A053222 -id:A053222 - cp's OEIS frontend

A053246 First differences of chowla(n).

0, 0, 2, -2, 5, -5, 6, -3, 4, -7, 15, -15, 9, -1, 6, -14, 20, -20, 21, -11, 3, -13, 35, -30, 10, -3, 15, -27, 41, -41, 30, -16, 5, -7, 42, -54, 21, -5, 33, -49, 53, -53, 39, -7, -7, -25, 75, -68, 35, -22, 25, -45, 65, -49, 47, -41, 9, -31, 107, -107, 33, 7, 22, -44, 59, -77, 57, -31

Offset: 1

Asher Auel, Jan 10 2000

sign

Second differences give A053223, for n>1.

If the first term is changed to 1, this is also the first differences of A001065. - N. J. A. Sloane, Jan 17 2023

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A048050, A053222, A053223.

Cf. also A001065.

[0] cat [DivisorSigma(1,n+1) - DivisorSigma(1,n) - 1: n in [2..100]]; // G. C. Greubel, Sep 03 2018

with(numtheory): seq( sigma(i+1) - sigma(i) - 1, i=2..100); # for n>1

Chowlan[n_] := If[n == 1, 0, DivisorSigma[1, n] - n - 1]; Table[Chowlan[n + 1] - Chowlan[n], {n, 1, 100}] (* G. C. Greubel, Sep 03 2018 *)
Differences[Join[{0},Table[DivisorSigma[1,n]-n-1,{n,2,100}]]] (* Harvey P. Dale, Dec 19 2022 *)

concat([0], vector(100, n, n++; sigma(n+1) - sigma(n) -1)) \\ G. C. Greubel, Sep 03 2018

a(n) = A053222(n) - 1, for n>1

A072611 Numbers k such that phi(k) divides sigma(k+1) - sigma(k).

Original entry on oeis.org

1, 2, 6, 14, 30, 40, 70, 140, 170, 174, 206, 215, 238, 390, 459, 518, 923, 957, 1334, 1364, 1540, 1634, 2685, 2974, 4364, 5180, 5934, 6048, 6467, 6510, 6623, 8028, 8094, 8260, 8814, 12136, 12954, 14099, 14841, 15416, 16472, 17094, 17835, 17927, 18873

Offset: 1

Benoit Cloitre, Aug 07 2002

nonn

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

Cf. A000010, A053222.

Select[Range[20000],Divisible[DivisorSigma[1,#+1]-DivisorSigma[1,#], EulerPhi[#]]&] (* Harvey P. Dale, Sep 15 2011 *)

Conjecture : Limit_{n -> infinity} log(a(n))/log(n) exists and = 2.6... .

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

A248211 First differences of omega(n), the number of distinct prime factors function (A001221).

Original entry on oeis.org

1, 0, 0, 0, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, -1, 1, 0, 0, -1, 1, -1, 1, -1, 1, -1, 2, -2, 0, 1, 0, 0, 0, -1, 1, 0, 0, -1, 2, -2, 1, 0, 0, -1, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, -1, 2, -2, 1, 0, -1, 1, 1, -2, 1, 0, 1, -2, 1, -1, 1, 0, 0, 0, 1, -2, 1

Offset: 1

Wesley Ivan Hurt, Oct 04 2014

First instance of abs(a(n)) > 2 is for n = 210. - Alonso del Arte, Oct 05 2014

Eric M. Schmidt, Table of n, a(n) for n = 1..10000
Paul Erdős, Carl Pomerance and András Sárközy, On locally repeated values of certain arithmetic functions. II, Acta Math. Hungar. 49 (1987), 251-259.

Cf. A001221 (omega).
Cf. A053222: first differences of sigma(n) = A000203.
Cf. A076191: first differences of bigomega(n) = A001222.
Cf. A127440: first differences of mobius(n) = A008683.

with(numtheory): A248211:=n->nops(factorset(n+1))-nops(factorset(n)): seq(A248211(n), n=1..100);

Table[PrimeNu[n + 1] - PrimeNu[n], {n, 100}] (* Hurt *)
Differences[PrimeNu[Range[100]]] (* Alonso del Arte, Oct 04 2014 *)

a(n) = omega(n+1) - omega(n); \\ Michel Marcus, Dec 29 2022

a(n) = omega(n+1) - omega(n) = A001221(n+1) - A001221(n).

G.f.: (1 - x)*Sum_{k>=1} x^(prime(k)-1)/(1 - x^prime(k)). - Ilya Gutkovskiy, Mar 15 2017

A227305 Numbers n such that sigma(n) - sigma(n-1) divides n.

Original entry on oeis.org

2, 3, 5, 6, 10, 19, 52, 118, 1054, 3201, 8586, 9802, 16120, 60556, 140698, 145216, 11273536, 29886160, 44868748, 8748377956, 325377469696, 2368739714188

Offset: 1

Alex Ratushnyak, Jul 05 2013

a(23) > 2.5*10^12. - Giovanni Resta, Jul 13 2013

Cf. A053222, A067816, A223138, A225211.

a(21)-a(22) from Giovanni Resta, Jul 13 2013

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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A053246 First differences of chowla(n).

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A072611 Numbers k such that phi(k) divides sigma(k+1) - sigma(k).

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

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A248211 First differences of omega(n), the number of distinct prime factors function (A001221).

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A227305 Numbers n such that sigma(n) - sigma(n-1) divides n.

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

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