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.

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A053232 Numbers for which values not equal to 2 occur in the expansion of A053230.

Original entry on oeis.org

1, 2, 23, 31, 32, 38, 39, 54, 59, 67, 68, 74, 75, 84, 98, 99, 114, 128, 129, 138, 149, 157, 158, 172, 177, 197, 198, 201, 202, 228, 229, 232, 240, 241, 247, 248, 257, 262, 263, 277, 278, 281, 283, 284, 293, 307, 308, 311, 313, 314, 332, 333, 339, 340, 349, 350
Offset: 1

Views

Author

Asher Auel, Jan 10 2000

Keywords

Crossrefs

Cf. A000203, A053224, A053230, A053231, A053233, A053234, A053235, A053236, A053237.

Programs

  • Maple
    with(numtheory): f := [seq( `if`((sigma(i+1) > sigma(i)),i,print( )), i=1..5000)];
seq( `if`(f[i+1] - f[i] <> 2,i,print( )), i=1..100);

A053237 Numbers n such that both A053230(n) and A053230(n+1) = 1.

Original entry on oeis.org

1, 31, 38, 67, 74, 98, 128, 197, 228, 240, 247, 277, 283, 307, 313, 332, 339, 369, 382, 400, 413, 430, 459, 489, 502, 520, 551, 609, 622, 646, 664, 729, 759, 771, 823, 830, 843, 908, 915, 940, 969, 1038, 1057, 1086, 1117, 1124, 1148, 1206, 1247, 1266, 1290
Offset: 1

Views

Author

Asher Auel, Jan 10 2000

Keywords

Comments

Pairs of consecutive 1's occur uncommonly often in A053230.

Crossrefs

Cf. A000203, A053224, A053230, A053231, A053232, A053233, A053234, A053235, A053236.

Programs

  • Maple
    with(numtheory): f := [seq( `if`((sigma(i+1) > sigma(i)),i,print( )), i=1..5000)];
seq(`if`((f[i+2]-f[i+1]=1) and (f[i+1]-f[i]=1),i,print( )), i=1..1500);

A364659 Numbers k such that sigma(k) < sigma(k+1) < sigma(k+2).

Original entry on oeis.org

1, 2, 61, 62, 73, 74, 133, 134, 145, 146, 193, 194, 253, 254, 313, 397, 398, 403, 457, 458, 481, 482, 493, 494, 523, 553, 554, 565, 566, 613, 614, 625, 626, 661, 662, 673, 674, 691, 733, 734, 757, 758, 763, 793, 794, 817, 818, 853, 854, 913, 914, 943, 973, 974, 997, 998
Offset: 1

Views

Author

Seiichi Manyama, Aug 01 2023

Keywords

Links

Crossrefs

Cf. A053224, A364657, A364662.

Programs

  • PARI
    isok(n) = sigma(n)

A364662 Numbers k such that sigma(k) < sigma(k+1) < sigma(k+2) < sigma(k+3).

Original entry on oeis.org

1, 61, 73, 133, 145, 193, 253, 397, 457, 481, 493, 553, 565, 613, 625, 661, 673, 733, 757, 793, 817, 853, 913, 973, 997, 1033, 1093, 1213, 1237, 1285, 1321, 1453, 1513, 1537, 1645, 1657, 1681, 1813, 1825, 1873, 1933, 2077, 2113, 2173, 2233, 2245, 2293, 2413, 2497, 2533, 2581, 2593, 2653, 2713
Offset: 1

Views

Author

Seiichi Manyama, Aug 01 2023

Keywords

Links

Crossrefs

Cf. A050944, A053224, A364659.

Programs

  • PARI
    isok(n) = sigma(n)

A067825 Even values of k such that sigma(k + 1) > sigma(k).

Original entry on oeis.org

2, 62, 74, 134, 146, 194, 254, 314, 398, 404, 458, 482, 494, 524, 554, 566, 614, 626, 662, 674, 692, 734, 758, 764, 794, 818, 854, 914, 944, 974, 998, 1034, 1094, 1124, 1154, 1214, 1238, 1286, 1322, 1394, 1454, 1514, 1538, 1574, 1646, 1658, 1682, 1754
Offset: 1

Views

Author

Benoit Cloitre, Feb 08 2002

Keywords

Comments

Most terms are == 2 (mod 6), first term == 4 (mod 6) is a(365) = 13474. - Zak Seidov, Apr 18 2013
First term == 0 (mod 6) may be 296527606374. - Jianing Song, Apr 01 2018

Links

Crossrefs

Cf. A000203 (sigma), A053224.

Programs

  • GAP
    Filtered([2,4..2000],n->Sigma(n+1)>Sigma(n)); # Muniru A Asiru, Apr 03 2018
  • Mathematica
    Select[Range[2, 2000, 2], DivisorSigma[1, #] < DivisorSigma[1, # + 1] &] (* Zak Seidov, Apr 18 2013 *)
  • PARI
    is(n) = !(n%2) && sigma(n + 1) > sigma(n); \\ Amiram Eldar, Apr 23 2024

A053225 First differences of sigma(n) that are positive.

Original entry on oeis.org

2, 1, 3, 6, 7, 5, 16, 10, 7, 21, 22, 4, 36, 11, 16, 42, 31, 6, 43, 22, 34, 54, 40, 76, 36, 26, 66, 48, 10, 108, 34, 8, 23, 60, 58, 48, 123, 40, 10, 16, 72, 106, 5, 140, 24, 60, 144, 56, 16, 132, 73, 61, 114, 106, 172, 106, 96, 126, 66, 216, 53, 56, 156, 127, 76, 204, 44, 36
Offset: 1

Views

Author

Asher Auel, Jan 06 2000

Keywords

Crossrefs

Cf. A000203, A053222, A053227.

Programs

  • Maple
    with (numtheory): seq( `if`((sigma(i) < sigma(i+1)),(sigma(i+1)-sigma(i)),print( )), i=1..139);

Formula

a(n) = A053222(A053224(n))

A067828 Odd numbers k such that sigma(k+1) < sigma(k).

Original entry on oeis.org

45, 105, 117, 165, 225, 273, 297, 315, 345, 357, 405, 465, 513, 525, 561, 585, 621, 693, 705, 765, 777, 825, 837, 861, 885, 945, 1005, 1113, 1125, 1155, 1185, 1197, 1281, 1305, 1365, 1395, 1425, 1485, 1521, 1545, 1575, 1593, 1617, 1701, 1725, 1755, 1785, 1845
Offset: 1

Views

Author

Benoit Cloitre, Feb 08 2002

Keywords

Comments

Most terms are == 3 (mod 6), first term == 1 (mod 6) is a(130) = 5005. First term == 5 (mod 6) may be 247818996425. - Jianing Song, Apr 01 2018

Links

Crossrefs

Cf. A000203, A053224, A067825.

Programs

A323726 Odd numbers k such that sigma(k-1) < sigma(k) < sigma(k+1), sigma(n) = A000203.

Original entry on oeis.org

3, 63, 75, 135, 147, 195, 255, 399, 459, 483, 495, 555, 567, 615, 627, 663, 675, 735, 759, 795, 819, 855, 915, 975, 999, 1035, 1095, 1215, 1239, 1287, 1323, 1455, 1515, 1539, 1647, 1659, 1683, 1815, 1827, 1875, 1935, 2079, 2115, 2175, 2235, 2247, 2295, 2415, 2499
Offset: 1

Views

Author

K. D. Bajpai, Nov 19 2019

Keywords

Comments

It appears that most of the terms are divisible by 3; the smallest exception is 13475.
Up to 10^9, 223182 of 20606497 (about 1%) of the terms are not divisible by 3. - Charles R Greathouse IV, Nov 28 2019

Examples

			sigma(62) = 96, sigma(63) = 104, sigma(64) = 127. Hence, 63 is in the sequence.
sigma(74) = 114, sigma(75) = 124, sigma(76) = 140. Hence, 75 is in the sequence.

Links

Crossrefs

Cf. A000203, A005408, A053224, A067828, A323380.

Programs

  • Magma
    f:=func; [k:k in [3..2500 by 2]| f(k-1) and f(k)] // Marius A. Burtea, Nov 19 2019
  • Maple
    Sigmas:= map(numtheory:-sigma, [$1..3000]):
select(t -> Sigmas[t-1] < Sigmas[t] and Sigmas[t] < Sigmas[t+1],
[seq(i,i=3..3000,2)]); # Robert Israel, Nov 23 2019
  • Mathematica
    Select[Range[1,8000,2], DivisorSigma[1, # - 1] < DivisorSigma[1, (#)] && DivisorSigma[1, #] < DivisorSigma[1, (# + 1)] &]

A333038 Numbers m such that sigma(m) <= sigma(m-1).

Original entry on oeis.org

5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 46, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 106, 107, 109, 111, 113, 115, 117, 118, 119, 121, 123
Offset: 1

Views

Author

Bernard Schott, Mar 06 2020

Keywords

Comments

This sequence is infinite because all primes p >= 5 are terms with sigma(p) < sigma(p-1).
The integer m is a term iff A053222(m-1) <= 0.
The numbers m such that sigma(m) = sigma(m-1) are in A231546.

Examples

			Sigma(9) = 1+3+9 = 13 < sigma(8) = 1+2+4+8 = 15 so 9 is a term.
Sigma(15) = 1+3+5+15 = 24 = sigma(14) = 1+2+7+14 = 24 so 15 is a term.
Sigma(63) = 1+3+7+9+21+63 = 104 > sigma(62) = 1+2+31+62 = 96 and 63 is not a term.

References

  • J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 620 pp. 82, 280, Ellipses Paris 2004

Crossrefs

Cf. A000203, A053222, A231546 (subsequence: sigma(m) = sigma(m-1)).
Cf. A053224 (sigma(m) < sigma(m+1)), A053226 (sigma(m) > sigma(m+1)).

Programs

A067827 Even numbers k such that k/2 is nonprime and sigma(k+1) > sigma(k).

Original entry on oeis.org

2, 404, 494, 524, 692, 764, 854, 944, 1034, 1124, 1394, 1682, 1784, 1826, 1844, 2114, 2204, 2294, 2414, 2534, 2564, 2714, 2774, 2804, 2834, 2924, 3002, 3014, 3044, 3074, 3266, 3284, 3374, 3434, 3464, 3644, 3674, 3794, 3842, 3854, 3884, 3914, 3926, 4094
Offset: 1

Views

Author

Benoit Cloitre, Feb 08 2002

Keywords

Links

Crossrefs

Cf. A000203, A053224.

Programs

  • Mathematica
    Select[Range[2, 4000, 2], !PrimeQ[#/2] && DivisorSigma[1, # + 1] > DivisorSigma[1, #] &] (* Amiram Eldar, Apr 29 2022 *)
Previous Showing 11-20 of 23 results. Next

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