cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

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

Original entry on oeis.org

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

Views

Author

Timothy L. Tiffin, Aug 16 2016

Keywords

Comments

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.

Examples

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

Links

Crossrefs

Subsequence of A005101.
Cf. A002025, A063990, A275997, A088831, A088832, A088833, A141547, A175989, A275701, A066539, A259180.

Programs

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

Extensions

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

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

Views

Author

Timothy L. Tiffin, Aug 16 2016

Keywords

Comments

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

Examples

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

Links

Crossrefs

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.

Programs

  • Mathematica
    Select[Range[10^7], 2 # - DivisorSigma[1, #] == 64 &] (* Michael De Vlieger, Jan 10 2017 *)
  • PARI
    isok(n) = 2*n - sigma(n) == 64; \\ Michel Marcus, Dec 30 2016

Extensions

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

Views

Author

Fabian Schneider, Sep 20 2017

Keywords

Links

Crossrefs

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).

Programs

  • Mathematica
    fQ[n_] := DivisorSigma[1, n] == 2 n + 128; Select[ Range@ 10^8, fQ] (* Robert G. Wilson v, Nov 19 2017 *)
  • PARI
    isok(n) = sigma(n) - 2*n == 128; \\ Michel Marcus, Sep 20 2017

Extensions

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

Views

Author

XU Pingya, Sep 19 2017

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    Table[k = 1; While[Log[2, DivisorSigma[1, k] - 2k] != n, k++]; k, {n, 30}]
  • PARI
    a(n) = my(k=1); while(sigma(k) - 2*k != 2^n, k++); k; \\ Michel Marcus, Sep 19 2017

Extensions

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

Views

Author

Max Alekseyev, Aug 27 2025

Keywords

Comments

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

Links

Crossrefs

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.

A181601 Numbers m with divisor 32 | m and abundance sigma(m)-2*m = 32.

Original entry on oeis.org

992, 28544, 122624, 507392, 537248, 698528, 791264, 1081568, 1279136, 2279072, 5029184, 307801856, 623799776, 712023296, 11196261056, 14809750016, 34355412992, 59640734144, 340536203264, 637707589184, 1091487733184, 1473169206272, 1709840369984, 2526522709184
Offset: 1

Views

Author

Vladimir Shevelev, Nov 01 2010

Keywords

Comments

A subsequence of A175989. - R. J. Mathar, Nov 04 2010

Links

Crossrefs

Cf. A000203, A005101, A005820, A045768, A097498, A118372, A153501, A175989, A181595, A181596, A181597, A181598, A181599.

Programs

  • Mathematica
    Select[32Range[1000000],DivisorSigma[1,#]-2#==32&] (* Harvey P. Dale, Aug 16 2011 *)

Extensions

Definition rephrased, a(5)-a(11) appended - R. J. Mathar, Nov 04 2010
a(12)-a(24) from Donovan Johnson, Dec 08 2011

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

Views

Author

Robert G. Wilson v, Jun 20 2015

Keywords

Comments

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.

Links

Crossrefs

Contains as subsequences A088831, A088832, A088833, A141547, A175989, A275996, A292626.
Cf. A005101, A292558.

Programs

  • Mathematica
    fQ[n_] := IntegerQ@ Log2[DivisorSigma[1, n] - 2 n]; Select[ Range@ 15000, fQ]
  • PARI
    isok(n) = isprimepower(sigma(n)-2*n, &p) && (p==2); \\ Michel Marcus, Mar 25 2017

A181707 Numbers of the form m=2^(t-1)*(2^t-33), where 2^t-33 is prime.

Original entry on oeis.org

992, 28544, 122624, 507392, 34355412992, 8796023816192, 140737211531264, 144115179217485824, 9671406556844465630216192, 162259276829213066154002603835392, 11417981541647679048463794346093005918389141504
Offset: 1

Views

Author

Vladimir Shevelev, Nov 06 2010

Keywords

Comments

Generated by t= 6, 8, 9, 10, 18, 22, 24, 29, 42, 54, 77, 90, 102, 137,...
A subsequence of A181595 because the abundance of m is 32, and 32 divides 2^(t-1) and therefore divides m.

Crossrefs

Cf. A181595, A181701, A000396, A181703, A181704, A181705, A181706, A175989.

Extensions

Definition simplified and more terms added by R. J. Mathar, Nov 18 2010
Showing 1-8 of 8 results.

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