A088832 -id:A088832 - cp's OEIS frontend

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

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

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

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.

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

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

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

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

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

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

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