A051444 -id:A051444 - cp's OEIS frontend

A063973 a(n) is the largest m such that usigma(m) = n (or 0 if no such m).

1, 0, 2, 3, 4, 5, 0, 7, 8, 9, 0, 11, 0, 13, 0, 0, 16, 17, 0, 19, 0, 0, 0, 23, 0, 25, 0, 27, 0, 29, 0, 31, 32, 0, 0, 24, 0, 37, 0, 28, 0, 41, 0, 43, 0, 0, 0, 47, 0, 49, 0, 0, 0, 53, 0, 39, 0, 0, 0, 59, 0, 61, 0, 0, 64, 0, 0, 67, 0, 52, 0, 71, 0, 73, 0, 0, 0, 50, 0, 79, 0, 81, 0, 83, 0, 0, 0, 0

Offset: 1

Labos Elemer, Sep 05 2001

nonn

usigma(m) is the sum of unitary divisors of m, A034448.

			a(12) = 11 because the unitary divisors of 11 are 1 and 11, and their sum is 12.

Daniel Suteu, Table of n, a(n) for n = 1..10000

Cf. A034444, A034448, A051444, A057637, A063972.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); a[n_] := Module[{k = n}, While[k > 0 && usigma[k] != n, k--]; k]; Array[a, 100]  (* Amiram Eldar, Aug 27 2024 *)

Corrected by Don Reble, May 14 2006

A070016 Least m such that Chowla's function value of m [A048050(m)] equals n or 0 if no such number exists.

Original entry on oeis.org

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, 117

Offset: 1

Labos Elemer, Apr 12 2002

nonn

Remark that A070016(n)=A070015(n+1) in accordance with A048995(k)+1=A005114(k).

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

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

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

f1[x_] := DivisorSigma[1, x]-x-1; t=Table[0, {128}]; Do[b=f1[n]; If[b<129&&t[[b]]==0, t[[b]]=n], {n, 1, 1000000}]; t

a(n)=Min{x; A048050(x)=n} or a(n)=0 if n is from A048995.

A167485 Smallest positive integer m such that n can be expressed as the sum of an initial subsequence of the divisors of m, or 0 if no such m exists.

Original entry on oeis.org

1, 1, 0, 2, 3, 0, 5, 4, 7, 15, 12, 21, 6, 9, 13, 8, 12, 30, 10, 42, 19, 18, 20, 57, 14, 36, 46, 30, 12, 102, 29, 16, 21, 42, 62, 84, 22, 36, 37, 18, 27, 63, 20, 50, 43, 66, 52, 129, 33, 75, 40, 78, 48, 220, 34, 36, 28, 49, 60, 265, 24, 132, 61, 32, 56, 117, 54, 100, 67, 90, 84

Offset: 0

Franklin T. Adams-Watters, Nov 04 2009

It appears that 2 and 5 are the only zeros in this sequence. This would follow from a slightly stronger version of the Goldbach conjecture: every even integer > 22 can be expressed as the sum of two primes p and q, with 5 < p < q < 5p. Then odd numbers can be obtained for pq and even numbers for 5pq.
Is a(n) = o(n)? - Arkadiusz Wesolowski, Nov 09 2013
The above question has been posed by Erdős. See Guy. - Stefano Spezia, Sep 25 2024
a(A000203(n)) <= n. Since A000203(n)/n can be arbitrarily large, that shows that lim inf_{n -> oo} a(n)/n = 0.  - Robert Israel, Sep 26 2024

			The divisors of 15 are 1,3,5,15, with cumulative sums 1,4,9,24. Since this is the smallest number where 9 occurs in the sums, a(9) = 15.

R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B2.

Robert Israel, Table of n, a(n) for n = 0..20000 (n = 0 .. 1000 from Michel Marcus)
Robert Israel, Plot of a(n)/n for n = 1 .. 10^6

Cf. A000203, A001065, A078587, A051444.

N:= 100: # for a(0) .. a(N)
count:= 1: V:= Array(0..N): V[0]:= 1:
for m from 1 while count < N-1 do
  L:= ListTools:-PartialSums(sort(convert(numtheory:-divisors(m),list)));
  for x in L do
    if x > N then break fi;
    if V[x] = 0 then V[x]:= m; count:= count+1 fi;
od od:
convert(V,list); # Robert Israel, Sep 26 2024

{u=vector(100); for(n=1,1000,ds=divisors(n);s=0; for(k=1,#ds,s+=ds[k];if(s>#u,break);if(!u[s],u[s]=n))); u}

A063975 Smallest numbers such that the number of terms in inverse set usigma equals n; where usigma = A034448.

Original entry on oeis.org

1, 12, 24, 60, 120, 72, 216, 288, 1320, 480, 240, 840, 1296, 2700, 960, 1512, 1080, 720, 1728, 2016, 3840, 3240, 3456, 2520, 3360, 3024, 1440, 3600, 6912, 2160, 19152, 2880, 7920, 13680, 9072, 12600, 6048, 5040, 18000, 6480, 27216, 13440, 7200, 27648, 5760

Offset: 1

Labos Elemer, Sep 05 2001

nonn

			usigma(x) = 288, invusigma(288) = {138, 154, 165, 168, 213, 235, 248, 253}, so a(288) = 8, the number of all terms in the inverse set and all similar numbers are larger: {288, 648, 672, 900}.

Donovan Johnson, Table of n, a(n) for n = 1..3000

Cf. A034444, A034448, A051444, A054973, A057637.

with(numtheory): A034448 := proc(n) option remember: local ans,i: ans:=1: for i from 1 to nops(ifactors(n)[2]) do ans := ans*(1+ifactors(n)[ 2 ][ i ][ 1 ]^ifactors(n)[ 2 ] [ i ] [ 2 ]): od: return ans: end: for n from 1 to 5000 do m:=A034448(n): if(type(ct[m],integer))then ct[m]:=ct[m]+1: else ct[m]:=1: fi:od: for m from 1 to 28 do for n from 1 to 5000 do if(ct[n]=m)then printf("%d, ",n):break: fi: od:od: # Nathaniel Johnston, Apr 29 2011

a(9) - a(45) from Nathaniel Johnston, Apr 29 2011

A196225 Smallest number k such that sigma(tau(k)) = n, or 0 if there is no such k.

Original entry on oeis.org

1, 0, 2, 4, 0, 16, 6, 64, 0, 0, 0, 12, 36, 4096, 24, 0, 0, 48, 0, 262144, 0, 0, 0, 144, 0, 0, 0, 60, 0, 268435456, 120, 576, 0, 0, 0, 3072, 0, 68719476736, 180, 900, 0, 240, 0, 4398046511104, 0, 0, 0, 5184, 0, 0, 0, 0, 0, 196608, 0, 960, 46656, 0, 0, 360, 0, 1152921504606846976

Offset: 1

Jaroslav Krizek, Jan 02 2013

nonn

Smallest number k such that A062069(k) = A000203(A000005(k)) = n, or 0 if there is no such k.

			a(6) = 16 because number 16 is the smallest number k such that sigma(tau(k)) = 6; (tau(16) = 5, sigma(5) = 6).

Amiram Eldar, Table of n, a(n) for n = 1..3323
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: Inversion of Multiplicative Functions (invphi.gp).

Cf. A062069 (sigma(tau(n))), A000203(sigma(n)), A000005(tau(n)), A005179, A051444.

a(n) = {my(v = invsigma(n), kmin = 0); for(i = 1, #v, k = A005179(v[i]); if(kmin == 0 || k < kmin, kmin = k)); kmin;} \\ Amiram Eldar, Jan 21 2025, using Max Alekseyev's invphi.gp and R. J. Mathar's A005179(n)

a(n) = 0 iff A051444(n) = 0.

a(24) and a(48) corrected by Amiram Eldar, Jan 21 2025

A291503 a(n) is the smallest k such that sigma(k) = phi(n), or 0 if no such k exists.

Original entry on oeis.org

1, 1, 0, 0, 3, 0, 5, 3, 5, 3, 0, 3, 6, 5, 7, 7, 0, 5, 10, 7, 6, 0, 0, 7, 19, 6, 10, 6, 12, 7, 29, 0, 19, 0, 14, 6, 22, 10, 14, 0, 27, 6, 20, 19, 14, 0, 0, 0, 20, 19, 21, 14, 0, 10, 27, 14, 22, 12, 0, 0, 24, 29, 22, 21, 33, 19, 0, 21, 43, 14, 0, 14, 30, 22, 27, 22, 24, 14, 45

Offset: 1

Altug Alkan, Aug 25 2017

			a(5) = 3 because sigma(3) = phi(5) and 3 is the smallest number with this property.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A051444, A069825.

N:= 100: # to get a(1)..a(N)
R:= Vector(N):
for k from 1 to N-1 do
  s:= numtheory:-sigma(k);
  if s <= N and R[s] = 0 then R[s]:= k fi;
od:
seq(R[numtheory:-phi(n)],n=1..N); # Robert Israel, Aug 25 2017

a(n) = for(k=1, eulerphi(n), if(sigma(k)==eulerphi(n), return(k))); 0 \\ after Charles R Greathouse IV at A051444

a(A069825(n)) = 0 for n > 1.

a(n) = A051444(A000010(n)). - Michel Marcus, Aug 25 2017

A329821 Largest k for which sigma(k) = A002191(n), where A002191 = range of sigma, the sum-of-divisors function A000203.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 11, 9, 13, 8, 17, 19, 23, 12, 29, 25, 31, 22, 37, 18, 27, 41, 43, 47, 53, 39, 49, 59, 61, 32, 67, 71, 73, 45, 79, 83, 89, 36, 50, 77, 97, 101, 103, 107, 109, 91, 113, 95, 81, 75, 82, 64, 127, 131, 121, 137, 139, 119, 149, 151, 125

Offset: 1

M. F. Hasler, Nov 22 2019

nonn

			The possible values of sigma(x) are A002191 = {1, 3, 4, 6, 7, 8, 12, ...}.
The 7th value is 12 = sigma(x) for x = 6 or 11. Since 11 is the largest such value, a(7) = 11.

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

Cf. A000203 (sigma), A002191 (range of sigma), A085790 (table of pre-images of x in A002191), A054973 (number of solutions of sigma(x) = n).

Cf. A051444 (smallest k such that sigma(k) = n).

A329821(n)=vecmax(invsigma(A002191(n))) \\ see Alekseyev link for invsigma(). An invsigmaMax() function is announced.

a(n) = A085790(m,A054973(m)) with m = A002191(n).

cp's OEIS Frontend

A063973 a(n) is the largest m such that usigma(m) = n (or 0 if no such m).

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A070016 Least m such that Chowla's function value of m [A048050(m)] equals n or 0 if no such number exists.

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A167485 Smallest positive integer m such that n can be expressed as the sum of an initial subsequence of the divisors of m, or 0 if no such m exists.

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A063975 Smallest numbers such that the number of terms in inverse set usigma equals n; where usigma = A034448.

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Maple

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A196225 Smallest number k such that sigma(tau(k)) = n, or 0 if there is no such k.

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A291503 a(n) is the smallest k such that sigma(k) = phi(n), or 0 if no such k exists.

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Maple

PARI

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A329821 Largest k for which sigma(k) = A002191(n), where A002191 = range of sigma, the sum-of-divisors function A000203.

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