A072627 -id:A072627 - cp's OEIS frontend

A072628 Number of divisors d of n such that d-1 is not prime.

1, 2, 1, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 3, 3, 3, 2, 3, 2, 4, 3, 4, 2, 2, 3, 4, 3, 4, 2, 5, 2, 3, 3, 4, 4, 4, 2, 3, 3, 5, 2, 4, 2, 4, 5, 4, 2, 3, 3, 6, 3, 5, 2, 4, 4, 5, 3, 4, 2, 5, 2, 3, 5, 4, 4, 6, 2, 4, 3, 7, 2, 4, 2, 3, 5, 4, 4, 6, 2, 6, 4, 4, 2, 5, 4, 4, 3, 5, 2, 7, 4, 5, 3, 4, 4, 4, 2, 4, 5, 7, 2, 5, 2, 5, 7

Offset: 1

Labos Elemer, Jun 28 2002

nonn

			If n = p is prime then divisors - 1 = {1, p} - 1 = {0, p-1} so a(p) = 2 if p <> 3.
240 has 20 divisors, of them 8 divisors d have nonprime value of d-1, {0, 1, 4, 9, 14, 15, 39, 119}, so a(240) = 8.

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

Cf. A000005, A072627.

di[x_] := Divisors[x]; dp[x_] := Part[di[x], Flatten[Position[PrimeQ[ -1+di[x]], True]]]-1; Table[DivisorSigma[0, w]-Length[dp[w]], {w, 1, 128}]
a[n_] := DivisorSum[n, 1 &, !PrimeQ[#-1] &]; Array[a, 100] (* Amiram Eldar, Apr 13 2024 *)

a(n) = sumdiv(n, d, !isprime(d-1)); \\ Amiram Eldar, Apr 13 2024

a(n) = A000005(n) - A072627(n) < A000005(n).

A322676 Numbers k > 0 such that k has more divisors d, such that d-1 is prime, than any other smaller positive number, with a(1) = 1.

Original entry on oeis.org

1, 3, 6, 12, 24, 48, 72, 120, 168, 240, 360, 720, 1440, 2160, 2520, 4320, 5040, 7560, 10080, 15120, 20160, 27720, 30240, 55440, 75600, 83160, 110880, 151200, 166320, 332640, 665280, 831600, 1330560, 1441440, 1663200, 2162160, 2882880, 3326400, 4324320, 6486480, 8648640

Offset: 1

Daniel Suteu, Dec 23 2018

nonn

Position of records in A072627.

Cf. A072627, A202727, A322678.

DeleteDuplicates[Table[{n,Count[Divisors[n]-1,?PrimeQ]},{n,87*10^5}],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]] (* _Harvey P. Dale, Dec 17 2024 *)

r=-1; for(n=1, 10^6, t=sumdiv(n, d, isprime(d-1)); if(t>r, r=t; print1(n, ", ")));

A341119 a(n) is the least positive number that has exactly n divisors d such that d-1 is prime.

Original entry on oeis.org

1, 3, 6, 18, 12, 36, 24, 48, 72, 120, 168, 336, 240, 540, 360, 960, 840, 1080, 720, 1680, 3024, 1440, 2160, 2880, 2520, 6480, 4320, 14040, 8640, 5040, 9240, 7560, 23520, 12600, 18480, 10080, 33600, 22680, 15120, 20160, 36960, 27720, 47880, 40320, 37800, 47520, 30240, 80640, 85680, 65520, 60480

Offset: 0

J. M. Bergot and Robert Israel, Feb 05 2021

nonn

a(n) is the least positive solution to A072627(k) = n.

The conjectured terms are exact if for 0 <= n <= 10000 we have a(n) / A046523(A000005(a(n))) <= 9. For the found terms, a(n) / A046523(A000005(a(n))) <= 7.3. - David A. Corneth, Jun 15 2022

			a(3) = 18 has 3 such divisors: 2+1=3, 5+1=6, 17+1=18, and is the least number with exactly 3.

Robert G. Wilson v, Table of n, a(n) for n = 0..1740 (first 216 terms from _Robert Israel_)
David A. Corneth, Conjectured first 10001 terms.

Cf. A000005, A008864, A046523, A067846, A072627, A341099.

f:= proc(n) nops(select(t -> isprime(t-1), numtheory:-divisors(n))) end proc:
N:= 60: count:= 0:
V:= Array(0..N):
for n from 1 while count < N+1 do
  v:= f(n);
  if v <= N and V[v] = 0 then
    count:=count+1;
    V[v]:= n;
  fi;
od:
convert(V,list);

With[{s = Array[DivisorSum[#, 1 &, PrimeQ[# - 1] &] &, 10^5]}, Array[FirstPosition[s, #][[1]] &, 51, 0]] (* Michael De Vlieger, Feb 05 2021 *)

a(n) = my(k=1); while (sumdiv(k, d, isprime(d-1)) != n, k++); k; \\ Michel Marcus, Feb 05 2021

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A072628 Number of divisors d of n such that d-1 is not prime.

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A322676 Numbers k > 0 such that k has more divisors d, such that d-1 is prime, than any other smaller positive number, with a(1) = 1.

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A341119 a(n) is the least positive number that has exactly n divisors d such that d-1 is prime.

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