A350936 -id:A350936 - cp's OEIS frontend

A350934 a(n) is the smallest number m such that tau(m - 1) = tau(m + 1) = tau(m) + n or 0 if no such m exists, where tau(k) = A000005(k).

34, 9, 7, 964324, 19, 3822025, 41, 15129, 341, 427166224, 199, 700569, 1241, 11923111249, 919, 12376324, 6641, 34539129, 12221, 363016809, 3401, 56776225, 5741, 199809, 52865, 48045571249, 47081, 3764067904, 21113, 19035769, 18089, 145371249, 59291, 2219069449

Offset: 0

Jaroslav Krizek, Jan 25 2022

nonn

Corresponding values of tau(a(n)): 4, 3, 2, 9, 2, 27, 2, 9, 4, 15, 2, 21, 4, 27, 2, 9, 4, 15, 6, 45, 4, 27, 2, 9, 8, 15, 6, 21, 4, 3, 2, 9, 4, 15, ...
Triples of [tau(a(n) - 1), tau(a(n)), tau(a(n) + 1)] = [tau(a(n)) + n, tau(a(n)), tau(a(n)) + n]: [4, 4, 4], [4, 3, 4], [4, 2, 4], [12, 9, 12], [6, 2, 6], [32, 27, 32], [8, 2, 8], [16, 9, 16], [12, 4, 12], ...
If n is odd then a(n) is a square. - Amiram Eldar, Jan 26 2022

			a(3) = 964324 because 964324 is the smallest number m such that tau(m-1) = tau(m+1) = tau(m)+3; tau(964323) = tau(964325) = tau(964324)+3 = 9+3 = 12.

Cf. A000005, A190646, A350132, A350935, A350936.

Magma
```
Ax:=func; [Ax(n): n in [0..8]]
```

seq[m_, nmax_] := Module[{s = Table[0, {m + 1}], c = 0, d1 = 1, d2 = 2, n = 3, d, k}, While[c < m + 1 && n < nmax, d = DivisorSigma[0, n]; If[d1 == d, k = d - d2 + 1; If[k >= 1 && k <= m + 1 && s[[k]] == 0, s[[k]] = n - 1; c++]]; n++; d1 = d2; d2 = d]; TakeWhile[s, # > 0 &]]; seq[8, 10^7] (* Amiram Eldar, Jan 26 2022 *)

More terms from Amiram Eldar, Jan 26 2022

A350935 a(n) is the smallest number m such that tau(m-1) = tau(m+1) = n*tau(m) or 0 if no such m exists, where tau(k) = A000005(k).

Original entry on oeis.org

34, 7, 19, 41, 6641, 199, 640063, 919, 17299, 22193, 350632961, 5741, 57394565119, 2345921, 3568049, 18089, 55171346530303, 41651, 193405731995647, 252881, 88099649, 1439024129, 916791443027132417, 90271, 821128751, 20969598977, 3959299, 2319679, 190190725057515297439745, 7860401

Offset: 1

Jaroslav Krizek, Jan 25 2022

nonn

Corresponding values of tau(a(n)): 4, 2, 2, 2, 4, 2, 4, 2, 2, 2, 4, 2, 8, 2, 4, 2, 8, 2, 4, 2, 4, 4, 8, 2, 4, 4, 2, 2, 8, 2, ...

Triples of [tau(a(n) - 1), tau(a(n)), tau(a(n) + 1)] = [n * tau(a(n)), tau(a(n)), n * tau(a(n))]: [4, 4, 4], [4, 2, 4], [6, 2, 6], [8, 2, 8], [20, 4, 20], [12, 2, 12], [28, 4, 28], [16, 2, 16], [18, 2, 18], [20, 2, 20], ...

			a(3) = 19 because 19 is the smallest number m such that tau(m-1) = tau(m+1) = 3 * tau(m); tau(18) = tau(20) = 3 * tau(19) = 3 * 2 = 6.

Cf. A000005, A190646, A350132, A350934, A350936.

Magma
```
Ax:=func; [Ax(n): n in [1..10]]
```

a(11)-a(16) from Jon E. Schoenfield, Jan 26 2022

More terms from Jon E. Schoenfield and David A. Corneth, Jan 27 2022

cp's OEIS Frontend

A350934 a(n) is the smallest number m such that tau(m - 1) = tau(m + 1) = tau(m) + n or 0 if no such m exists, where tau(k) = A000005(k).

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