A007369 -id:A007369 - cp's OEIS frontend

A083532 First difference sequence of A007369. Differences between impossible values for sum of divisors of n.

3, 4, 1, 1, 5, 1, 2, 2, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 4, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 3, 1, 2, 4, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 1, 2, 1

Offset: 1

Labos Elemer, May 20 2003

nonn

			29 and 33 are the 15th and 16th nonsense values for sigma(x), since there exist no numbers n of which they are sums of divisors, while {30,31,32} equal sigma(x); e.g., for x = 29, 16, 31, respectively, thus 33 - 29 = 4 = a(15) = A007369(16) - A007369(15).

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

Cf. A002191, A007369, A083235, A083531, A083533, A083534, A083536.

t0[x_] := Table[j, {j, 1, x}]; t=Table[DivisorSigma[1, w], {w, 1, 25000}]; u=Union[%]; c=Complement[t0[25000], u]; Delete[c-RotateRight[c], 1]

a(n) = A007369(n+1) - A007369(n).

A094505 Powers of 2 which are not the sum of divisors of any other number. Powers of 2 present in A007369.

Original entry on oeis.org

2, 16, 64, 2048

Offset: 1

Labos Elemer, Jun 02 2004

nonn

The next term, 2^470, is too large to include.

Cf. A007369, A000203, A046528, A094502.

a(n) = 2^A078426(n).

A070240 Duplicate of A007369.

Original entry on oeis.org

1, 2, 5, 9, 10, 11, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 33, 34, 35, 37, 41, 43, 45, 46, 47

Offset: 1

dead

A002191 Possible values for sum of divisors of n.

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 12, 13, 14, 15, 18, 20, 24, 28, 30, 31, 32, 36, 38, 39, 40, 42, 44, 48, 54, 56, 57, 60, 62, 63, 68, 72, 74, 78, 80, 84, 90, 91, 93, 96, 98, 102, 104, 108, 110, 112, 114, 120, 121, 124, 126, 127, 128, 132, 133, 138, 140, 144, 150, 152, 156

Offset: 1

N. J. A. Sloane

nonn

Distinct values attained by the sigma(n) function, in ascending order.

The asymptotic density of this sequence is 0 (Niven, 1951, Rao and Murty, 1979). - Amiram Eldar, Jul 23 2020

			a(100) = 272, a(10^3) = 3696, a(10^4) = 44496, a(10^5) = 510356, a(10^6) = 5691216. - _M. F. Hasler_, Nov 22 2019

J. W. L. Glaisher, Number-Divisor Tables. British Assoc. Math. Tables, Vol. 8, Camb. Univ. Press, 1940, p. 85.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 840.
Ivan Niven, The asymptotic density of sequences, Bull. Amer. Math. Soc., Vol. 57 (1951), pp. 420-434.
R. Sita Rama Chandra Rao and G. Sri Rama Chandra Murty, On a theorem of Niven, Canadian Mathematical Bulletin, Vol 22, No. 1 (1979), pp. 113-115.

Cf. A000203, A007609.
Complement of A007369. A175192(a(n)) = 1, A054973(a(n)) >= 1. - Jaroslav Krizek, Mar 01 2010
See A083531 for the gaps, i.e., first differences. - M. F. Hasler, Mar 12 2018
Subsequence of A211347.

N:= 1000: # to get all entries <= N
select(`<=`,{seq(numtheory[sigma](i),i=1..N)},N); # Robert Israel, Jun 16 2014

lim=1000; Select[Union[DivisorSigma[1,Range[lim]]], #<=lim &] (* T. D. Noe, May 06 2010 *)

list(lim)=select(n->n<=lim,Set(vector(lim\=1,n,sigma(n)))) \\ Charles R Greathouse IV, Nov 12 2013

A002191_upto(N,M=N\1+1)=Set(apply(t->min(sigma(t),M), [1..N\1-1]))[^-1] \\ Needs big stack for N >= 10^6; slower alternative: {A002191_upto(N)= my(L=List(1),s); for(n=2,N\=1,N<(s=sigma(n))||listput(L,s));Set(L)}
A2191=A002191_upto(1e4); A002191(n)={#A2191A002191_upto(n*logint(n,10)+n); A2191[n]} \\ - M. F. Hasler, Nov 22 2019

a(n)/n < log_10(n) + O(1) with O(1) <= 1 for all n. - M. F. Hasler, Nov 22 2019

A007368 Smallest k such that sigma(x) = k has exactly n solutions.

Original entry on oeis.org

2, 1, 12, 24, 96, 72, 168, 240, 336, 360, 504, 576, 1512, 1080, 1008, 720, 2304, 3600, 5376, 2520, 2160, 1440, 10416, 13392, 3360, 4032, 3024, 7056, 6720, 2880, 6480, 10800, 13104, 5040, 6048, 4320, 13440, 5760, 18720, 20736, 19152, 22680, 43680

Offset: 0

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

It's not obvious that a(n) exists for all n; I'd like to see a proof. - David Wasserman, Jun 07 2002
Note that k-1 is frequently prime. See A115374 for the least prime. For each n, it appears that there are an infinite number of k such that sigma(x)=k has exactly n solutions. - T. D. Noe, Jan 21 2006
According to Sierpiński, H. J. Kanold proved that there is a k such that sigma(x)=k has n or more solutions. Sierpiński states that Erdős proved that if, for some k, sigma(x)=k has exactly n solutions, then there are an infinite number of such k. - T. D. Noe, Oct 18 2006
Index of the first occurrence of n in A054973. - Jaroslav Krizek, Apr 25 2009

			a(10) = 504; {204, 220, 224, 246, 284, 286, 334, 415, 451, 503} is the set of x such that sigma(x) = 504.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe and Donovan Johnson, Table of n, a(n) for n = 0..5000 (terms up to a(429) from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Kevin Ford, Sergei Konyagin, On two conjectures of Sierpiński concerning the arithmetic functions σ and ϕ, arXiv:1910.08452 [math.NT], 2019.
W. Sierpiński, Elementary Theory of Numbers, Warszawa 1964, page 166.
R. G. Wilson, V, Letter to N. J. A. Sloane, Jul. 1992

Cf. A000203, A054973, A002191, A007609.
Cf. A115374 (least prime p such that sigma(x)=sigma(p) has exactly n solutions).
Cf. A007369, A007370, A007371, A007372 (n such that sigma(x)=k has 0, 1, 2 and 3 solutions).
Cf. A184393, A184394, A201915 (smallest solution, largest solution, triangle of solutions for sigma(x)=a(n)).

Needs["Statistics`DataManipulation`"]; s=DivisorSigma[1, Range[10^5]]; f=Frequencies[s]; fs=Sort[f]; tfs=Transpose[fs][[1]]; utfs=Union[tfs]; firstMissing=First[Complement[Range[Last[utfs]], utfs]]; pos=1; Table[While[tfs[[pos]]T. D. Noe *)
terms = 100; cnt = DivisorSigma[1, Range[terms^3]] // Tally // Sort; a[0] = 2; a[n_] := SelectFirst[cnt, #[[2]] == n&][[1]]; Table[a[n], {n, 0, terms - 1}] (* Jean-François Alcover, Jul 18 2017 *)

More terms from David W. Wilson

A007370 Numbers k such that sigma(x) = k has a unique solution.

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 13, 14, 15, 20, 28, 30, 36, 38, 39, 40, 44, 57, 62, 63, 68, 74, 78, 91, 93, 102, 110, 112, 121, 127, 133, 138, 150, 158, 160, 162, 164, 171, 174, 176, 183, 194, 195, 198, 200, 204, 212, 217, 222, 230, 242, 255, 256, 258, 260, 266, 278, 282

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
Wacław Sierpiński, Elementary Theory of Numbers, Państ. Wydaw. Nauk., Warsaw, 1964, p. 165.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Wacław Sierpiński, Elementary Theory of Numbers, Warszawa, 1964.
Robert G. Wilson v, Letter to N. J. A. Sloane, Jul. 1992.

Cf. A000203.

Number of solutions: A007369 (0), this sequence (1), A007371 (2), A007372 (3), A060660 (4), A060661 (5), A060662 (6), A060663 (7), A060664 (8), A060665 (9), A060666 (10), A060678 (11), A060676 (12).

a = Table[ 0, {250} ]; Do[ s = DivisorSigma[ 1, n ]; If[ s < 251, a[ [ s ] ]++ ], {n, 1, 250} ]; Select[ Range[ 250 ], a[ [ # ] ] == 1 & ]

list(lim)=my(v=vectorsmall(lim\1), u=List(), s); for(k=1,#v,s=sigma(k); if(s<=#v, v[s]++)); for(k=1,#v,if(v[k]==1, listput(u,k))); Vec(u) \\ Charles R Greathouse IV, Jun 15 2015

is(k) = invsigmaNum(k) == 1 \\ Amiram Eldar, Nov 18 2024, using Max Alekseyev's invphi.gp

A051444 Smallest k such that sigma(k) = n, or 0 if there is no such k, where sigma = A000203 = sum of divisors.

Original entry on oeis.org

1, 0, 2, 3, 0, 5, 4, 7, 0, 0, 0, 6, 9, 13, 8, 0, 0, 10, 0, 19, 0, 0, 0, 14, 0, 0, 0, 12, 0, 29, 16, 21, 0, 0, 0, 22, 0, 37, 18, 27, 0, 20, 0, 43, 0, 0, 0, 33, 0, 0, 0, 0, 0, 34, 0, 28, 49, 0, 0, 24, 0, 61, 32, 0, 0, 0, 0, 67, 0, 0, 0, 30, 0, 73, 0, 0, 0, 45, 0, 57, 0, 0, 0, 44, 0, 0, 0, 0, 0

Offset: 1

Jud McCranie

Column 1 of A299762. - Omar E. Pol, Mar 14 2018

This is a right inverse of sigma = A000203 on A002191 = range(sigma): if n is in A002191, then there is some x with sigma(x) = n, and by definition a(n) is the smallest such x, so sigma(a(n)) = n. - M. F. Hasler, Nov 22 2019

			sigma(1) = 1, so a(1) = 1.
There is no k with sigma(k) = 2, since sigma(k) >= k + 1 for all k > 1 and sigma(1) = 1, so a(2) = 0.
sigma(4) = 7, and 4 is the smallest (since only) such number, so a(7) = 4.
6 and 12 are the only k with sigma(k) = 12, so 6 is the smallest and a(12) = 6.

R. K. Guy, Unsolved Problems Theory of Numbers, B1.

T. D. Noe, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: invphi.gp, Oct. 2005

Cf. A000203, A002192, A007626, A007369 (positions of zeros), A299762.

Do[ k = 1; While[ DivisorSigma[ 1, k ] != n && k < 10^4, k++ ]; If[ k != 10^4, Print[ k ], Print[ 0 ] ], {n, 1, 100} ]

a(n)=for(k=1,n,if(sigma(k)==n,return(k))); 0 \\ Charles R Greathouse IV, Mar 09 2014

A051444(n)=if(n=invsigma(n),vecmin(n)) \\ See Alekseyev link for invsigma(). An update including invsigmaMin = A051444 is planned. - M. F. Hasler, Nov 21 2019

Edited by M. F. Hasler, Nov 22 2019

A007371 Numbers k such that sigma(x) = k has exactly 2 solutions.

Original entry on oeis.org

12, 18, 31, 32, 54, 56, 80, 98, 104, 108, 114, 124, 126, 128, 132, 140, 152, 156, 182, 186, 210, 264, 272, 280, 308, 320, 342, 378, 390, 392, 399, 403, 408, 416, 440, 444, 448, 492, 522, 532, 570, 572, 594, 608, 630, 632, 726, 762, 770, 774, 780, 784, 800

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Donovan Johnson)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Robert G. Wilson v, Letter to N. J. A. Sloane, Jul. 1992.

Cf. A000203.

Number of solutions: A007369 (0), A007370 (1), this sequence (2), A007372 (3), A060660 (4), A060661 (5), A060662 (6), A060663 (7), A060664 (8), A060665 (9), A060666 (10), A060678 (11), A060676 (12).

a = Table[ 0, {750} ]; Do[ s = DivisorSigma[ 1, n ]; If[ s < 751, a[ [ s ] ]++ ], {n, 1, 750} ]; Select[ Range[ 750 ], a[ [ # ] ] == 2 & ]

is(n)=sum(k=1,n,sigma(k)==n)==2 \\ Charles R Greathouse IV, Mar 09 2014

is(k) = invsigmaNum(k) == 2 \\ Amiram Eldar, Nov 17 2024, using Max Alekseyev's invphi.gp

A007372 Numbers k such that sigma(x) = k has exactly 3 solutions.

Original entry on oeis.org

24, 42, 48, 60, 84, 90, 224, 228, 234, 248, 270, 294, 324, 450, 468, 528, 558, 620, 640, 660, 810, 882, 888, 896, 968, 972, 1020, 1050, 1104, 1116, 1140, 1216, 1232, 1240, 1274, 1332, 1392, 1400, 1452, 1456, 1464, 1482, 1524, 1530, 1600, 1694, 1716, 1760

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Donovan Johnson)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Robert G. Wilson v, Letter to N. J. A. Sloane, Jul. 1992.

Cf. A000203.

Number of solutions: A007369 (0), A007370 (1), A007371 (2), this sequence (3), A060660 (4), A060661 (5), A060662 (6), A060663 (7), A060664 (8), A060665 (9), A060666 (10), A060678 (11), A060676 (12).

a = Table[ 0, {2500} ]; Do[ s = DivisorSigma[ 1, n ]; If[ s < 2501, a[ [ s ] ]++ ], {n, 1, 2500} ]; Select[ Range[ 2500 ], a[ [ # ] ] == 3 & ]

is(n)=sum(k=1,n,sigma(k)==n)==3 \\ Charles R Greathouse IV, Mar 09 2014

is(k) = invsigmaNum(k) == 3 \\ Amiram Eldar, Nov 17 2024, using Max Alekseyev's invphi.gp

A070015 Least m such that the sum of the aliquot parts of m (A001065) equals n, or 0 if no such number exists.

Original entry on oeis.org

1, 2, 0, 4, 9, 0, 6, 8, 10, 15, 14, 21, 121, 27, 22, 16, 12, 39, 289, 65, 34, 18, 20, 57, 529, 95, 46, 69, 28, 115, 841, 32, 58, 45, 62, 93, 24, 155, 1369, 217, 44, 63, 30, 50, 82, 123, 52, 129, 2209, 75, 40, 141, 0, 235, 42, 36, 106, 99, 68, 265, 3481, 371, 118, 64, 56

Offset: 0

Labos Elemer, Apr 12 2002

For odd n >= 9, a(n) <= A073046((n-1)/2). - Max Alekseyev, Sep 01 2025

			For n=128: a(128)=16129, divisors={1,127,16129}, 1+127=sigma(n)-n=128 and 16129 is the smallest.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 9884 terms from Richard J. Mathar)
Mersenne Forum, Given sigma(n)-n, find the smallest possible n

Cf. A000203, A001065, A048050, A051444, A007369, A070016, A005114, A048995.

See A359132 for another version.

f[x_] := DivisorSigma[1, x]-x; t=Table[0, {128}]; Do[c=f[n]; If[c<129&&t[[c]]==0, t[[c]]=n], {n, 1000000}]; t

a(n) = min(x, A001065(x)=n) or a(n)=0 if n is an untouchable number (i.e., if from A005114).

a(0)=1 prepended by Max Alekseyev, Sep 01 2025

cp's OEIS Frontend

A083532 First difference sequence of A007369. Differences between impossible values for sum of divisors of n.

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A094505 Powers of 2 which are not the sum of divisors of any other number. Powers of 2 present in A007369.

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A070240 Duplicate of A007369.

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A002191 Possible values for sum of divisors of n.

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A007368 Smallest k such that sigma(x) = k has exactly n solutions.

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A007370 Numbers k such that sigma(x) = k has a unique solution.

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A051444 Smallest k such that sigma(k) = n, or 0 if there is no such k, where sigma = A000203 = sum of divisors.

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A007371 Numbers k such that sigma(x) = k has exactly 2 solutions.

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