A036454 -id:A036454 - cp's OEIS frontend

A009087 Numbers whose number of divisors is prime (i.e., numbers of the form p^(q-1) for primes p,q).

2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239

Offset: 1

Simon Colton (simonco(AT)cs.york.ac.uk)

Invented by the HR Automatic Concept Formation Program. If the sum of divisors is prime, then the number of divisors is prime, i.e., this is a supersequence of A023194.

A010055(a(n)) * A010051(A100995(a(n))+1) = 1. - Reinhard Zumkeller, Jun 06 2013

			tau(16)=5 and 5 is prime.

S. Colton, Automated Theory Formation in Pure Mathematics. New York: Springer (2002)

Indranil Ghosh, Table of n, a(n) for n = 1..12546 (terms 1..1000 from T. D. Noe)
S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.

Subsequence of A000961.

Cf. A023194, A036454, A203967.

a009087 n = a009087_list !! (n-1)
a009087_list = filter ((== 1) . a010051 . (+ 1) . a100995) a000961_list
-- Reinhard Zumkeller, Jun 05 2013

Select[Range[250],PrimeQ[DivisorSigma[0,#]]&] (* Harvey P. Dale, Sep 28 2011 *)

is(n)=isprime(isprimepower(n)+1) \\ Charles R Greathouse IV, Sep 16 2015

from sympy import primepi, integer_nthroot, primerange
def A009087(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x,k-1)[0]) for k in primerange(x.bit_length()+1)))
    return bisection(f,n,n) # Chai Wah Wu, Feb 22 2025

p^(q-1), p, q primes.

A036455 Numbers n such that d(d(n)) is an odd prime, where d(k) is the number of divisors of k.

Original entry on oeis.org

6, 8, 10, 14, 15, 21, 22, 26, 27, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 120, 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 168, 177, 178, 183

Offset: 1

Labos Elemer

nonn

Compare with sequence A007422 and A030513 -- the resemblance is rather strong. Still this sequence is different. For example, 36, 100, 120, and 168 are here.

			a(15) = 39 and d(39) = 4, d(d(39)) = d(4) = 3 and d(d(d(39))) = 2. After 3 iteration the equilibrium is reached.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000005, A007422, A030513, A036450, A036452, A036454, A036456, A036457, A036458.

filter:= proc(n) local r;
  r:= numtheory:-tau(numtheory:-tau(n));
  r::odd and isprime(r)
end proc:
select(filter, [$1..1000]); # Robert Israel, Feb 02 2016

fQ[n_] := Module[{d2 = DivisorSigma[0, DivisorSigma[0, n]]}, d2 > 2 && PrimeQ[d2]]; Select[Range[200], fQ] (* T. D. Noe, Jan 22 2013 *)

is(n)=isprime(n=numdiv(numdiv(n))) && n>2 \\ Charles R Greathouse IV, Jan 22 2013

d(d(d(a(n)))) = 2 for all n.

A036459(a(n)) = 3. - Ivan Neretin, Jan 25 2016

Definition clarified by R. J. Mathar and Charles R Greathouse IV, Jan 22 2013

A053470 a(n) is the cototient of n (A051953) iterated twice.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 2, 1, 4, 0, 4, 0, 4, 1, 4, 0, 8, 0, 8, 3, 8, 0, 8, 1, 8, 3, 8, 0, 12, 0, 8, 1, 12, 1, 16, 0, 12, 7, 16, 0, 22, 0, 16, 9, 16, 0, 16, 1, 22, 1, 16, 0, 24, 7, 16, 9, 22, 0, 24, 0, 16, 9, 16, 1, 24, 0, 24, 5, 24, 0, 32, 0, 20, 11, 24, 1, 36, 0, 32, 9, 30, 0, 44, 9, 24, 1, 32, 0

Offset: 1

Labos Elemer, Jan 14 2000

nonn

Iteration of A051953 is ended at fixed point 0. Analogous 2nd iterates for number of divisors (A000005) and Euler-Phi (A000010) are A036454 and A010554.

			n=50, n_1 = n - phi(n) = 50 - 20 = 30, n_2 = n_1 - Phi(n_1) = 30 - 8 = 22, so the 50th term is 22.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A051953, A000005, A000010, A036454, A010554.

Cf. A053471, A053472, A053473.

Array[Nest[# - EulerPhi@ # &, #, 2] &, 89] (* Michael De Vlieger, Dec 23 2017 *)

A051953(n) = (n-eulerphi(n));
A053470(n) = if(1==n,0,A051953(A051953(n))); \\ Antti Karttunen, Dec 22 2017

a(1) = 0; for n > 1, a(n) = A051953(A051953(n)).

A053477 Sum of iterates of divisor number function A000005.

Original entry on oeis.org

1, 2, 5, 9, 7, 15, 9, 17, 14, 19, 13, 27, 15, 23, 24, 23, 19, 33, 21, 35, 30, 31, 25, 41, 30, 35, 36, 43, 31, 47, 33, 47, 42, 43, 44, 50, 39, 47, 48, 57, 43, 59, 45, 59, 60, 55, 49, 67, 54, 65, 60, 67, 55, 71, 64, 73, 66, 67, 61, 87, 63, 71, 78, 73, 74, 83, 69, 83, 78, 87, 73

Offset: 1

Labos Elemer, Jan 14 2000

nonn

			If n is prime then the iteration sequence is {p,2} and the sum is p+2. If n=30, then iterations of the d function are {30,8,4,3,2} and their sum is a(30)=47.

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

Cf. A000005, A036452, A036453, A036454, A036455, A036459.

f:= proc(n) option remember;
  if n <= 2 then n
  else n + procname(numtheory:-tau(n));
  fi
end proc:
map(f, [$1..80]); # Robert Israel, Nov 14 2016

g[n_] := DivisorSigma[0, n]; f[n_] := Plus @@ Drop[ FixedPointList[g, n], -1]; Table[ f[n], {n, 71}] (* Robert G. Wilson v, Dec 16 2004 *)

A135430 Numbers k for which Ramanujan's function tau(k)=A000594(k) is an odd prime.

Original entry on oeis.org

63001, 458329, 942841, 966289, 1510441, 2961841, 4879681, 14280841, 29019769, 46117681, 49182169, 51652969, 56957209, 75047569, 80120401, 86136961, 93644329, 97752769, 104509729, 162384049, 164378041, 177235969, 193571569

Offset: 1

Giovanni Resta, Dec 12 2007

nonn

Here, negative integers whose absolute value is prime are considered prime.
a(1) = 63001 was found by Lehmer in 1965. It is known that tau(n) is odd if and only if n is an odd square. Indeed, a(1)=251^2, a(2)=677^2, ..., a(7)=47^4. The first sixth power in the sequence is 1151^6.
From Olivier Rozier, Feb 03 2016 (Start)
a(n) = p^(q-1) for p,q odd primes, and p not included in A007659, so that a(n) is a subsequence of A036454. Consequence of the arithmetical properties: (i) tau function is multiplicative, (ii) for p prime, tau(p^(k-1)) is the k-th term of a Lucas sequence.
It is conjectured that the equation |tau(n)|=2 has no solution. (End)

			tau(63001) = -80561663527802406257321747 which is prime.

Dana Jacobsen, Table of n, a(n) for n = 1..1000
Michael Bennett, Adela Gherga, Vandita Patel, and Samir Siksek, Odd values of the Ramanujan tau function, arXiv:2101.02933 [math.NT], 2021.
D. H. Lehmer, The Primality of Ramanujan's Tau-Function, The American Mathematical Monthly, Vol. 72, No. 2, Part 2 (Feb., 1965), pp. 15-18.
N. Lygeros and O. Rozier, Odd prime values of the Ramanujan tau function, Ramanujan Journal, Vol. 32 (2013), pp. 269-280.
Eric Weisstein's World of Mathematics, Tau Function Prime

Cf. A000594 (Ramanujan's tau function tau(n)).

Cf. A265913 (tau(a(n))).

Select[Range[1, 7000, 2]^2, PrimeQ@RamanujanTau@# &]

for(x=1,1000, n=(2*x+1)^2; if(isprime(abs(ramanujantau(n))), print1(n", "))) \\ Dana Jacobsen, Sep 07 2015

use ntheory ":all"; for (0..1000) { my $n = (2*$+1)**2; say $n if is_prime(abs(ramanujan_tau($n))); } # _Dana Jacobsen, Sep 07 2015

A186637 Semiprime powers with special exponents: k^(j-1) where both j and k are arbitrary semiprime numbers.

Original entry on oeis.org

64, 216, 729, 1000, 1024, 2744, 3375, 7776, 9261, 10648, 15625, 17576, 35937, 39304, 42875, 54872, 59049, 59319, 65536, 97336, 100000, 117649, 132651, 166375, 185193, 195112, 238328, 262144, 274625, 328509, 405224, 456533, 537824, 551368, 614125, 636056, 658503, 753571, 759375, 804357, 830584, 857375

Offset: 1

Jonathan Vos Post, Feb 24 2011

Semiprime analog of A036454: prime powers with special exponents: q^(p-1) where both p and q are arbitrary prime numbers.

			a(1) = smallest semiprime to power of (smallest semiprime - 1) = 4^(4-1) = 4^3 = 64.

Cf. A001358, A036454, A113877, A186621.

from math import isqrt
from sympy import primepi, primerange, integer_nthroot, factorint
def A186637(n):
    def A072000(n): return int(-((t:=primepi(s:=isqrt(n)))*(t-1)>>1)+sum(primepi(n//p) for p in primerange(s+1)))
    def f(x): return int(n+x-sum(A072000(integer_nthroot(x, p-1)[0]) for p in range(4,x.bit_length()+1) if sum(factorint(p).values())==2))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f,n,n) # Chai Wah Wu, Sep 12 2024

{a(n)} = {A001358(i) ^ A186621(j)}.

{a(n)} = {a^b where a and b are elements of A001358} = {(p*q)^((r*s)-1) for primes p, q, r, s, not necessarily distinct}.

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A009087 Numbers whose number of divisors is prime (i.e., numbers of the form p^(q-1) for primes p,q).

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A036455 Numbers n such that d(d(n)) is an odd prime, where d(k) is the number of divisors of k.

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A053470 a(n) is the cototient of n (A051953) iterated twice.

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A053477 Sum of iterates of divisor number function A000005.

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A135430 Numbers k for which Ramanujan's function tau(k)=A000594(k) is an odd prime.

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A186637 Semiprime powers with special exponents: k^(j-1) where both j and k are arbitrary semiprime numbers.

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Formula