A073757 -id:A073757 - cp's OEIS frontend

A357898 a(n) is the least k such that phi(k) + d(k) = 2^n, or -1 if there is no such k, where phi(k) = A000010(k) is Euler's totient function and d(k) = A000005(k) is the number of divisors of k.

1, 3, 7, 21, 31, 77, 127, 301, 783, 1133, 3399, 4781, 8191, 16637, 37367, 101601, 131071, 305837, 524287, 1073581, 3220743, 4201133, 8544103, 18404669, 34012327, 67139117, 135255431, 300528877, 824583699, 1073862029, 2147483647, 4295564381, 8603449703, 25807607829

Offset: 1

J. M. Bergot and Robert Israel, Oct 19 2022

nonn

All primes in this sequence are primes of the form 2^n - 1. This is true because phi(p) = 2^n - 2 if p = 2^n - 1 is a Mersenne prime. - Thomas Scheuerle, Oct 19 2022
274878976349 = a(38) < a(37) = 274881227398. - Martin Ehrenstein, Oct 24 2022
d(k) <= A070319(2^n). - David A. Corneth, Oct 25 2022

			a(3) = 7 because phi(7)+d(7) = 6+2 = 2^3, and 7 is the least number that works.

Max Alekseyev, Table of n, a(n) for n = 1..160 (a(35)..a(38) from Martin Ehrenstein; a(39)..a(49) from David A. Corneth)

Cf. A000005, A000010, A061468, A070319, A073757.

V:= Array(0..23): count:= 0:
for n from 1 while count < 23 do
  s:= phi(n)+tau(n);
  t:= padic:-ordp(s,2);
  if V[t] = 0 and s = 2^t then
     V[t]:= n; count:= count+1;
  fi
od:
convert(V,list)[2..-1];

a(27)-a(33) from Giorgos Kalogeropoulos, Oct 22 2022

a(34) from Martin Ehrenstein, Oct 24 2022

A073764 a(n) = least number x such that A045763(x)=n or 0 if no such number exists.

Original entry on oeis.org

6, 0, 10, 15, 14, 21, 18, 35, 22, 33, 26, 39, 0, 65, 30, 36, 38, 57, 44, 95, 46, 63, 42, 115, 50, 64, 58, 87, 54, 75, 68, 155, 60, 99, 74, 111, 72, 185, 66, 117, 86, 129, 92, 215, 94, 141, 78, 235, 84, 105, 98, 100, 96, 265, 90, 135, 118, 147, 122, 183, 108, 305, 102

Offset: 1

Labos Elemer, Aug 08 2002

nonn

A070297 without its term a(0). [From R. J. Mathar, Sep 23 2008]

			A045763(x)=5 first holds if x=14 because unrelated set of 14={4,6,8,10,12} has 5 entries. No solutions were found for n=2,13,67,93 when tested at <100000.

Cf. A045763, A073757-A073766.

t=Table[0, {100}]; Do[s=n+1-DivisorSigma[0, n]-EulerPhi[n]; If[s<101&&t[[s]]==0, t[[s]]=n], {n, 1, 100000}]; t

A082513 a(n)=A000720(n)-A000005(n).

Original entry on oeis.org

-1, -1, 0, -1, 1, -1, 2, 0, 1, 0, 3, -1, 4, 2, 2, 1, 5, 1, 6, 2, 4, 4, 7, 1, 6, 5, 5, 3, 8, 2, 9, 5, 7, 7, 7, 2, 10, 8, 8, 4, 11, 5, 12, 8, 8, 10, 13, 5, 12, 9, 11, 9, 14, 8, 12, 8, 12, 12, 15, 5, 16, 14, 12, 11, 14, 10, 17, 13, 15, 11, 18, 8, 19, 17, 15, 15, 17, 13, 20, 12, 17, 18, 21, 11, 19, 19, 19, 15, 22, 12, 20, 18, 20, 20, 20, 12, 23, 19, 19, 16

Offset: 1

Labos Elemer, Apr 29 2003

Cf. A000010, A000005, A000720, A073757, A037228, A063070, A045763.

Table[ -DivisorSigma[0, w]+PrimePi[w], {w, 1, 128}]

cp's OEIS Frontend

A357898 a(n) is the least k such that phi(k) + d(k) = 2^n, or -1 if there is no such k, where phi(k) = A000010(k) is Euler's totient function and d(k) = A000005(k) is the number of divisors of k.

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A073764 a(n) = least number x such that A045763(x)=n or 0 if no such number exists.

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A082513 a(n)=A000720(n)-A000005(n).

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