A231546 -id:A231546 - cp's OEIS frontend

A231545 Numbers n such that n = sigma(n) - sigma(n-1).

2, 6, 8586, 16120, 29886160

Offset: 1

Jaroslav Krizek, Nov 11 2013

nonn

Also, numbers n such that antisigma(n) = antisigma(n-1), where antisigma(n) = A024816(n) = the sum of the non-divisors of n that are between 1 and n.

Numbers n such that A163553(n-1) = 0.

			6 is in sequence because antisigma(6) = antisigma(5) = 9.

Cf. A067816, A024816 (antisigma(n)), A231546 (numbers n such that sigma(n) = sigma(n-1)).

a(n) = A067816(n) + 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

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

A333039 Composites m such that sigma(m) < sigma(m-1).

Original entry on oeis.org

9, 21, 25, 27, 33, 35, 39, 45, 46, 49, 51, 55, 57, 65, 69, 77, 81, 85, 87, 91, 93, 95, 99, 105, 106, 111, 115, 117, 118, 119, 121, 123, 125, 129, 133, 141, 143, 145, 153, 155, 159, 161, 165, 166, 169, 171, 175, 177, 183, 185, 187, 189, 201

Offset: 1

Bernard Schott, Mar 12 2020

nonn

As all primes p >= 5 satisfy sigma(p) < sigma(p-1) [see A333038], this sequence is reserved for composite numbers.
This sequence is infinite because all squares of primes p, p >= 3 are terms.
Composites such that sigma(m) = sigma(m-1) are in A231546.

			sigma(77) = 1+7+11+77 = 96 < sigma(76) = 1+2+4+19+38+76 = 140, hence composite 77 is a term.
sigma(135) = 1+3+5+9+15+27+45+135 = 240 > sigma(134) = 1+2+67+134 = 204, hence composite 135 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

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000203, A053222, A231546, A333038.

filter:= m -> not isprime(m) and numtheory:-sigma(m) < numtheory:-sigma(m-1) : select(filter, [$1..500]);

Select[Range[200], CompositeQ[#] && DivisorSigma[1, #] < DivisorSigma[1, # - 1] &] (* Amiram Eldar, Mar 12 2020 *)

isok(m) = (m>1) && !isprime(m) && (sigma(m) < sigma(m-1)); \\ Michel Marcus, Mar 15 2020

A231547 Numbers n such that n < sigma(n) - sigma(n-1).

Original entry on oeis.org

12, 18, 20, 24, 30, 36, 42, 48, 54, 60, 72, 80, 84, 90, 96, 102, 104, 108, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 180, 192, 198, 200, 204, 210, 216, 224, 228, 234, 240, 252, 258, 260, 264, 270, 272, 276, 280, 282, 288, 294, 300, 306

Offset: 1

Jaroslav Krizek, Nov 12 2013

nonn

Also numbers n such that antisigma(n) < antisigma(n-1), where antisigma(n) = A024816(n) = the sum of the non-divisors of n that are between 1 and n.
Numbers n such that A163553(n-1) < 0.
Numbers n such that antisigma(n) > antisigma(n-1) = A231711.
Numbers n such that antisigma(n) = antisigma(n-1) = A231545.
Complement of union of A231711, A231545 and number 1.
Does this sequence have a density? - Charles R Greathouse IV, Jul 14 2024

			12 is in sequence because antisigma(12) = 50 < antisigma(11) = 54.

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A231545, A231546, A231711, A231548, A024816, A163553.

Select[Range[350],#<(DivisorSigma[1,#]-DivisorSigma[1,#-1])&] (* Harvey P. Dale, May 26 2016 *)

is(n)=nCharles R Greathouse IV, Jul 14 2024

A231711 Numbers n such that n > sigma(n) - sigma(n-1).

Original entry on oeis.org

3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82

Offset: 1

Jaroslav Krizek, Nov 12 2013

nonn

Numbers n such that antisigma(n) > antisigma(n-1), where antisigma(n) = A024816(n) = the sum of the non-divisors of n that are between 1 and n.
Numbers n such that A163553(n-1) > 0.
Numbers n such that antisigma(n) < antisigma(n-1) = A231547.
Numbers n such that antisigma(n) = antisigma(n-1) = A231545.
Complement of union of A231547, A231545 and number 1.

			10 is in sequence because antisigma(10) = 37 > antisigma(9) = 32.

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A231545, A231546, A231547, A231548, A024816, A163553.

with(numtheory); A231711:=n->`if`(sigma(n)-sigma(n-1)A231711(n), n=1..100); # Wesley Ivan Hurt, Nov 14 2013

isok(n) = n > sigma(n) - sigma(n-1); \\ Michel Marcus, Nov 14 2013

A333040 Even numbers m such that sigma(m) < sigma(m-1).

Original entry on oeis.org

46, 106, 118, 166, 226, 274, 298, 316, 346, 358, 406, 466, 514, 526, 562, 586, 622, 694, 706, 766, 778, 826, 838, 862, 886, 946, 1006, 1114, 1126, 1156, 1186, 1198, 1282, 1306, 1366, 1396, 1426, 1486, 1522, 1546, 1576, 1594, 1618, 1702, 1726, 1756

Offset: 1

Bernard Schott, Mar 22 2020

nonn

The even terms of A333039 represent about only 7% of the data, so they are proposed in this sequence. Some of these integers are semiprimes with for example these two families:
1) m = 2*p with p prime of the form k^2+k+3 is in A027753. The first few terms are: 46, 118, 226, 766, ... but not all the integers of this form are terms; the first 3 counterexamples are 6, 10, 1018 (see examples).
2) m = 2*p with p prime = (r*s*t+1)/2 and 2A234103. The first few terms are: 106, 166, 274, 346, 358, ... but not all the integers of this form are terms; the first 3 counterexamples are 386, 898 and 958 (see examples).
There is also this subsequence of even m = 2^2*p where p prime, congruent to 34 mod 45, is in A142330. The first few terms are: 316, 1396, 1756, 2416, ... but not all the integers of this form are terms; the first counterexample that comes from the 8th term of A142330 is 5716.
Even (and odd) numbers such that sigma(m) = sigma(m-1) are in A231546.

			166 = 2*83 and 165 = 3*5*11, as 252 = sigma(166) < sigma(165) = 288, hence 166 is a term.
386 = 2*193 and 385 = 5*7*11, as 582 = sigma(386) > sigma(385)= 576, hence 386 is not a term.
766 = 2*383 where 383 = 19^2+19+3 and 765 = 3^2*5*13, as 1152 = sigma(766) < sigma(765) = 1404, hence 766 is a term.
1018 = 2*509 where 509 = 22^2+22+3, and 1017 = 3^2*113, as 1530 = sigma(1018) > sigma(1017) = 1482, hence 1018 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.

Robert Israel, Table of n, a(n) for n = 1..10000

Intersection of A005843 and A333039.
Subsequence of A333038.
Cf. A231546.

filter:= n -> numtheory:-sigma(n) < numtheory:-sigma(n-1):
select(filter, [seq(i,i=2..2000,2)]); # Robert Israel, Mar 29 2020

Select[2 * Range[1000], DivisorSigma[1, #] < DivisorSigma[1, #-1] &] (* Amiram Eldar, Mar 24 2020 *)

isok(m) = !(m%2) && (sigma(m) < sigma(m-1)); \\ Michel Marcus, Mar 22 2020

A227307 Numbers k that divide sigma(k) - sigma(k-1).

Original entry on oeis.org

2, 6, 15, 19, 207, 958, 1335, 1365, 1635, 2686, 2975, 3201, 4365, 4536, 8586, 14842, 16120, 18874, 19359, 20146, 24958, 33999, 36567, 42819, 53580, 56565, 64666, 74919, 79827, 79834, 84135, 92686, 109215, 111507, 116938, 122074, 138238, 147455, 161002, 162603, 166935

Offset: 1

Alex Ratushnyak, Jul 05 2013

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000203, A053222, A223136, A223137, A227306.

A231546 is a subsequence.

With[{m = 200000}, 1 + Position[Differences[DivisorSigma[1, Range[m]]]/Range[2, m], ?IntegerQ] // Flatten] (* _Amiram Eldar, Dec 31 2024 *)

list(lim) = {my(s1 = 1, s2); for(k = 2, lim, s2 = sigma(k); if(!((s2-s1) % k), print1(k, ", ")); s1 = s2);} \\ Amiram Eldar, Dec 31 2024

A333041 Odd numbers m such that sigma(m) > sigma(m-1).

Original entry on oeis.org

3, 63, 75, 135, 147, 195, 255, 315, 399, 405, 459, 483, 495, 525, 555, 567, 615, 627, 663, 675, 693, 735, 759, 765, 795, 819, 855, 915, 945, 975, 999, 1035, 1095, 1125, 1155, 1215, 1239, 1287, 1323, 1395, 1455, 1515, 1539, 1575, 1647, 1659, 1683, 1755, 1785, 1815, 1827, 1845, 1875

Offset: 1

Bernard Schott, Apr 14 2020

nonn

The odd terms of A333038 [sigma(m) <= sigma(m-1)] represent about 95% of the data, so the odd integers that do not satisfy this relation are proposed here.
Except for 3, there are no prime powers in this sequence.
It appears that most of the terms are divisible by 3; the two smallest exceptions are 13475 and 17255 (see A323726).
Odd (and even) numbers such that sigma(m) = sigma(m-1) are in A231546.

			sigma(63) = 1+3+7+9+21+63 = 104 > sigma(62) = 1+2+31+62=96 and 63 is in the sequence.
sigma(77) = 1+7+11+77 = 96 < sigma(76) = 1+2+4+19+38+76 = 140 and 77 is not a term.

A323726 is a subsequence.
Apart from the first term, a subsequence of A334117.
Cf. A000203, A053222, A231546, A333038, A333039, A333040.

Select[2 * Range[1000] + 1, DivisorSigma[1, #] > DivisorSigma[1, # - 1] &] (* Amiram Eldar, Apr 14 2020 *)

is(n)=n%2 && sigma(n)>sigma(n-1) \\ Charles R Greathouse IV, Apr 14 2020

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A231545 Numbers n such that n = sigma(n) - sigma(n-1).

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

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A333039 Composites m such that sigma(m) < sigma(m-1).

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A231547 Numbers n such that n < sigma(n) - sigma(n-1).

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A231711 Numbers n such that n > sigma(n) - sigma(n-1).

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A333040 Even numbers m such that sigma(m) < sigma(m-1).

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A227307 Numbers k that divide sigma(k) - sigma(k-1).

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A333041 Odd numbers m such that sigma(m) > sigma(m-1).

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