A275701 -id:A275701 - cp's OEIS frontend

A275996 Numbers n whose abundance is 64: sigma(n) - 2n = 64.

108, 220, 6808, 8968, 14008, 24448, 66928, 552568, 786208, 1020568, 5303488, 8229568, 10001848, 133685248, 499722448, 2608895488, 4733164768, 7163795488, 13707973408, 14468025568, 16122444736, 27339731968, 34351218688, 34672397728, 35371084288, 69657461248

Offset: 1

Timothy L. Tiffin, Aug 16 2016

Any term x = a(m) of this sequence can be used with any term y of A275997 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.
The smallest amicable pair is (220, 284) = (a(2), A275997(2)) = (A063990(1), A063990(2)), where 284 - 220 = 64 is the abundance of 220 and the deficiency of 284.
The amicable pair (66928, 66992) = (a(7), A275997(11)) = (A063990(18), A063990(19)), and 66992 - 66928 = 64 is the abundance of 66928 and the deficiency of 66992.

			a(1) = 108, since sigma(108) - 2*108 = 280 - 216 = 64.

Max Alekseyev, Table of n, a(n) for n = 1..63

Subsequence of A005101.

Cf. A002025, A063990, A275997, A088831, A088832, A088833, A141547, A175989, A275701, A066539, A259180.

isok(n) = sigma(n) - 2*n == 64; \\ Michel Marcus, Dec 30 2016

a(14)-a(15) from Michel Marcus, Dec 30 2016
a(16)-a(21) from Lars Blomberg, Jan 12 2017
Terms a(22) onward from Max Alekseyev, Aug 27 2025

A275702 Numbers n whose deficiency is 26: 2n - sigma(n) = 26.

Original entry on oeis.org

58, 75, 328, 850, 1210, 2848, 35968, 537088, 549768921088, 8796145451008

Offset: 1

Timothy L. Tiffin, Aug 05 2016

Any term x = a(m) can be combined with any term y = A275701(n) to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2.  Although this property is a necessary condition for two numbers to be amicable, it is not a sufficient one.  So far, these two sequences have produced only one amicable pair: (1210,1184) = (a(5),A275701(2)) = (A063990(4),A063990(3)).  If more are ever found, then they will also exhibit x-y = 26.
Notice that:
a(1) =     58 =   29*  2 = (4^1+25)*(4^1)/2
a(3) =    328 =   41*  8 = (4^2+25)*(4^2)/2
a(6) =   2848 =   89* 32 = (4^3+25)*(4^3)/2
a(7) =  35968 =  281*128 = (4^4+25)*(4^4)/2
a(8) = 537088 = 1049*512 = (4^5+25)*(4^5)/2.
If p = 4^k+25 is prime and n = p*(p-25)/2, then it is not hard to show that 2*n - sigma(n) = 26. The values of k in A204388 will guarantee that p is prime (A104072). Similarly, if q = 2*4^k+25 is prime and n = q*(q-25)/2, then 2*n - sigma(n) = 26. However, q will never be prime since it will always be divisible by 3: 2*4^k+25 == (2*1^k+25) mod 3 == 27 mod 3 == 0 mod 3. So, the following values will be in this sequence and provide upper bounds for the next seven terms:
(4^10+25)*(4^10)/2 = 549768921088 >= a(9)
(4^11+25)*(4^11)/2 = 8796145451008 >= a(10)
(4^17+25)*(4^17)/2 = 147573952804424777728 >= a(11)
(4^35+25)*(4^35)/2 = 696898287454081973187748591279228938354688 >= a(12)
(4^46+25)*(4^46)/2 = 12259964326927110866866776279099475433218926722425028608 >= a(13)
(4^56+25)*(4^56)/2 = 13479973333575319897333507543509880240529303896615642871755920375808 >= a(14)
(4^59+25)*(4^59)/2 = 55213970774324510299478046898216207773446358605225195265697257166471168 >= a(15).
The rightmost digit of n = p*(p-25)/2 will always be 8. [Proof: If k is odd, then 4^k+25 == 9 mod 10 and (4^k)/2 == 2 mod 10, which implies that p*(p-25)/2 == 8 mod 10. If k is even, then 4^k+25 == 1 mod 10 and (4^k)/2 == 8 mod 10, which implies that p*(p-25)/2 == 8 mod 10.]
a(10) > 2.3*10^12. - Giovanni Resta, Aug 07 2016
a(11) > 10^18. - Hiroaki Yamanouchi, Aug 21 2018

			a(1) = 58, since 2*58-sigma(58) = 116-90 = 26.
a(2) = 75, since 2*75-sigma(75) = 150-124 = 26.
a(3) = 328, since 2*328-sigma(328) = 656-630 = 26.

D. Alpern, Factorization using the Elliptic Curve Method.

Cf. A033879, A063990, A104072, A204388, A275701 (abundance 26).

[n: n in [1..2*10^6] | (2*n-SumOfDivisors(n)) eq 26]; // Vincenzo Librandi, Aug 06 2016

Select[Range[10^6], 2 # - (DivisorSigma[1, #]) == 26 &] (* Vincenzo Librandi, Aug 06 2016 *)

is(n) = 2*n-sigma(n)==26 \\ Felix Fröhlich, Aug 06 2016

a(9) from Giovanni Resta, Aug 07 2016

a(10) from Hiroaki Yamanouchi, Aug 21 2018

A275749 Prime numbers of the form 2*4^k - 27.

Original entry on oeis.org

5, 101, 524261, 8388581

Offset: 1

Timothy L. Tiffin, Aug 07 2016

nonn

Values of the exponent k are given in A275767, and every exponent (except for the first one) is odd. Consequently, after a(1) = 5, the rightmost digit of each term in this sequence will be 1.
As seen in the link below, a(5) = 2*4^291 - 27 > 3.1658 * 10^175. As a result of the recent extensions to A275767 by Vincenzo Librandi,
a(6) = 2*4^1263 - 27 > 5.0442 * 10^760
a(7) = 2*4^2661 - 27 > 2.4136 * 10^1602
a(8) = 2*4^3165 - 27 > 6.6206 * 10^1905
a(9) > 2*4^5000 - 27 > 3.9901 * 10^3010.
These primes a(m) can be used to generate numbers having abundance 26. The formula a(m)*(a(m)+27)/2 produces some of the terms in A275701.

			a(1) = 2*4^A275767(1) - 27 = 2*4^2  - 27 =      32 - 27 =       5.
a(2) = 2*4^A275767(2) - 27 = 2*4^3  - 27 =     128 - 27 =     101.
a(3) = 2*4^A275767(3) - 27 = 2*4^9  - 27 =  524288 - 27 =  524261.
a(4) = 2*4^A275767(4) - 27 = 2*4^11 - 27 = 8388608 - 27 = 8388581.

Timothy L. Tiffin, Table of n, a(n) for n = 1..5
D. Alpern, Factorization using the Elliptic Curve Method.

Cf. A275701, A275750, A275767.

Select[2*4^Range[2, 200] - 27, PrimeQ] (* Michael De Vlieger, Aug 08 2016 *)

a(n) = 2*4^A275767(n) - 27.

A275997 Numbers k whose deficiency is 64: 2k - sigma(k) = 64.

Original entry on oeis.org

134, 284, 410, 632, 1292, 1628, 4064, 9752, 12224, 22712, 66992, 72944, 403988, 556544, 2161664, 2330528, 8517632, 13228352, 14563832, 15422912, 20732792, 89472632, 134733824, 150511232, 283551872, 537903104, 731670272, 915473696, 1846850576, 2149548032, 2159587616

Offset: 1

Timothy L. Tiffin, Aug 16 2016

Any term x = a(m) in this sequence can be used with any term y in A275996 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.
The smallest amicable pair is (220, 284) = (A275996(2), a(2)) = (A063990(1), A063990(2)), where 284 - 220 = 64 is the abundance of 220 and the deficiency of 284.
The amicable pair (66928, 66992) = (A275996(7), a(11)) = (A063990(18), A063990(19)), where 66992 - 66928 = 64 is the deficiency of 66992 and the abundance of 66928.
Contains numbers 2^(k-1)*(2^k + 63) whenever 2^k + 63 is prime. - Max Alekseyev, Aug 27 2025

			a(1) = 134, since 2*134 - sigma(134) = 268 - 204 = 64.

Max Alekseyev, Table of n, a(n) for n = 1..89

Deficiency k: A191363 (k=2), A125246 (k=4), A141548 (k=6), A125247 (k=8), A101223 (k=10), A141549 (k=12), A141550 (k=14), A125248 (k=16), A223608 (k=18), A223607 (k=20), A223606 (k=22), A385255(k=24), A275702 (k=26), A387352 (k=32).
Abundance k: A088831 (k=2), A088832 (k=4), A087167 (k=6), A088833 (k=8), A223609 (k=10), A141545 (k=12), A141546 (k=14), A141547 (k=16), A223610 (k=18), A223611 (k=20), A223612 (k=22), A223613 (k=24), A275701 (k=26), A175989 (k=32), A275996 (k=64), A292626 (k=128).
Cf. A002025, A063990.

Select[Range[10^7], 2 # - DivisorSigma[1, #] == 64 &] (* Michael De Vlieger, Jan 10 2017 *)

isok(n) = 2*n - sigma(n) == 64; \\ Michel Marcus, Dec 30 2016

a(23)-a(31) from Jinyuan Wang, Mar 02 2020

A292626 Numbers k whose abundance is 128: sigma(k) - 2*k = 128.

Original entry on oeis.org

860, 5336, 6536, 9656, 16256, 55796, 70864, 98048, 361556, 776096, 2227616, 4145216, 4498136, 4632896, 8124416, 13086016, 34869056, 38546576, 150094976, 172960856, 196066256, 962085536, 1080008576, 1733780336, 1844788112, 2143256576, 2531343872, 2986104064, 9677743616, 11276687456, 17104503968, 20680182272, 21568135616

Offset: 1

Fabian Schneider, Sep 20 2017

Max Alekseyev, Table of n, a(n) for n = 1..87

Subsequence of A259174.
Deficiency k: A191363 (k=2), A125246 (k=4), A141548 (k=6), A125247 (k=8), A101223 (k=10), A141549 (k=12), A141550 (k=14), A125248 (k=16), A223608 (k=18), A223607 (k=20), A223606 (k=22), A385255(k=24), A275702 (k=26), A387352 (k=32), A275997 (k=64).
Abundance k: A088831 (k=2), A088832 (k=4), A087167 (k=6), A088833 (k=8), A223609 (k=10), A141545 (k=12), A141546 (k=14), A141547 (k=16), A223610 (k=18), A223611 (k=20), A223612 (k=22), A223613 (k=24), A275701 (k=26), A175989 (k=32), A275996 (k=64).

fQ[n_] := DivisorSigma[1, n] == 2 n + 128; Select[ Range@ 10^8, fQ] (* Robert G. Wilson v, Nov 19 2017 *)

isok(n) = sigma(n) - 2*n == 128; \\ Michel Marcus, Sep 20 2017

a(9)-a(18) from Michel Marcus, Sep 20 2017
a(19)-a(24), a(26), a(29)-a(30), a(33) from Robert G. Wilson v, Nov 20 2017
Missing terms a(25), a(27)-a(28), a(31)-a(32) inserted and terms a(34) onward added by Max Alekseyev, Aug 30 2025

A275750 Prime numbers of the form 4^k - 27.

Original entry on oeis.org

37, 229, 997, 1048549, 4194277, 67108837, 1125899906842597, 72057594037927909, 288230376151711717, 1361129467683753853853498429727072845797, 1393796574908163946345982392040522594123749, 1725436586697640946858688965569256363112777243042596638790631055949797

Offset: 1

Timothy L. Tiffin, Aug 07 2016

nonn

Values of the exponent k are given in A274519. If the exponent is odd, then the rightmost digit of a(n) will be 7. If the exponent is even, then the rightmost digit of a(n) will be 9.
As a result of the recent extensions to A274519 by Vincenzo Librandi,
a(13) = 4^305  - 27 > 4.2491 * 10^183
a(14) = 4^515  - 27 > 1.1505 * 10^310
a(15) = 4^2029 - 27 > 3.7994 * 10^1221
a(16) = 4^2393 - 27 > 5.3648 * 10^1440
a(17) = 4^2605 - 27 > 2.3242 * 10^1568
a(18) = 4^3530 - 27 > 1.8696 * 10^2125
a(19) = 4^4036 - 27 > 8.2058 * 10^2429
a(20) = 4^4750 - 27 > 6.0947 * 10^2859
a(21) > 4^5000 - 27 > 1.9950 * 10^3010.
These primes a(m) can be used to generate numbers having abundance 26. The formula a(m)*(a(m)+27)/2 produces some of the terms in A275701.

			a(1) = 4^A274519(1) - 27 = 4^3  - 27 =       64 - 27 =       37.
a(2) = 4^A274519(2) - 27 = 4^4  - 27 =      256 - 27 =      229.
a(3) = 4^A274519(3) - 27 = 4^5  - 27 =     1024 - 27 =      997.
a(4) = 4^A274519(4) - 27 = 4^10 - 27 =  1048576 - 27 =  1048549.
a(5) = 4^A274519(5) - 27 = 4^11 - 27 =  4194304 - 27 =  4194277.
a(6) = 4^A274519(6) - 27 = 4^13 - 27 = 67108864 - 27 = 67108837.

D. Alpern, Factorization using the Elliptic Curve Method.

Cf. A274519, A275701, A275749.

Select[4^Range[3, 120] - 27, PrimeQ] (* Michael De Vlieger, Aug 08 2016 *)

a(n) = 4^A274519(n) - 27.

A385255 Numbers m whose deficiency is 24: sigma(m) - 2*m = -24.

Original entry on oeis.org

124, 9664, 151115727458150838697984

Offset: 1

Max Alekseyev, Jul 29 2025

Contains numbers 2^(k-1)*(2^k + 23) for k in A057203. First three terms have this form.

Deficiency k: A191363 (k=2), A125246 (k=4), A141548 (k=6), A125247 (k=8), A101223 (k=10), A141549 (k=12), A141550 (k=14), A125248 (k=16), A223608 (k=18), A223607 (k=20), A223606 (k=22), A275702 (k=26).
Abundance k: A088831 (k=2), A088832 (k=4), A087167 (k=6), A088833 (k=8), A223609 (k=10), A141545 (k=12), A141546 (k=14), A141547 (k=16), A223610 (k=18), A223611 (k=20), A223612 (k=22), A223613 (k=24), A275701 (k=26).
Cf. A057203.

A387352 Numbers m with deficiency 32: sigma(m) - 2*m = -32.

Original entry on oeis.org

250, 376, 1276, 12616, 20536, 396916, 801376, 1297312, 8452096, 33721216, 40575616, 59376256, 89397016, 99523456, 101556016, 150441856, 173706136, 269096704, 283417216, 500101936, 1082640256, 1846506832, 15531546112, 34675557856, 136310177392, 136783784608

Offset: 1

Max Alekseyev, Aug 27 2025

Contains numbers 2^(k-1)*(2^k + 31) for k in A247952.

Max Alekseyev, Table of n, a(n) for n = 1..54

Deficiency k: A191363 (k=2), A125246 (k=4), A141548 (k=6), A125247 (k=8), A101223 (k=10), A141549 (k=12), A141550 (k=14), A125248 (k=16), A223608 (k=18), A223607 (k=20), A223606 (k=22), A385255(k=24), A275702 (k=26), A275997 (k=64).
Abundance k: A088831 (k=2), A088832 (k=4), A087167 (k=6), A088833 (k=8), A223609 (k=10), A141545 (k=12), A141546 (k=14), A141547 (k=16), A223610 (k=18), A223611 (k=20), A223612 (k=22), A223613 (k=24), A275701 (k=26), A175989 (k=32), A275996 (k=64), A292626 (k=128).
Cf. A247952.

A063788 Numbers k such that sigma(k) = 2k + Omega(k), where Omega(n) is the number of prime divisors of n (with repetition).

Original entry on oeis.org

18, 88, 4030, 5830, 518656, 13174976, 134094848, 2146926592, 2251798907715584, 12504224434300196, 324257317741920256

Offset: 1

Jason Earls, Aug 16 2001

Includes terms 633825300114085990300727115776 and 2596148429267411760623818083663872. - Donovan Johnson, Dec 19 2008; edited by Max Alekseyev, May 27 2025

Terms a(2)-a(4) come from A088832, a(5) from A223609, a(6) and a(10) from A088833, a(7) from A141546, a(8) from A141547, a(9) from A275701, a(11) from A223611. Also includes the following terms k with Omega(k) = 56: 246434407522188377975875310632234056969345758857269346304, 15937923506379504700185810932457673797717574263217988829184, 264936582814027097239593278653623212574863771975442952634761216, 7948097484419456643668355219907727481405487440330234556835692544. - Max Alekseyev, May 27 2025

Cf. A000203 (sigma), A001222 (Omega), A088832, A223609, A088833, A141546, A141547, A275701, A223611.

for(n=1,10^8, if(sigma(n)==2*n+bigomega(n),print(n)))

Numbers k such that A000203(k) = 2k + A001222(k). - Wesley Ivan Hurt, Oct 30 2022

a(7)-a(8) from Donovan Johnson, Dec 19 2008

a(9) from Donovan Johnson confirmed and a(10)-a(11) added by Max Alekseyev, May 27 2025

cp's OEIS Frontend

A275996 Numbers n whose abundance is 64: sigma(n) - 2n = 64.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A275702 Numbers n whose deficiency is 26: 2n - sigma(n) = 26.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

A275749 Prime numbers of the form 2*4^k - 27.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A275997 Numbers k whose deficiency is 64: 2k - sigma(k) = 64.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A292626 Numbers k whose abundance is 128: sigma(k) - 2*k = 128.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A275750 Prime numbers of the form 4^k - 27.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A385255 Numbers m whose deficiency is 24: sigma(m) - 2*m = -24.

Views

Author

Keywords

Comments

Crossrefs

A387352 Numbers m with deficiency 32: sigma(m) - 2*m = -32.

Views

Author

Keywords

Comments