers n such that sigma(x) = n has at least two distinct solutions, with each x having the same squarefree kernel, where sigma(x) is the sum of divisor function (A000203).
Equally, sequence A000203(A255335(n)) sorted into ascending order, with duplicates removed.

			4800 is the sum of divisors of 1512 and 2058, and rad(1512) = rad(2058) = 42, hence 4800 is in the sequence with j=1512 and k=2058.

Subsequence of A159886.
Cf. A000203 (sum of divisors of n), A007947 (squarefree kernel of n).
Cf. A254791 (a subsequence).
Cf. A252997, A255334, A255335, A255412.

a(n) = A000203(A255334(n)) = A000203(A255335(n)) for n = 1 .. 7. - Antti Karttunen, Apr 05 2015

More terms from Antti Karttunen, Apr 13 2015

A300869 Odd numbers m such that sigma(x) = m has more than 1 solution.

Original entry on oeis.org

31, 399, 403, 1767, 3751, 4123, 5187, 5673, 9517, 11811, 12369, 17143, 22971, 27001, 30783, 33883, 34671, 43617, 48279, 53413, 53599, 54873, 58683, 68859, 69967, 73017, 73749, 80199, 86831, 88753, 109771, 117273, 122493, 123721, 141267, 152019, 153543, 158503, 160797

Offset: 1

M. F. Hasler, following a suggestion from Altug Alkan, Mar 16 2018

nonn

Goormaghtigh conjecture implies that 31 is the only prime in this sequence. - Jianing Song, Apr 27 2019

			a(1) = 31 = A123523(2), the smallest odd number m for which sigma(x) = m has (at least, and also exactly) two solutions, x = 16 and x = 25.
a(56) = 347529 = A123523(3) is the smallest odd m for which sigma(x) = m has (at least, and also exactly) three solutions, x = 406^2, x = 2*319^2 and x = 489^2.

Robert Israel, Table of n, a(n) for n = 1..9260
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Wikipedia, Goormaghtigh conjecture.

Odd terms in A159886.
Cf. A000203 (sigma), A002191, A007368.
A123523 is a subsequence, except for the initial 1.
Cf. A331036.

N:= 200000: # for terms <= N
Res:= NULL: count:= 0:
for m from 1 to floor(sqrt(N)) by 2 do
sm:= numtheory:-sigma(m^2);
for k from 1 to floor(log[2](N/sm+1)) do
   v:= sm*(2^k-1);
   if v <= N then Res:= Res, v; count:= count+1 fi;
od
od:
B:= sort([Res]):
Dups:= select(t -> B[t+1]=B[t], [$1..nops(B)-1]):
sort(convert(convert(B[Dups],set),list)); # Robert Israel, Jan 15 2020

With[{s = PositionIndex@ Array[DivisorSigma[1, #] &, 10^6]}, Keys@ KeySort@ KeySelect[s, And[OddQ@ #, Length@ Lookup[s, #] > 1] &]] (* Michael De Vlieger, Mar 16 2018 *)

MAX=1e6; LIM=1e4; b=0; A300869=[]; for(x=1,LIM, for(i=1,2, (s=sigma(i*x^2))>MAX && next(2); bittest(b,s\2) && (setsearch(A300869,s) || S=setunion(A300869,[s])) || b+=1<<(s\2)))

is(k) = k%2 && invsigmaNum(k) > 1; \\ Amiram Eldar, Dec 16 2024, using Max Alekseyev's invphi.gp

A258931 Numbers k such that card({x|sigma(x)=k}) > 1 and (Sum_{sigma(x)=k} x) < k.

Original entry on oeis.org

124, 378, 403, 1904, 3751, 4064, 5187, 5456, 6188, 9296, 9800, 11532, 12369, 13664, 14378, 15210, 16256, 16352, 17654, 18018, 18536, 19110, 19304, 19376, 20336, 21450, 22971, 23240, 23478, 24056, 24584, 24986, 25298, 26754, 28616, 28938, 31640, 33883, 34398

Offset: 1

Michel Marcus, Jun 15 2015

nonn

By definition these terms do not belong to A007370 nor to A007369.
All terms so far appear to be in A007371, with 2 pre-images. Are there any terms with more?
Yes, I find six up to 10^8 with 3 pre-images: 10714158, 12093224, 17315298, 30507906, 54891018, 81629262. - Charles R Greathouse IV, Jun 15 2015

			For k=124, the x's such that sigma(x)=124 are 48 and 75, and 48 + 75 = 123 < 124.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A159886.
Cf. A000203 (sigma, the sum of divisors), A085790.
Cf. A007369 (sigma(x)=n has no solution), A007370 (exactly 1 solution),
Cf. A007371 (exactly 2 solutions), A007372 (exactly has 3 solutions).
Cf. A258913 (Sum_{sigma(x)=n} x).

isok(n) = my(v = select(x->sigma(x)==n, vector(n, i, i))); (#v > 1) && (vecsum(v) < n);

list(lim)=my(v=vector(lim\1), u=List(), s); for(k=1,#v,s=sigma(k); if(s>#v,next); v[s]=if(v[s]==0, -k, abs(v[s])+k)); for(i=1,#v, if(v[i]>0 && v[i]Charles R Greathouse IV, Jun 15 2015

A159953 Values in A054973 larger than 1.

Original entry on oeis.org

2, 2, 3, 2, 2, 3, 3, 2, 2, 3, 5, 2, 3, 3, 4, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 5, 2, 2, 6, 4, 2, 2, 5, 2, 5, 3, 3, 3, 7, 3, 6, 2, 3, 2, 2, 6, 3, 2, 4, 2, 3, 8, 2, 9, 4, 2, 6, 2, 2, 2, 2, 2, 2, 4, 8, 4, 2, 2, 2, 3, 4, 3, 9, 2, 10, 2, 3, 2, 4, 4, 3, 4, 2, 2, 11, 5, 2, 5, 2, 3, 4, 2, 2, 3, 5, 3, 8, 7, 4, 15, 2, 4, 7, 8

Offset: 1

Jaroslav Krizek, Apr 27 2009

nonn

This is a survey of how many solutions the equation sigma(x)=k has for k in A159886, or about the lengths of the plateaus in A007609.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1095 from Jean-François Alcover)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203, A007609, A054973, A159886.

read("transforms3") ; a054973 := BFILETOLIST("b054973.txt") ;
for i from 1 to 1000 do if op(i,a054973) > 1 then printf("%d,", op(i,a054973)) ; fi; od: # R. J. Mathar, May 22 2009

b[n_] := Sum[Boole[DivisorSigma[1, k] == n], {k, 1, n}];
Select[Array[b, 1000], # > 1&] (* Jean-François Alcover, Apr 06 2020 *)

list(lim) = {my(s); for(k = 1, lim, s = invsigmaNum(k); if(s > 1, print1(s, ", ")));} \\ Amiram Eldar, Dec 25 2024, using Max Alekseyev's invphi.gp

Edited and extended by R. J. Mathar, May 22 2009

A206421 Corresponding values of sigma(m) of numbers in A206036.

Original entry on oeis.org

12, 18, 12, 24, 24, 31, 18, 42, 32, 24, 60, 31, 42, 56, 72, 32, 48, 54, 48, 60, 56, 90, 42, 96, 84, 72, 48, 124, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 96, 104, 84, 144, 126, 96, 144, 72, 114, 124, 140, 96, 168, 80, 186, 126, 84, 224, 108, 132, 120, 180

Offset: 1

Jaroslav Krizek, Feb 07 2012

nonn

			a(1) = 12 because sigma(A206036(1)) = sigma(6) = 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203, A159886 (values k such that sigma(x)= k has more than one solution), A206036 (numbers m such that sigma(m) = sigma(k) has solution for distinct k).

list(lim) = my(s); for(k = 1, lim, s = sigma(k); if(invsigmaNum(s) > 1, print1(s, ", "))); \\ Amiram Eldar, Dec 15 2024, using Max Alekseyev's invphi.gp

a(n) = A000203(A206036(n)). - Amiram Eldar, Dec 15 2024

A258912 Numbers k such that A000203(x) = k has more than one solution and they all share the same largest prime factor.

Original entry on oeis.org

1178, 1364, 1408, 1656, 1767, 1836, 1922, 1984, 2108, 2196, 2328, 2368, 3162, 3336, 3410, 3996, 4096, 4123, 4144, 4278, 4898, 5064, 5076, 5084, 5248, 5456, 5488, 5673, 6014, 6208, 6504, 6784, 6816, 7416, 7998, 8618, 8896, 9088, 9184, 9517, 10048, 10292, 10864

Offset: 1

Michel Marcus, Jun 14 2015

nonn

By definition this is a subsequence of A159886.

Pollack shows that the density of such integers relative to A002191 is 1.

			The pre-image of 1178 is [592, 925], and both have greatest prime factor 37, so 1178 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Pollack, Remarks on fibers of the sum-of-divisors function, Analytic number theory (volume in honor of H. Maier), M. Rassias and C. Pomerance, eds., Springer, 2015, alternative link.

Cf. A000203 (sum of divisors), A002191 (possible values of sum of divisors), A159886 (sigma(x)=n has more than one solution).

isok(n) = {my(v = select(x->sigma(x)==n, vector(n, i, i))); if (#v < 2, return (0)); vgpf = vector(#v, k, fvk = factor(v[k]); fvk[#fvk~,1]); vecmin(vgpf) == vecmax(vgpf);}

A229953 Numbers k for which k = sigma(sigma(x)) = sigma(sigma(y)) for some x and y such that k = x + y.

Original entry on oeis.org

4, 8, 32, 60, 128, 8192, 43200, 69360, 120960, 131072, 524288, 4146912, 6549984, 12927600, 13335840, 16329600, 34715520, 51603840, 57879360, 59633280, 107775360, 160797000, 169155840, 252067200, 371226240, 391789440, 436230144, 439883136, 489888000, 657296640

Offset: 1

Paolo P. Lava, Oct 04 2013

nonn

A072868 is a subsequence of this sequence. Any term x of A072868 can be expressed as x = 2*sigma(sigma(x/2)).
Note the analogy with amicable pair sums (A180164) which satisfy a similar condition: k = sigma(x) = sigma(y) where k = x + y. - Michel Marcus, Oct 07 2013
When terms do not belong to A072868, then they belong to A159886, and the (x,y) couples are (23,37), (14999,28201), (34673,34687), (55373,65587), (2056961,2089951), (2458187,4091797), (4586987,8340613), (5174363,8161477), (6204767,10124833), (15788453,18927067), (25748273,25855567), (20699927,37179433), (22239647,37393633), ... - Michel Marcus, Oct 08 2013

			4 = 2 + 2 = 2*sigma(sigma(2)).
8 = 4 + 4 = 2*sigma(sigma(4)).
32 = 16 + 16 = 2*sigma(sigma(16)).
60 = 23 + 37 = sigma(sigma(23)) = sigma(sigma(37)).
128 = 64 + 64 = 2*sigma(sigma(64)).
8192 = 4096 + 4096 = 2*sigma(sigma(4096)).

Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experiment. Math. (1996) vol. 5, no. 2, pp. 91-100 (see merge at n=60 in tree of section 4 page 97).

Cf. A000203, A072868.

with(numtheory); P:=proc(q) local j,n;
for n from 1 to q do for j from 1 to trunc(n/2) do
if sigma(sigma(j))=sigma(sigma(n-j)) and sigma(sigma(j))=n then print(n);
fi; od; od; end: P(10^6);

a(7)-a(20) from Giovanni Resta, Oct 06 2013

a(21)-a(30) from Donovan Johnson, Oct 08 2013

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A254035 Sequence A255412 sorted into ascending order, with duplicates removed.

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A300869 Odd numbers m such that sigma(x) = m has more than 1 solution.

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A258931 Numbers k such that card({x|sigma(x)=k}) > 1 and (Sum_{sigma(x)=k} x) < k.

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A159953 Values in A054973 larger than 1.

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A206421 Corresponding values of sigma(m) of numbers in A206036.

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A258912 Numbers k such that A000203(x) = k has more than one solution and they all share the same largest prime factor.

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A229953 Numbers k for which k = sigma(sigma(x)) = sigma(sigma(y)) for some x and y such that k = x + y.

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