A292626 -id:A292626 - cp's OEIS frontend

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

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

A292558 a(n) is the smallest number k such that sigma(k) - 2k = 2^n.

Original entry on oeis.org

20, 12, 56, 550, 572, 108, 860, 952, 1232, 6328, 3708, 40540, 37072, 79288, 327260, 357112, 302000, 527296, 1764056, 6506512, 38559776, 21893248, 42257216, 167771740, 90798560, 469761208, 508198064, 490304800, 1353048560, 2951488480, 5067417200, 32648918272, 40086360272

Offset: 1

XU Pingya, Sep 19 2017

For n > 31, a(n) > 1.724 * 10^10.

a(1) = A088831(1), a(2) = A088832(1), a(3) = A088833(1), a(4) = A141547(1), a(5) = A175989(1), a(6) = A275996(1), a(7) = A292626(1). - Max Alekseyev, Aug 27 2025

			sigma(20) - 2*20 = 2^1, a(1) = 20.
sigma(108) - 2*108 = 64 = 2^6, a(6) = 108.

Max Alekseyev, Table of n, a(n) for n = 1..50

Cf. A000203, A033880, A088831, A088832, A088833, A141547, A175989, A275996.

Table[k = 1; While[Log[2, DivisorSigma[1, k] - 2k] != n, k++]; k, {n, 30}]

a(n) = my(k=1); while(sigma(k) - 2*k != 2^n, k++); k; \\ Michel Marcus, Sep 19 2017

Terms a(32) onward from Max Alekseyev, Aug 27 2025

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.

A259174 Numbers whose abundance is a power of 2.

Original entry on oeis.org

12, 20, 56, 70, 88, 104, 108, 220, 368, 464, 550, 572, 650, 748, 836, 860, 952, 992, 1232, 1504, 1672, 1888, 1952, 2140, 2392, 2744, 3708, 4030, 5336, 5830, 6328, 6536, 6808, 7192, 7304, 7544, 7912, 8968, 9656, 9820, 10184, 10792, 11096, 13496, 14008

Offset: 1

Robert G. Wilson v, Jun 20 2015

Subsequence of A005101 whose abundance is a term of A000079 except 1.
Below 35*10^8, only 236925 is odd and its abundance is 2^9.
Least terms with abundance 2^e for e = 1, 2, ... are listed in A292558.

Robert G. Wilson v, Table of n, a(n) for n = 1..482

Contains as subsequences A088831, A088832, A088833, A141547, A175989, A275996, A292626.

Cf. A005101, A292558.

fQ[n_] := IntegerQ@ Log2[DivisorSigma[1, n] - 2 n]; Select[ Range@ 15000, fQ]

isok(n) = isprimepower(sigma(n)-2*n, &p) && (p==2); \\ Michel Marcus, Mar 25 2017

cp's OEIS Frontend

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

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A292558 a(n) is the smallest number k such that sigma(k) - 2k = 2^n.

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

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A259174 Numbers whose abundance is a power of 2.

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