A053226 -id:A053226 - cp's OEIS frontend

A053244 Numbers n such that A053238(n) = 4.

5, 29, 34, 65, 70, 94, 99, 123, 198, 227, 239, 244, 274, 280, 304, 310, 327, 332, 362, 373, 391, 402, 421, 452, 481, 492, 510, 539, 601, 612, 636, 654, 659, 719, 749, 761, 811, 816, 821, 832, 899, 904, 927, 958, 1033, 1050, 1081, 1110, 1115, 1139, 1201

Offset: 1

Asher Auel, Jan 10 2000

nonn

Cf. A000203, A053226, A053238, A053239, A053240, A053241, A053242, A053243, A053245.

with(numtheory): f := [seq( `if`((sigma(i) > sigma(i+1)),i,print( )), i=1..8000)];
seq( `if`(f[i+1] - f[i] = 4,i,print( )), i=1..3000);

A050944 Numbers k such that sigma(k) > sigma(k+1) > sigma(k+2) > sigma(k+3).

Original entry on oeis.org

44, 104, 116, 164, 224, 272, 296, 344, 356, 464, 512, 560, 584, 620, 704, 776, 824, 836, 860, 884, 1004, 1112, 1184, 1196, 1280, 1304, 1424, 1484, 1520, 1544, 1592, 1616, 1700, 1724, 1880, 1904, 1952, 1964, 2024, 2036, 2120, 2144, 2276, 2324, 2372, 2384, 2444

Offset: 1

Jud McCranie, Jan 02 2000

nonn

			sigma(44)=84 > sigma(45)=78 > sigma(46)=72 > sigma(47)=48.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, entry 44.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000203, A050945, A053226.

Position[Partition[DivisorSigma[1,Range[2500]],4,1],?(Max[ Differences[ #]]<0&),1,Heads->False]//Flatten (* _Harvey P. Dale, Sep 13 2020 *)

Name simplified by Sean A. Irvine, Aug 20 2021

A364657 Numbers k such that sigma(k) > sigma(k+1) > sigma(k+2).

Original entry on oeis.org

44, 45, 104, 105, 116, 117, 164, 165, 224, 225, 272, 273, 296, 297, 315, 344, 345, 356, 357, 405, 464, 465, 512, 513, 525, 560, 561, 584, 585, 620, 621, 693, 704, 705, 765, 776, 777, 824, 825, 836, 837, 860, 861, 884, 885, 945, 1004, 1005, 1112, 1113, 1125, 1155

Offset: 1

Seiichi Manyama, Aug 01 2023

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A050944, A053226, A364659.

isok(n) = sigma(n)>sigma(n+1) && sigma(n+1)>sigma(n+2);

A053227 First differences of sigma(n) that are negative.

Original entry on oeis.org

1, 4, 2, 6, 14, 13, 19, 10, 12, 29, 2, 26, 40, 15, 6, 53, 4, 48, 52, 6, 6, 24, 67, 21, 44, 48, 40, 30, 106, 43, 76, 30, 72, 121, 44, 88, 65, 42, 116, 12, 90, 122, 40, 24, 154, 15, 115, 112, 18, 30, 54, 170, 64, 134, 96, 28, 2, 36, 227, 18, 68, 184, 79, 120, 176, 132, 148

Offset: 1

Asher Auel, Jan 06 2000

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000203, A053222, A053225.

with(numtheory): seq( `if`((sigma(i) > sigma(i+1)),(sigma(i) - sigma(i+1)),print( )), i=1..200);

Abs[Select[Differences[DivisorSigma[1,Range[200]]],#<0&]] (* Harvey P. Dale, Apr 17 2022 *)

a(n) = A053222(A053226(n))

A050945 Numbers k such that sigma(k) > sigma(k+1) > sigma(k+2) > sigma(k+3) > sigma(k+4) > sigma(k+5).

Original entry on oeis.org

20021154, 28503474, 53728674, 224539314, 242917674, 304178874, 341295954, 370845474, 377692314, 443998554, 533007474, 610484874, 709445274, 855529674, 964885074, 978052074, 1069943874, 1114988874, 1167961794

Offset: 1

Jud McCranie, Jan 02 2000

nonn

There are many examples of 4 consecutive numbers having decreasing values of sigma (see cross references) and there are 83 cases of 6 consecutive numbers with decreasing values of sigma < 4,293,000,000. Interestingly (and for unknown reasons) there are no examples of exactly 5 such consecutive numbers in that range! All known members of the sequence are multiples of 6 and one less than a multiple of 35.

			sigma(20021154)=40894848 > sigma(20021155)=35458560 > ... > sigma(20021159)=20021160.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, entry 44.

Seiichi Manyama, Table of n, a(n) for n = 1..100

Cf. A000203, A050944, A053226, A364657, A364680.

Name simplified by Seiichi Manyama, Aug 03 2023

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

Bernard Schott, Mar 06 2020

nonn

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.

			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.

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

Cf. A000203, A053222, A231546 (subsequence: sigma(m) = sigma(m-1)).

Cf. A053224 (sigma(m) < sigma(m+1)), A053226 (sigma(m) > sigma(m+1)).

filter:= m -> sigma(m) <= sigma(m-1): select(filter, [$1..500]);

Select[Range[2, 123], DivisorSigma[1, #] <= DivisorSigma[1, # - 1] &] (* Amiram Eldar, Mar 06 2020 *)
Flatten[Position[Partition[DivisorSigma[1,Range[200]],2,1],?(#[[2]]<= #[[1]]&),1,Heads->False]]+1 (* _Harvey P. Dale, Mar 28 2020 *)

isok(m) = (m>1) && (sigma(m) <= sigma(m-1)); \\ Michel Marcus, Mar 09 2020

A364680 Smallest initial number k of n consecutive numbers satisfying sigma(k) > sigma(k+1) > ... > sigma(k+n-1).

Original entry on oeis.org

1, 4, 44, 44, 20021154, 20021154

Offset: 1

Seiichi Manyama, Aug 02 2023

Cf. A000203, A075029.

Cf. A050944, A050945, A053226, A364657.

b(n) = my(k=n); while(sigma(k)>sigma(k+1), k++); k-n+1;
a(n) = my(k=1); while(b(k)

cp's OEIS Frontend

A053244 Numbers n such that A053238(n) = 4.

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A050944 Numbers k such that sigma(k) > sigma(k+1) > sigma(k+2) > sigma(k+3).

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A364657 Numbers k such that sigma(k) > sigma(k+1) > sigma(k+2).

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A053227 First differences of sigma(n) that are negative.

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A050945 Numbers k such that sigma(k) > sigma(k+1) > sigma(k+2) > sigma(k+3) > sigma(k+4) > sigma(k+5).

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

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A364680 Smallest initial number k of n consecutive numbers satisfying sigma(k) > sigma(k+1) > ... > sigma(k+n-1).

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