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 1-5 of 5 results.

A191218 Odd numbers n such that sigma(n) is congruent to 2 modulo 4.

Original entry on oeis.org

5, 13, 17, 29, 37, 41, 45, 53, 61, 73, 89, 97, 101, 109, 113, 117, 137, 149, 153, 157, 173, 181, 193, 197, 229, 233, 241, 245, 257, 261, 269, 277, 281, 293, 313, 317, 325, 333, 337, 349, 353, 369, 373, 389, 397, 401, 405, 409, 421, 425, 433, 449, 457, 461, 477
Offset: 1

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Author

Luis H. Gallardo, May 26 2011

Keywords

Comments

Exactly the numbers of the form p^{4k+1}*m^2 with p a prime congruent to 1 modulo 4 and m a positive integer coprime with p. The odd perfect numbers are all of this form.
See A228058 for the terms where m > 1. - Antti Karttunen, Apr 22 2019

Examples

			For n=3 one has a(3)=17 since sigma(17) = 18 = 4*4 +2 is congruent to 2 modulo 4

Links

Crossrefs

Subsequence of A191217.
Cf. A228058, A324898 (subsequences).
Cf. A000203, A191219, A324647, A324718, A324719.

Programs

  • Maple
    with(numtheory): genodd := proc(b) local n,s,d; for n from 1 to b by 2 do s := sigma(n);
if modp(s,4)=2 then print(n); fi; od; end;
  • Mathematica
    Select[Range[1,501,2],Mod[DivisorSigma[1,#],4]==2&] (* Harvey P. Dale, Nov 12 2017 *)
  • PARI
    forstep(n=1,10^3,2,if(2==(sigma(n)%4),print1(n,", "))) \\ Joerg Arndt, May 27 2011
  • PARI
    list(lim)=my(v=List()); forstep(e=1,logint(lim\=1,5),4, forprimestep(p=5,sqrtnint(lim,e),4, my(pe=p^e); forstep(m=1,sqrtint(lim\pe),2, if(m%p, listput(v,pe*m^2))))); Set(v) \\ Charles R Greathouse IV, Feb 16 2022

A324647 Odd numbers k such that 2*k is equal to bitwise-AND of 2*k and sigma(k).

Original entry on oeis.org

1116225, 1245825, 1380825, 2127825, 10046025, 16813125, 203753025, 252880425, 408553425, 415433025, 740361825, 969523425, 1369580625, 1612924425, 1763305425, 2018027025, 2048985225, 2286684225, 3341556225, 3915517725, 3985769025, 4051698525, 7085469825, 7520472225
Offset: 1

Views

Author

Antti Karttunen, Mar 14 2019

Keywords

Comments

If this sequence has no terms common with A324649 (A324897, A324898), or no terms common with A324727, then there are no odd perfect numbers.
First 22 terms factored:
1116225 = 3^2 * 5^2 * 11^2 * 41
1245825 = 3^2 * 5^2 * 7^2 * 113
1380825 = 3^2 * 5^2 * 19^2 * 17 [Here the unitary prime is not the largest]
2127825 = 3^2 * 5^2 * 7^2 * 193
10046025 = 3^4 * 5^2 * 11^2 * 41
16813125 = 3^2 * 5^4 * 7^2 * 61
203753025 = 3^2 * 5^2 * 7^2 * 18481
252880425 = 3^2 * 5^2 * 7^2 * 22937
408553425 = 3^2 * 5^2 * 7^2 * 37057
415433025 = 3^2 * 5^2 * 7^4 * 769
740361825 = 3^2 * 5^2 * 7^2 * 67153
969523425 = 3^4 * 5^2 * 13^2 * 2833
1369580625 = 3^2 * 5^4 * 7^2 * 4969
1612924425 = 3^2 * 5^2 * 7^2 * 146297
1763305425 = 3^2 * 5^2 * 7^2 * 159937
2018027025 = 3^2 * 5^2 * 7^2 * 183041
2048985225 = 3^2 * 5^2 * 7^2 * 185849
2286684225 = 3^2 * 5^2 * 7^2 * 207409
3341556225 = 3^2 * 5^2 * 7^2 * 303089
3915517725 = 3^4 * 5^2 * 7^2 * 39461
3985769025 = 3^4 * 5^2 * 7^2 * 40169
4051698525 = 3^2 * 5^2 * 7^2 * 367501.
Compare the above factorizations to the various constraints listed for odd perfect numbers in the Wikipedia article. However, this is NOT a subsequence of A191218 (A228058), see below.
The first terms that do not belong to A191218 are 399736269009 = (3 * 7^2 * 11 * 17 * 23)^2 and 1013616036225 = (3^2 * 5 * 13 * 1721)^2, that both occur instead in A325311. The first terms with omega(n) <> 4 are 9315603297, 60452246925, 68923392525, and 112206463425. They factor as 3^2 * 7^2 * 11^2 * 13^2 * 1033, 3^2 * 5^2 * 7^2 * 17^2 * 18973, 3^2 * 5^2 * 13^2 * 19^2 * 5021, 3^2 * 5^2 * 7^2 * 199^2 * 257. - Giovanni Resta, Apr 21 2019
From Antti Karttunen, Jan 13 2025: (Start)
Because of the "monotonic property" of bitwise-and, this is a subsequence of nondeficient numbers (A023196).
Both odd perfect numbers, and quasiperfect numbers, if such numbers exist at all, would satisfy the condition for being included in this sequence. Furthermore, any term must be either an odd square with an odd abundancy (in A156942), which subset is given in A379490 (where quasiperfect numbers must thus reside, if they exist), or be included in A228058, i.e., satisfy the Euler's criteria for odd perfect numbers.
(End)

Links

Crossrefs

Odd terms of A324652.
Cf. A191218, A228058, A318468, A324649, A324659, A324718, A324719, A324722, A324727, A324880, A324897, A324898, A325311, A379490 (square terms).

Programs

  • PARI
    for(n=1,oo,if((n%2)&&((2*n)==bitand(2*n,sigma(n))),print1(n,", ")));

Formula

{Odd k such that 2k = A318468(k)}.

Extensions

a(23)-a(24) from Giovanni Resta, Apr 21 2019

A324727 Odd numbers such that 2n is equal to A318466(n), bitor(2*n,sigma(n)), where bitor is A003986.

Original entry on oeis.org

3, 7, 15, 21, 31, 55, 57, 63, 93, 105, 111, 127, 171, 189, 201, 213, 215, 217, 231, 237, 249, 253, 255, 315, 351, 357, 363, 369, 381, 393, 447, 465, 469, 473, 483, 489, 497, 501, 511, 651, 705, 747, 759, 789, 813, 831, 833, 879, 889, 895, 917, 959, 987, 989, 1001, 1015, 1023, 1155, 1365, 1377, 1407, 1467, 1491, 1503, 1505, 1515, 1533, 1595
Offset: 1

Views

Author

Antti Karttunen, Mar 15 2019

Keywords

Links

Crossrefs

Odd terms of A324726.
Cf. A000203, A003986, A318466, A324647, A324718, A324719.

Programs

  • Mathematica
    Select[Range[1, 2000, 2], 2*# == BitOr[2*#, DivisorSigma[1, #]] &] (* Paolo Xausa, Mar 11 2024 *)
  • PARI
    for(n=1,oo,if((n%2)&&((2*n)==bitor(2*n,sigma(n))),print1(n,", ")));

A324719 Odd numbers n for which bitor(2n,sigma(n)) = 2*bitor(n,sigma(n)-n), where bitor is bitwise-OR, A003986.

Original entry on oeis.org

3, 7, 15, 27, 31, 51, 55, 63, 111, 119, 123, 125, 127, 219, 255, 411, 447, 485, 493, 495, 505, 511, 735, 765, 771, 831, 879, 927, 959, 965, 985, 1011, 1023, 1563, 1587, 1611, 1731, 1779, 1791, 1799, 1887, 1921, 1923, 1945, 1975, 1983, 1991, 2019, 2031, 2041, 2043, 2045, 2047, 3099, 3183, 3231, 3279, 3291, 3327, 3459, 3535, 3579
Offset: 1

Views

Author

Antti Karttunen, Mar 14 2019

Keywords

Comments

Odd numbers n for which 2*A318456(n) = A318466(n).
If there are no common terms with A324718, then there are no odd perfect numbers.
The following subsequence of terms k are those with sigma(k) == 2 (mod 4): 3725, 7281, 15325, 24525, 25713, 32481, 51633, 52209, 59121, 63553, 114417, 117009, 120753, 121725, 122725, 123245, 130833, 208881, 236925, 241325, 245725, 253325, 261297, 384993, 411633, 457713, 468081, 482481, 482525, 482725, 483325, ..., and are thus present in A191218.

Links

Crossrefs

Cf. A003986, A191218, A318456, A318466, A324718, A324727.

Programs

  • Mathematica
    Select[Range[1, 10^4, 2], Block[{s = DivisorSigma[1, #]}, BitOr[2*#, s] == 2* BitOr[#, s-#]] &] (* Paolo Xausa, Mar 11 2024 *)
  • PARI
    for(n=1,oo,if((n%2) && (2*(bitor(n, sigma(n)-n))==bitor(n+n, sigma(n))),print1(n,", ")));

A324639 Numbers k such that bitand(2k,sigma(k)) = 2*bitand(k,sigma(k)-k), where bitand is bitwise-AND, A004198.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 9, 10, 16, 17, 20, 26, 28, 32, 36, 37, 38, 41, 44, 50, 64, 73, 74, 88, 98, 100, 104, 128, 130, 134, 136, 137, 149, 152, 153, 164, 172, 184, 256, 257, 261, 262, 264, 272, 277, 284, 293, 294, 304, 328, 337, 368, 392, 405, 410, 424, 442, 464, 496, 512, 520, 521, 522, 528, 529, 538, 548, 549, 550, 556, 560, 577
Offset: 1

Views

Author

Antti Karttunen, Mar 14 2019

Keywords

Comments

Numbers k for which 2*A318458(k) = A318468(k).

Links

Crossrefs

Cf. A004198, A318458, A318468.
Subsequences: A324643, A324718 (odd terms).

Programs

  • Mathematica
    Select[Range[1000], Block[{s = DivisorSigma[1, #]}, BitAnd[2*#, s] == 2* BitAnd[#, s-#]] &] (* Paolo Xausa, Mar 11 2024 *)
  • PARI
    for(n=1,oo,if( (2*(bitand(n, sigma(n)-n))==bitand(n+n, sigma(n))),print1(n,", ")));
Showing 1-5 of 5 results.

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