A045768 -id:A045768 - cp's OEIS frontend

A050414 Numbers k such that 2^k - 3 is prime.

3, 4, 5, 6, 9, 10, 12, 14, 20, 22, 24, 29, 94, 116, 122, 150, 174, 213, 221, 233, 266, 336, 452, 545, 689, 694, 850, 1736, 2321, 3237, 3954, 5630, 6756, 8770, 10572, 14114, 14400, 16460, 16680, 20757, 26350, 30041, 34452, 36552, 42689, 44629, 50474, 66422, 69337, 116926, 119324, 123297, 189110, 241004, 247165, 284133, 354946, 394034, 702194, 750740, 840797, 1126380, 1215889, 1347744, 1762004, 2086750

Offset: 1

Jud McCranie, Dec 22 1999

nonn

With 65 known primes corresponding to k < 1762005, these primes appear to be more common than Mersenne primes.  Of course at this time, the larger terms correspond only to probable primes. - Paul Bourdelais, Feb 04 2012
The numbers 2^k-3 and 2^k-1 are both primes for k = 3, 5, ? The lesser number 2^p-3 is prime for primes p = 3, 5, 29, 233, 42689, 69337, ... (see A283266). - Thomas Ordowski, Sep 18 2015
The terms a(43)-a(49) were found by Paul Underwood, a(50)-a(51) found by M. Frind and P. Underwood, a(52) found by Gary Barnes, a(53)-a(58) found by M. Frind and P. Underwood, and a(59)-a(66) found by Paul Bourdelais (see link Henri Lifchitz and Renaud Lifchitz). - Elmo R. Oliveira, Dec 02 2023

			k = 22, 2^22 - 3 = 4194301 is prime.
k = 24, 2^24 - 3 = 16777213 is prime.

Keith Conrad, Square patterns and infinitude of primes, University of Connecticut, 2019.
Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n-3, PRP Top Records.

Cf. A045768, A050415, A057732 (numbers k such that 2^k + 3 is prime).
For prime terms see A283266.
Sequences of numbers k such that 2^k - d is prime: A000043 (d=1), A050414 (d=3), A059608 (d=5), A059609 (d=7), A059610 (d=9), A096817 (d=11), A096818 (d=13), A059612 (d=15), A059611 (d=17), A096819 (d=19), A096820 (d=21), A057220 (d=23), A356826 (d=29).

Do[ If[ PrimeQ[ 2^n -3 ], Print[n]], { n, 1, 15000 }]

for(n=2, 10^5, if(ispseudoprime(2^n-3), print1(n, ", "))) \\ Felix Fröhlich, Jun 23 2014

More terms from Robert G. Wilson v, Sep 15 2000
More terms from Andrey V. Kulsha, Feb 11 2001
a(40) verified with 20 iterations of Miller-Rabin test, from Dmitry Kamenetsky, Jul 12 2008
a(41) a new PRP term, from Serge Batalov, Oct 20 2008
Corrected and extended by including two smaller (apparently known) PRP and 16 larger terms from PRP Top Records of this form, all discovered by M. Frind & P. Underwood, Gary Barnes, Oct 20 2008
a(59)-a(60) discovered by Paul Bourdelais, Mar 26 2012
a(61)-a(63) discovered by Paul Bourdelais, Jun 18 2019
a(64) discovered by Paul Bourdelais, Jul 16 2019
a(65) discovered by Paul Bourdelais, Apr 20 2020
a(66) discovered by Paul Bourdelais, May 28 2020

A054024 Sum of the divisors of n reduced modulo n.

Original entry on oeis.org

0, 1, 1, 3, 1, 0, 1, 7, 4, 8, 1, 4, 1, 10, 9, 15, 1, 3, 1, 2, 11, 14, 1, 12, 6, 16, 13, 0, 1, 12, 1, 31, 15, 20, 13, 19, 1, 22, 17, 10, 1, 12, 1, 40, 33, 26, 1, 28, 8, 43, 21, 46, 1, 12, 17, 8, 23, 32, 1, 48, 1, 34, 41, 63, 19, 12, 1, 58, 27, 4, 1, 51, 1, 40, 49, 64, 19, 12, 1, 26, 40

Offset: 1

Asher Auel, Jan 19 2000

If a(n) = 0, then n is a multiply-perfect number (A007691). - Alonso del Arte, Mar 30 2014

			a(12) = 4 because sigma(12) = 28 and 28 == 4 (mod 12).
a(13) = 1 because 13 is prime.
a(14) = 10 because sigma(14) = 24 and 24 == 10 (mod 14).

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from T. D. Noe, terms 1001..20000 from Alois P. Heinz).
Walter Nissen, Abundancy : Some Resources.

Cf. A000203 (sigma), A005114 (untouchable numbers), A007691 (positions of 0's), A045768, A045769, A088834, A045770, A076496, A159907.

a054024 n = mod (a000203 n) n  -- Reinhard Zumkeller, Mar 23 2013

with(numtheory): seq(sigma(i) mod i, i=1..100);

Table[Mod[DivisorSigma[1, n], n], {n, 80}] (* Alonso del Arte, Mar 30 2014 *)

a(n)=sigma(n)%n \\ Charles R Greathouse IV, Nov 04 2014

a(n) = sigma(n) mod n.

a(p) = 1 for p prime.

A045770 Numbers k such that sigma(k) == 8 (mod k).

Original entry on oeis.org

1, 7, 10, 49, 56, 368, 836, 11096, 17816, 45356, 77744, 91388, 128768, 254012, 388076, 2087936, 2291936, 13174976, 29465852, 35021696, 45335936, 120888092, 260378492, 381236216, 775397948, 3381872252, 4856970752, 6800228816, 8589344768, 44257207676, 114141404156, 1461083549696, 1471763808896, 2199013818368

Offset: 1

Dan Hoey

nonn

Every number of the form 2^(j-1)*(2^j - 9), where 2^j - 9 is prime, is a term. - Jon E. Schoenfield, Jun 02 2019

If m is a term of A045768 with gcd(m,3) = 1 and sigma(m) = 3*q*m + 2 for some integer q, then 3*m is a term of this sequence since sigma(3*m) = 4*q*(3*m) + 8. Some other large terms: 36893488108764397568, 877615520070055755776, 1700388548189538291286016, 85954979333046510417991676, 2081228720695521934665574252544. - Max Alekseyev, May 25 2025

Max Alekseyev, Table of n, a(n) for n = 1..41 (first 36 terms from Jud McCranie)

Cf. A054024, A045768, A045769, A088834, A076496.

q:= k-> nops(map(x-> x mod k, {8, numtheory[sigma](k)}))=1:
select(q, [$1..100000])[];  # Alois P. Heinz, Apr 07 2025

Select[Range[1000000], Mod[DivisorSigma[1, #] - 8, #] == 0 &] (* Pontus von Brömssen, Apr 07 2025 *)

isok(k) = Mod(sigma(k),k) == 8; \\ Pontus von Brömssen, Apr 07 2025

a(18)-a(26) from T. D. Noe, Apr 06 2011
Initial term 1 added and a(27)-a(31) from Donovan Johnson, Mar 01 2012
a(32)-a(34) from Giovanni Resta, Apr 02 2014
Term a(2)=7 inserted by Pontus von Brömssen, Apr 07 2025

A050415 Primes of the form 2^k - 3.

Original entry on oeis.org

5, 13, 29, 61, 509, 1021, 4093, 16381, 1048573, 4194301, 16777213, 536870909, 19807040628566084398385987581, 83076749736557242056487941267521533, 5316911983139663491615228241121378301, 1427247692705959881058285969449495136382746621

Offset: 1

Jud McCranie, Dec 22 1999

nonn

If p = 2^k - 3 is in this sequence, then p*2^(k-1) is abundant with abundance 2. - Claude Morin, Feb 01 2007

Equivalently, primes which give a prime number when 0's and 1's are interchanged in their binary representation; note that the resulting prime is always 10_2 = 2_10 (see A347476). - Bernard Schott, Nov 14 2021

Vincenzo Librandi, Table of n, a(n) for n = 1..30
Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025. See p. 12.

Cf. A045768, A050414.

Subsequence of A347476.

[a: n in [1..200] | IsPrime(a) where a is 2^n-3]; // Vincenzo Librandi, Dec 08 2011

lst={};Do[p=2^n-3;If[PrimeQ[p],AppendTo[lst,p]],{n,0,5*5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 27 2009 *)
Select[Table[2^n-3,{n,1,400}],PrimeQ] (* Vincenzo Librandi, Dec 08 2011 *)

for(n=3,999,if(ispseudoprime(t=2^n-3),print1(t", "))) \\ Charles R Greathouse IV, Jul 02 2013

a(n) = 2^A050414(n) - 3.

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

Original entry on oeis.org

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

A088831 Numbers k whose abundance is 2: sigma(k) - 2k = 2.

Original entry on oeis.org

20, 104, 464, 650, 1952, 130304, 522752, 8382464, 134193152, 549754241024, 8796086730752, 140737463189504, 144115187270549504

Offset: 1

Labos Elemer, Oct 28 2003

A subset of A045768.
If 2^k-3 is prime (k is a term of A050414) then 2^(k-1)*(2^k-3) is in the sequence; this fact is a result of the following interesting theorem that I have found. Theorem: If j is an integer and 2^k-(2j+1) is prime then 2^(k-1)*(2^k-(2j+1)) is a solution of the equation sigma(x)=2(x+j). - Farideh Firoozbakht, Feb 23 2005
Note that the fact "if 2^p-1 is prime then 2^(p-1)*(2^p-1) is a perfect number" is also a trivial result of this theorem. All known terms of this sequence are of the form 2^(k-1)*(2^k-3) where 2^k-3 is prime. Conjecture: There are no terms of other forms. So the next terms of this sequence are likely 549754241024, 8796086730752, 140737463189504, 144115187270549504, 2^93*(2^94-3), 2^115*(2^116-3), 2^121*(2^122-3), 2^149*(2^150-3), etc. - Farideh Firoozbakht, Feb 23 2005
The conjecture in the previous comment is incorrect. The first counterexample is 650, which has factorization 2*5^2*13. - T. D. Noe, May 10 2010
a(11) > 10^12. - Donovan Johnson, Dec 08 2011
a(12) > 10^13. - Giovanni Resta, Mar 29 2013
a(14) > 10^18. - Hiroaki Yamanouchi, Aug 23 2018
Any term x of this sequence can be combined with any term y of A191363 to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2, which is a necessary (but not sufficient) condition for two numbers to be amicable. - Timothy L. Tiffin, Sep 13 2016
Is there any odd term in this sequence? - Jenaro Tomaszewski, Jan 06 2021
If there exists any odd term in this sequence, it must be weird, so it must exceed 10^28. - Alexander Violette, Jan 02 2022

			Abundances of terms in A045768: {-1,2,2,2,2,2,2,2,2,2} so 1 is not here.

Singh, S. Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem. New York: Walker, p. 13, 1997.
Guy, R. K. "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers." Sec. B2 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 45-53, 1994.

P. Hagis and G. L. Cohen, Some Results Concerning Quasiperfect Numbers, J. Austral. Math. Soc. Ser. A 33, 275-286, 1982.
Tyler Ross, A Perfect Number Generalization and Some Euclid-Euler Type Results, Journal of Integer Sequences, Vol. 27 (2024), Article 24.7.5. See p. 3.

Cf. A033880, A045768, A050414, A191363 (deficiency 2).

Select[Range[10^6], DivisorSigma[1, #] - 2 # == 2 &] (* Michael De Vlieger, Feb 25 2017 *)

is(n)=sigma(n)==2*n+2 \\ Charles R Greathouse IV, Feb 21 2017

Solutions to sigma(x)-2*x = 2.

One more term from Farideh Firoozbakht, Feb 23 2005
Comment and example corrected by T. D. Noe, May 10 2010
a(10) from Donovan Johnson, Dec 08 2011
a(11) from Giovanni Resta, Mar 29 2013
a(12) from Jud McCranie, Jun 18 2017
a(13) from Hiroaki Yamanouchi, Aug 23 2018

A088832 Numbers k whose abundance is 4: sigma(k) - 2*k = 4.

Original entry on oeis.org

12, 70, 88, 1888, 4030, 5830, 32128, 521728, 1848964, 8378368, 34359083008, 66072609790, 549753192448, 259708613909470, 2251799645913088, 9223372026117357568

Offset: 1

Labos Elemer, Oct 28 2003

If 2^m-5 is prime (A059608) then n=2^(m-1)*(2^m-5) is in the sequence. 12, 88, 1888, 32128, 521728, 8378368 & 34359083008 are such terms. See comments in A088831. - Farideh Firoozbakht, Feb 15 2008
Any term x of this sequence can be combined with any term y of A125246 to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2, which is a necessary (but not sufficient) condition for two numbers to be amicable. - Timothy L. Tiffin, Sep 13 2016
Also contains 865268370658615254581248 = 2^23 * 16823249 * 6131278669. - Max Alekseyev, May 29 2025

			Abundances of terms in A045769: {-5,4,4,4,4,4,4,4,4,4} so A045769(1)=9 is not here.

Subsequence of A045769.

Cf. A033880, A045768, A088830, A059608, A125246 (deficiency 4).

Do[If[DivisorSigma[1,n]==2n+4,Print[n]],{n,650000000}] (* Farideh Firoozbakht, Feb 15 2008 *)

is(n)=sigma(n)==2*n+4 \\ Charles R Greathouse IV, Feb 21 2017

Solutions to sigma(x)-2*x=4.

One more terms from Farideh Firoozbakht, Feb 15 2008
a(11)-a(12) from Donovan Johnson, Dec 23 2008
a(13) from Donovan Johnson, Dec 08 2011
a(14)-a(15) from Hiroaki Yamanouchi, Aug 23 2018
a(16) from Max Alekseyev, May 29 2025

A181598 Numbers m with divisor 8 | m and abundance sigma(m)-2*m = 8.

Original entry on oeis.org

56, 368, 11096, 17816, 77744, 128768, 2087936, 2291936, 13174976, 35021696, 45335936, 381236216, 4856970752, 6800228816, 8589344768, 1461083549696, 1471763808896, 2199013818368, 19502341651712, 118123076415296, 933386556194816, 144141575952121856, 417857739454939136

Offset: 1

Vladimir Shevelev, Nov 01 2010

nonn

a(19) > 10^13. - Giovanni Resta, Apr 02 2014

Cf. A008590, A088833, A097498, A181595, A181596, A181597, A118372, A045768, A000396, A005101, A153501, A005820.

isok(n) = !(n % 8) && (sigma(n) - 2*n == 8); \\ Michel Marcus, Feb 08 2016

A088833 INTERSECT A008590. - R. J. Mathar, Nov 04 2010

Definition rephrased by R. J. Mathar, Nov 04 2010
a(16)-a(17) from Donovan Johnson, Dec 08 2011
a(18) from Giovanni Resta, Apr 02 2014
a(19)-a(23) from the b-file at A088833 added by Amiram Eldar, Mar 11 2024

A076496 Numbers k such that sigma(k) == 12 (mod k).

Original entry on oeis.org

1, 6, 11, 24, 30, 42, 54, 66, 78, 102, 114, 121, 138, 174, 186, 222, 246, 258, 282, 304, 318, 354, 366, 402, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 678, 762, 780, 786, 822, 834, 894, 906, 942, 978, 1002, 1038, 1074, 1086, 1146, 1158, 1182, 1194, 1266, 1338

Offset: 1

Labos Elemer, Oct 21 2002

nonn

			6*p is a solution if p > 3 is prime, since sigma(6*p) = 1 + 2 + 3 + 6 + p + 2*p + 3*p + 6*p = 12*(p+1) = 2*6*p + 12 = 2*k + 12. These are "regular" solutions. Also k = 121, 304 are "singular" solutions. See other remainders in cross-references.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000, corrected by _Sidney Cadot_, Feb 05 2023

Cf. A054024, A045768, A045769, A045770, A076495, A088834.

Cf. A141545 (a subsequence).

Select[Range[2000], Mod[DivisorSigma[1, #] - 12, #] == 0 &] (* Vincenzo Librandi, Mar 11 2014, corrected by Amiram Eldar, Jan 04 2023 *)

isok(k) = Mod(sigma(k), k) == 12; \\ Michel Marcus, Jan 04 2023

Initial term 1 added by Vincenzo Librandi, Mar 11 2014

Terms 6 and 11 inserted by Michel Marcus, Jan 04 2023

A045769 Numbers k such that sigma(k) == 4 (mod k).

Original entry on oeis.org

1, 3, 9, 12, 70, 88, 1888, 4030, 5830, 32128, 521728, 1848964, 8378368, 34359083008, 66072609790, 549753192448, 259708613909470, 2251799645913088, 9223372026117357568

Offset: 1

Dan Hoey

Every number of the form 2^(j-1)*(2^j - 5), where 2^j - 5 is prime, is a term. See A059608. - Jon E. Schoenfield, Jun 02 2019

Contains subsequence A088832.

Cf. A054024, A045768, A059608, A088834, A045770, A076496.

isok(k) = Mod(sigma(k), k) == 4; \\ Michel Marcus, Jan 04 2023

a(13) from Harvey P. Dale, Mar 20 2011
Initial term 1 inserted and a(14)-a(16) from Donovan Johnson, Mar 01 2012
Term 3 inserted by Michel Marcus, Jan 04 2023
a(18) from Jon E. Schoenfield confirmed, and a(17), a(19) added by Max Alekseyev, Jun 08 2025

cp's OEIS Frontend

A050414 Numbers k such that 2^k - 3 is prime.

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A054024 Sum of the divisors of n reduced modulo n.

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A045770 Numbers k such that sigma(k) == 8 (mod k).

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A050415 Primes of the form 2^k - 3.

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

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A088831 Numbers k whose abundance is 2: sigma(k) - 2k = 2.

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A088832 Numbers k whose abundance is 4: sigma(k) - 2*k = 4.

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