A097646 -id:A097646 - cp's OEIS frontend

A077084 Erroneous version of A097646.

1, 2, 6, 10, 20, 22, 46, 48, 58, 82, 106, 166, 178, 180, 208, 226, 262, 346, 358, 382, 466, 478, 502, 864, 1120, 3552

Offset: 1

Labos Elemer, Oct 28 2002

A018784 Numbers n such that sigma(phi(n)) = n.

1, 3, 15, 28, 255, 744, 2418, 20440, 65535, 548856, 2835756, 4059264, 4451832, 10890040, 13192608, 23001132, 54949482, 110771178, 220174080, 445701354, 4294967295, 16331433888, 18377794080, 94951936080, 204721968000, 386940247200, 601662398400, 1433565580920

Offset: 1

David W. Wilson

nonn

The numbers 2^2^n-1 for n=0,1,...,5 are in the sequence because 2^2^n-1=(2^2^0+1)*(2^2^1+1)*(2^2^2+1)*...*(2^2^(n-1)+1); 2^2^k+1 for k=0,1,2,3 & 4 are primes (Fermat primes); sigma(2^k)=2^(k+1)-1 and phi is a multiplicative function. Hence if p is a known Fermat prime (p=2^2^n+1 for n=0,1,2,3 & 4) then p-2 is in the sequence, note that this is not true for unknown Fermat primes if they exist. - Farideh Firoozbakht, Aug 27 2004

Graeme L. Cohen, On a conjecture of Makowski and Schinzel, Colloquium Mathematicae, Vol. 74, No. 1 (1997), pp. 1-8. See Notes p. 7.

Cf. A097645, A097646, A019434.

Select[Range[10^6], DivisorSigma[1, EulerPhi[#]] == # &] (* Amiram Eldar, Dec 10 2020 *)

is(n)=sigma(eulerphi(n))==n \\ Charles R Greathouse IV, Nov 27 2013

sigma(A001229), sorted.

Wilson's search was complete only through a(19) = 50319360. Jud McCranie reports Jun 15 1998 that the terms through a(24) are certain.

a(26)-a(28) added. Verified sequence is complete through a(28) by Donovan Johnson, Jun 30 2012

A097645 Numbers k such that k = sigma(phi(k) + pi(k)).

Original entry on oeis.org

1, 6, 54, 78, 1296, 1482, 6480, 6552, 14040, 20160, 36936, 1273896

Offset: 1

Farideh Firoozbakht, Sep 07 2004

Does this sequence have any odd terms > 1? There is no other term up to 3*10^7.

a(13) > 10350781218. - J.W.L. (Jan) Eerland, Dec 25 2021

			1273896 is in the sequence because pi(1273896) = 98190, phi(1273896) = 391680, and sigma(98190+391680) = 1273896.

Cf. A018784, A082515, A097646.

Do[If[n==DivisorSigma[1, EulerPhi[n]+PrimePi[n]], Print[n]], {n, 10000000}]
n=1;Monitor[Parallelize[While[True,If[DivisorSigma[1,EulerPhi[n]+PrimePi[n]]==n,Print[n]];n++];n],n] (* J.W.L. (Jan) Eerland, Dec 25 2021 *)

isok(k) = k == sigma(eulerphi(k) + primepi(k)); \\ Michel Marcus, Dec 25 2021

A097652 Numbers n such that n=phi(phi(n)+sigma(n)) and n is not of the form 2*p where p is a Sophie Germain odd prime.

Original entry on oeis.org

1, 2, 20, 48, 180, 208, 864, 1120, 1368, 3552, 58320, 76416, 79968, 95488, 107520, 338688, 570240, 595968, 653184, 1347840, 5199552, 7918848, 14592000, 93699072, 159138176, 167078784, 246688000, 281640960, 314548224, 323985408, 338411520, 347578368, 352002048

Offset: 1

Farideh Firoozbakht, Sep 09 2004

nonn

It is obvious that if n=2*p where p is a Sophie Germain odd prime then n=phi(phi(n)+sigma(n)). This sequence is a subsequence of A097646. Except for the first term all terms of this sequence are even. There is no other term up to 3*10^7.

			14592000 is in the sequence because 14592000=2*7296000, 7296000 is not a Sophie Germain odd prime and phi(phi(14592000)+sigma(14592000)) =14592000.

C. K. Caldwell, The Prime Glossary, Sophie Germain prime.

Cf. A097646, A005384.

Do[If[(!PrimeQ[n/2]||!PrimeQ[n+1])&&n==EulerPhi[EulerPhi[n]+ DivisorSigma[1, n]], Print[n]], {n, 30000000}]

a(24)-a(33) from Donovan Johnson, Jan 18 2012

A227623 Numbers n such that phi(n) + pi(n) is prime.

Original entry on oeis.org

2, 5, 6, 17, 18, 23, 24, 25, 31, 33, 36, 41, 47, 48, 51, 70, 75, 84, 87, 98, 106, 109, 111, 127, 130, 157, 160, 170, 180, 191, 192, 197, 211, 212, 217, 219, 220, 221, 233, 236, 241, 242, 244, 247, 248, 257, 258, 259, 260, 261, 278, 279, 286, 288, 290, 331, 333

Offset: 1

K. D. Bajpai, Jul 18 2013

			phi(786) + pi(786) = 260 + 137 = 397 which is prime, so 786 is in the sequence.

K. D. Bajpai, Table of n, a(n) for n = 1..1326

Cf. A097646.

with(numtheory):K := proc(n) local a; a :=(phi(n)+ pi(n)); if  isprime(a)  then RETURN (n) fi: end: seq(K(n), n=1..1000); # K. D. Bajpai, Jul 18 2013
with(numtheory):K:=proc()local n,a,c;  c:=1; for n from 1 to 10000 do; a:=(phi(n)+ pi(n));if  isprime(a)  then lprint(c,n); c:=c+1; fi;od; end: K(); # K. D. Bajpai, Jul 18 2013

Select[Range[200], PrimeQ[EulerPhi[#] + PrimePi[#]] &] (* G. C. Greubel, Jul 04 2017 *)

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A077084 Erroneous version of A097646.

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A018784 Numbers n such that sigma(phi(n)) = n.

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A097645 Numbers k such that k = sigma(phi(k) + pi(k)).

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A097652 Numbers n such that n=phi(phi(n)+sigma(n)) and n is not of the form 2*p where p is a Sophie Germain odd prime.

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A227623 Numbers n such that phi(n) + pi(n) is prime.

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