A156560 -id:A156560 - cp's OEIS frontend

A067702 Numbers k such that sigma(k) == 0 (mod k+2).

12, 70, 88, 180, 1888, 4030, 5830, 32128, 521728, 1848964, 8378368, 34359083008, 66072609790, 549753192448, 259708613909470, 2251799645913088

Offset: 1

Benoit Cloitre, Feb 05 2002

If 2^i-5 is prime for i > 2 then let x = (2^i-5)*2^(i-1). Then sigma(x)=2*(x+2), so x is in the sequence. There are other terms that are not of this form. - Jud McCranie, Jan 12 2019

Contains terms of A088832, terms m of A088834 with (sigma(m)-6)/m = 3, terms m of A045770 with (sigma(m)-8)/m = 4, terms m of A076496 with (sigma(m)-12)/m = 6. - Max Alekseyev, May 26 2025

			sigma(180) = 546 = 3(180+2), so 180 is in the sequence.

Contains subsequence A088832.

Cf. A045768, A156560, A059608, A045770, A076496, A088834.

Select[Range[84*10^5],Divisible[DivisorSigma[1,#],#+2]&] (* Harvey P. Dale, May 11 2018 *)

isok(n) = sigma(n) % (n+2) == 0; \\ Michel Marcus, Nov 22 2013

a(9)-a(11) from Michel Marcus, Nov 22 2013
a(12)-a(13) from Jud McCranie, Jan 12 2019
a(14) from Jud McCranie, Jan 13 2019
a(15) from Jud McCranie, Dec 02 2019
a(16) from Max Alekseyev, May 26 2025

A234504 Number of ways to write n = k + m with k > 0 and m > 0 such that 2^(phi(k) + phi(m)/4) - 5 is prime, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 2, 1, 1, 3, 2, 3, 2, 3, 2, 3, 4, 5, 5, 4, 5, 6, 7, 4, 5, 6, 7, 6, 5, 7, 8, 5, 7, 9, 8, 8, 6, 8, 7, 10, 7, 10, 10, 9, 9, 8, 9, 10, 5, 10, 10, 9, 10, 10, 9, 10, 9, 7, 12, 14, 10, 9, 5, 11, 7, 13, 8, 13, 6, 9, 11, 11, 14, 15, 9, 13

Offset: 1

Zhi-Wei Sun, Dec 26 2013

nonn

Conjecture: a(n) > 0 for all n > 10.

We have verified this for n up to 50000. The conjecture implies that there are infinitely many primes of the form 2^n - 5.

			a(15) = 2 since 2^(phi(2) + phi(13)/4) - 5 = 2^4 - 5 = 11 and 2^(phi(3) + phi(12)/4) - 5 = 2^3 - 5 = 3 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A000079, A050522, A156560, A234309, A234451, A234470, A234475, A234503, A236358.

f[n_,k_]:=2^(EulerPhi[k]+EulerPhi[n-k]/4)-5
a[n_]:=Sum[If[PrimeQ[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A181704 Numbers m=2^(t-1)*(2^t-5), where 2^t-5 is prime.

Original entry on oeis.org

12, 88, 1888, 32128, 521728, 8378368, 34359083008, 549753192448, 2251799645913088, 9223372026117357568, 2361183241263023915008, 2596148429267413634121263069790208, 2722258935367507707522529418717050175488

Offset: 1

Vladimir Shevelev, Nov 06 2010

nonn

All these numbers are in A181595 because their abundance is 4, a proper divisor of m.

Cf. A059608, A156560, A181595, A181701, A000396, A181703, A045769

Rest[2^(#-1) (2^#-5)&/@(Round[N[Log[#+5]/Log[2]]]&/@Select[Table[2^t-5,{t,120}],PrimeQ])] (* Harvey P. Dale, Dec 16 2010 *)

571728 replaced with 521728 by R. J. Mathar, Dec 05 2010

A238751 Lesser prime of third Mersenne prime pair {2^m - 5, 5*2^m - 1}.

Original entry on oeis.org

11, 251, 1019, 4091, 65531, 4294967291

Offset: 1

Ilya Lopatin and Juri-Stepan Gerasimov, Mar 04 2014

By comparing A059608 and A001770, the next term, if it exists, is larger than 2^350028. - Giovanni Resta, Mar 06 2014
Lesser prime of Mersenne prime pair of order k and of the form {2^m - (2k - 1), (2k - 1)*2^m - 1}:
for order k = 1: 3, 7, 31, 127, 8191, 131071, ... (Mersenne primes A000668)
for order k = 2: 5, 13, 61, ...
for order k = 3: 11, 251, 1019, 4091, 655531, 4294967291, ... (this sequence)
for order k = 4:
for order k = 5: 2097143, ...
for order k = 6: 3, ...
for order k = 7:
for order k = 8: 17, 1009, 16369, ...
for order k = 9: 47, 65519, 1048559, 68719476719, ...
for order k = 10: 13, 2097133, ...
for order k = 11: 107, 8171, ...
for order k = 12: 41, 233, 4073, ...
for order k = 13: 487, ...
for order k = 14: 5, 229, 997, ...
for order k = 15: 97, ...

			11 is in this sequence because Mersenne prime pair {2^4-(2*3-1) = 11, (2*3-1)*2^4-1 = 79} where 3 is order and 11 is lesser prime (for m = 4).

Cf. A237422, A238694, A238749, A059608, A001770, A156560, A050522.

2^Select[Range[1000], PrimeQ[2^# - 5] && PrimeQ[5*2^# - 1] &] - 5 (* Giovanni Resta, Mar 06 2014 *)

Numbers 2^m - 5 for m belonging to the intersection of A001770 and A059608. - Max Alekseyev, Feb 20 2024

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A067702 Numbers k such that sigma(k) == 0 (mod k+2).

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A234504 Number of ways to write n = k + m with k > 0 and m > 0 such that 2^(phi(k) + phi(m)/4) - 5 is prime, where phi(.) is Euler's totient function.

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A181704 Numbers m=2^(t-1)*(2^t-5), where 2^t-5 is prime.

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A238751 Lesser prime of third Mersenne prime pair {2^m - 5, 5*2^m - 1}.

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