A074625 -id:A074625 - cp's OEIS frontend

A070982 Smallest integer k such that n divides sigma(k).

1, 3, 2, 3, 8, 5, 4, 7, 10, 19, 43, 6, 9, 12, 8, 21, 67, 10, 37, 19, 20, 43, 137, 14, 149, 45, 34, 12, 173, 24, 16, 21, 86, 67, 76, 22, 73, 37, 18, 27, 163, 20, 257, 43, 40, 137, 281, 33, 52, 149, 101, 63, 211, 34, 109, 28, 49, 173, 353, 24, 169, 48, 32, 93, 72, 86, 401

Offset: 1

Benoit Cloitre, May 24 2002

Ivan Neretin, Table of n, a(n) for n = 1..10000
József Sándor, The sum-of-divisors minimum and maximum functions, Research Report Collection, Volume 8, Issue 1, 2005. See p. 3.

Right diagonal of A074625.

Cf. A005179 (analog for number of divisors), A061026 (analog for Euler totient).

a = ConstantArray[1, 67]; k = 1; While[Length[vac = Rest[Flatten[Position[a, 1]]]] > 0, k++; a[[Intersection[Divisors[DivisorSigma[1, k]], vac]]] *= k]; a (* Ivan Neretin, May 15 2015 *)
With[{dsk=Table[{k,DivisorSigma[1,k]},{k,500}]},Table[SelectFirst[ dsk, Divisible[#[[2]],n]&],{n,70}]][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 04 2018 *)

PARI
```
a(n)=my(s); while(sigma(s++)%n, ); s
```

a(n) = min( k : sigma(k) == 0 mod(n) ).
Sum(k=1, n, a(k)) seems to be asymptotic to c*n^2 with probably 1.1 < c < 1.2.
By Xylouris' form of Linnk's theorem, a(n) << n^5. Can this be improved? - Charles R Greathouse IV, Mar 09 2017

A084303 Smallest x such that sigma(x) mod 6 = n.

Original entry on oeis.org

5, 1, 7, 2, 3, 2401

Offset: 0

Labos Elemer, Jun 02 2003

Row 6 of A074625 (apart from different ordering). - Michel Marcus, Dec 19 2013

			n=5: sigma(2401) = 1+7+49+343+2401 = 2801 = 6*466+5, hence a(5)=2401.

Cf. A000203, A074384.

Table[k = 1; While[Mod[DivisorSigma[1, k], 6] != n, k++]; k, {n, 0, 5}] (* Michael De Vlieger, Mar 25 2017 *)

a(n) = {my(x = 1); while((sigma(x) % 6) != n, x++); x;} \\ Michel Marcus, Dec 18 2013

A097014 Smallest x such that sigma(x) mod 210 = n.

Original entry on oeis.org

104, 1, 211, 2, 3, 8311689, 5, 4, 7, 3698, 399, 130321, 6, 9, 13, 8, 1477, 2458624, 10, 725904, 19, 676, 6751, 18766224, 14, 52707600, 489, 24649, 12, 220433409, 29, 16, 21, 1250, 2779, 3694084, 22, 5184, 37, 18, 27, 1382976, 20, 729, 43, 128, 217

Offset: 0

Labos Elemer, Aug 19 2004

Compare with A084303 and A097013.

Row 210 of A074625 (apart from different ordering). - Michel Marcus, Dec 19 2013

			Several numbers belonging to odd residues are either large squares or twice-squares.
E.g.: r = n = 209: a(209) = 36324729 = 6027^2, sigma(6027^2) = 210*172974 + 209.
Full set is easily available running through squares or twice squares.

Michel Marcus, Table of n, a(n) for n = 0..209

Cf. A000203, A084301, A084303, A097011, A097012, A097013, A097015.

A097013 Smallest x such that sigma(x) mod 30 = n.

Original entry on oeis.org

24, 1, 21, 2, 3, 923521, 5, 4, 7, 18, 27, 2401, 6, 9, 13, 8, 217, 9604, 10, 1089, 19, 98, 91, 21609, 14, 14641, 28, 49, 12, 2614689

Offset: 0

Labos Elemer, Aug 19 2004

Compare with A084303.

Row 30 of A074625 (apart from different ordering). - Michel Marcus, Dec 19 2013

Cf. A000203, A084301, A084303, A097011, A097012, A097014, A097015.

t=Table[Mod[DivisorSigma[1, w], 30], {w, 1, 2700000}]; Table[Min[Flatten[Position[t, j]]], {j, 0, 29}]

A233929 Smallest x such that sigma(x) == n-1 (mod n).

Original entry on oeis.org

1, 1, 7, 2, 3, 2401, 5, 4, 7, 18, 21, 9604, 6, 9, 13, 8, 44, 21609, 10, 18, 19, 289, 36, 9604, 14, 162, 57, 72, 12, 2614689, 29, 16, 21, 625, 63, 38416, 22, 4608, 37, 18, 27, 21609, 20, 289, 43, 36, 50, 38416, 33, 196, 111, 162, 157, 28561, 34, 1296, 28, 49

Offset: 1

Michel Marcus, Dec 18 2013

nonn

Right subdiagonal of A074625.

Records values are: 1, 7, 2401, 9604, 21609, 2614689, 21215236, 36324729, 53304601, 338964921, 431642176, 528264256, 1307979556, ... obtained at indices: 1, 3, 6, 12, 18, 30, 60, 210, 288, 384, 534, 630, 732. - Michel Marcus, Dec 22 2013

Michel Marcus and Donovan Johnson, Table of n, a(n) for n = 1..3000 (first 800 terms from Michel Marcus)

Cf. A000203, A070982, A074625.

a(n) = {x = 1; while ((sigma(x) % n) != (n - 1), x++); x;} \\ Michel Marcus, Dec 18 2013

A074634 Cototient-remainder triangle: triangular array T(n,k) (n >= 1, 0 <= k < n) read by rows, where T(n,k) = smallest number x such that cototient(x) mod n = k.

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 2, 4, 9, 1, 2, 4, 9, 6, 1, 2, 4, 9, 6, 25, 1, 2, 4, 9, 6, 18, 10, 1, 2, 4, 9, 6, 25, 10, 15, 1, 2, 4, 9, 6, 25, 10, 15, 12, 1, 2, 4, 9, 6, 25, 10, 15, 12, 21, 1, 2, 4, 9, 6, 24, 10, 15, 12, 21, 45, 1, 2, 4, 9, 6, 25, 10, 15, 12, 21, 30, 35, 1, 2, 4, 9, 6, 25, 10, 15, 12

Offset: 1

Labos Elemer, Aug 29 2002

			1; 1,2; 1,2,4; 1,2,4,9; 1,2,4,9,6; 1,2,4,9,6,25; 1,2,4,9,6,18,10; 1,2,4,9,6,25,10,15; 1,2,4,9,6,25,10,15,12, ...

Cf. A051953, A074625.

{k=0, s=0, fl=1}; Table[Print["#"]; Table[fl=1; Print[{r, m}]; Do[s=Mod[n-EulerPhi[n], m]; If[(s==r)&&(fl==1), Print[n]; fl=0], {n, 1, 500}], {r, 0, m-1}], {m, 1, 50}]

Min{x; Mod[x-Phi[x], n]=r}, r=1..n, n=1, ...

Name modified to match data by Sean A. Irvine, Jan 22 2025

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A070982 Smallest integer k such that n divides sigma(k).

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A084303 Smallest x such that sigma(x) mod 6 = n.

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A097014 Smallest x such that sigma(x) mod 210 = n.

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A097013 Smallest x such that sigma(x) mod 30 = n.

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A233929 Smallest x such that sigma(x) == n-1 (mod n).

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A074634 Cototient-remainder triangle: triangular array T(n,k) (n >= 1, 0 <= k < n) read by rows, where T(n,k) = smallest number x such that cototient(x) mod n = k.

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