A015886 -id:A015886 - cp's OEIS frontend

A206768 a(n) = smallest number k such that sigma(k-n) = sigma(k) - n, with k > n+1.

3, 5, 5, 7, 7, 11, 81, 11, 11, 13, 13, 17, 4431, 17, 17, 19, 19, 23, 25, 23, 23, 29

Offset: 1

Paolo P. Lava, Jan 10 2013

This sequence begins
3, 5, 5, 7, 7, 11, 81, 11, 11, 13, 13, 17, 4431, 17, 17, 19, 19, 23, 25, 23, 23, 29, ?, 29, ?, 29, 29, 31, 31, 37, ?, 37, 51, 37, 37, 41, 81, 41, 41, 43, 43, 47, ?, 47, 47, 53, ?, 53, 3364, 53, 53, 59, ?, 59, ?, 59, 59, 61, 61, 67, ?, 67, ?, 67, 67, 71, ?, 71, 71, 73, 73, 79, 91, 79, ?, 79, 79, 83, ?, 83, 83, 89, ?, 89, ?, 89, 89, 101, ?, 97, ?, 97, 125, 97, 97, 101, ?, 101, 101, 103, 103, 107... where the other missing terms (designated by "?") are > 10^6, if they exist.
For a given n, n being even, among the integers k satisfying the property sigma(k-n) = sigma(k)-n, we will find prime numbers p, such that p and p-n are primes. This is because in that case sigma(p-n) = (p-n)+1 = (p+1)-n = sigma(p)-n. For instance, when n is even, for n=2 to 14, a(n) is the first term of A006512, A046132, A046117, A092402, A092146, A092216, A098933. If we restrict to composite numbers, then see A084293. - Michel Marcus, Feb 16 2013
For the missing terms mentioned in first comment, a(n) is > 10^7. - Michel Marcus, Sep 21 2013

			a(13) = 4431 because 4431 is the minimum number for which sigma(4431-13) = sigma(4418)= 6771 and sigma(4431) - 13 = 6784 -13 = 6771.
a(19) = 25 because 25 is the minimum number for which sigma(25-19) = sigma(6) = 12 and sigma(25) - 19 = 31 -19 = 12.

Cf. A015886.

A206768:=proc(q)
local k,n;
for n from 1 to q do
  for k from n+1 to q do
  if sigma(-n+k)=sigma(k)-n then print(k); break; fi;
od; od; end:
A206768(1000000000);

A063680 Solutions to sigma(k) + 7 = sigma(k+7).

Original entry on oeis.org

74, 531434, 387420482, 2541865828322

Offset: 1

Jud McCranie, Jul 28 2001

No other solutions < 4290000000. Sequence A063679 shows how to generate more solutions, but there may be solutions other than those produced by A063679.
No others < 10^17. - Seth A. Troisi, Oct 25 2022
k or k+7 must be a square or twice a square (A028982). See comment in A015886. - Seth A. Troisi, Oct 26 2022
From Jon E. Schoenfield, Oct 26 2022: (Start)
Each of the first 4 terms of the sequence is of the form k = 9^j - 7:
             74 = 9^2  - 7,
         531434 = 9^6  - 7,
      387420482 = 9^9  - 7,
  2541865828322 = 9^13 - 7.
The next terms of this form are 9^53 - 7 and 9^82 - 7.
Does the sequence contain any terms that are not of this form?
(End)
No other terms < 2.7*10^15. - Jud McCranie, Jul 27 2025

			sigma(74) + 7 = 121 = sigma(74+7), so 74 is in the sequence.

Cf. A000203, A063679.

Cf. A054799, A015913, A015914, A015915, A015916, A015917, A054905.

isok(k) = sigma(k) + 7 == sigma(k+7); \\ Michel Marcus, Oct 25 2022

a(4) from Seth A. Troisi, Oct 24 2022

A305196 a(n) is the smallest number k such that tau(k + n) = tau(k) + n where tau(n) is the number of divisors of n (A000005).

Original entry on oeis.org

1, 1, 10, 9, 26, 25, 74, 29, 82, 441, 170, 133, 348, 131, 166, 3025, 344, 559, 1602, 557, 820, 9979, 986, 4333, 1236, 9191, 694, 3249, 1652, 3481, 9378, 34969, 3118, 249967, 5636, 36829, 3324, 51947, 3994, 6561, 5000, 15835, 16806, 3557, 6436, 119025, 6254, 589777, 7512, 1768851

Offset: 0

Michel Marcus, May 27 2018

nonn

			10 and 12 have respectively 4 and 6 divisors, that is, 12-10 = 6-4, so a(2)=10.
9 and 12 have respectively 3 and 6 divisors, that is, 12-9 = 6-3, so a(3)=9.

Michel Marcus and Giovanni Resta, Table of n, a(n) for n = 0..244 (terms up to a(106) from Michel Marcus)

Cf. A000005, A099642, A015886 (similar, with sigma).

f:= proc(n) local k;
  for k from 1 do
      if numtheory:-tau(k+n)=numtheory:-tau(k)+n then return k fi
    od
end proc:
map(f, [$0..50]); # Robert Israel, May 28 2018

Array[Block[{k = 1}, While[DivisorSigma[0, k + #] != DivisorSigma[0, k] + #, k++]; k] &, 40, 0] (* Michael De Vlieger, May 27 2018 *)

a(n) = {my(k=1); while(numdiv(k+n) != numdiv(k) + n, k++); k;}

A217259 Numbers n such that sigma(n+1) - sigma(n-1) = 2; sigma(n) = A000203(n) = sum of divisors of n.

Original entry on oeis.org

4, 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, 240, 270, 282, 312, 348, 420, 432, 435, 462, 522, 570, 600, 618, 642, 660, 810, 822, 828, 858, 882, 1020, 1032, 1050, 1062, 1092, 1152, 1230, 1278, 1290, 1302, 1320, 1428, 1452, 1482, 1488

Offset: 1

Jaroslav Krizek, Mar 17 2013

nonn

Also numbers n such that antisigma(n+1) - antisigma(n-1) = 2*n - 1.
Antisigma(n) = A024816(n) = sum of nondivisors of n.
Union of A014574 (average of twin prime pairs) and sequence 435, 8576, 8826, … (= all terms < 100000).
If n = average of twin prime pairs (q < p) then antisigma(p) - antisigma(q) = 2*n - 1 = p + q - 1.
No term found below 2*10^9 to continue sequence 435, 8576, 8826, ... - Michel Marcus, Mar 19 2013

			Number 435 is in sequence because antisigma(436) - antisigma(434) = 94496 - 93627 = 869 = 2*435 - 1.

Jaroslav Krizek, Table of n, a(n) for n = 1..1227 (all terms < 100000)

Cf. A014574, A015886, A024816, A050507, A206768.

Equals A054799 + 1. - Michel Marcus, May 21 2018

Flatten[Position[Partition[DivisorSigma[1,Range[1500]],3,1],?(#[[3]]- #[[1]] == 2&),1,Heads->False]]+1 (* _Harvey P. Dale, May 03 2018 *)

isok(n) = (sigma(n+1) - sigma(n-1)) == 2; \\ Michel Marcus, May 20 2018

A290125 Square array read by antidiagonals T(n,k) = sigma(k + n) - sigma(k) - n, with n>=0 and k>=1.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 2, 2, 3, 0, -2, 0, 0, 1, 0, 5, 3, 5, 5, 6, 0, -5, 0, -2, 0, 0, 1, 0, 6, 1, 6, 4, 6, 6, 7, 0, -3, 3, -2, 3, 1, 3, 3, 4, 0, 4, 1, 7, 2, 7, 5, 7, 7, 8, 0, -7, -3, -6, 0, -5, 0, -2, 0, 0, 1, 0, 15, 8, 12, 9, 15, 10, 15, 13, 15, 15, 16

Offset: 0

Michel Marcus, Jul 20 2017

A015886(n) gives the position of the first zero in the n-th row of this array.

			Array begins:
  0, 0, 0,  0,  0,  0,  0, ...
  1, 0, 2, -2,  5, -5,  6, ...
  1, 2, 0,  3,  0,  1,  3, ...
  3, 0, 5, -2,  6, -2,  7, ...
  1, 5, 0,  4,  3,  2,  0, ...
  6, 0, 6,  1,  7, -5, 15, ...
  1, 6, 3,  5,  0, 10,  0, ...
  7, 3, 7, -2, 15, -5,  9, ...
  ...

Michel Marcus, Table of n, a(n) for n = 0..5049

Cf. A000203 (sigma), A015886.

Table[Function[n, If[k + n == 0, 0, DivisorSigma[1, k + n]] - If[k == 0, 0, DivisorSigma[1, k]] - n][m - k], {m, 12}, {k, m, 1, -1}] // Flatten (* Michael De Vlieger, Jul 20 2017 *)

PARI
```
T(n,k) = sigma(k + n) - sigma(k) - n;
```

a(n) = n++; my(s = ceil((-1+sqrt(1+8*n))/2));r=n-binomial(s,2)-1;k=s-r;T(r,k) \\ David A. Corneth, Jul 20 2017

from sympy import divisor_sigma
l=[]
def T(n, k):
    return 0 if n==0 or k==0 else divisor_sigma(k + n) - divisor_sigma(k) - n
for n in range(11): l+=[T(k, n - k + 1) for k in range(n + 1)]
print(l) # Indranil Ghosh, Jul 21 2017

T(0, k) = 0 for all k.

A304994 a(n) is the smallest number k such that sigma(k + n) = sigma(k) - n, or -1 if no such number exists.

Original entry on oeis.org

1, 4, 6976, 8, 15

Offset: 0

Michel Marcus, May 23 2018

Further terms of the sequence: a(5) > 10^9 (if it exists), a(6) = 35, a(7) = 16, a(8) = 39, a(10) = 51, a(12) = 95, a(14) = 831, a(15) = 32, a(16) = 45, a(18) = 88, a(19) = 36, a(20) = 63, a(21) = 50, a(22) = 222, a(24) = 135, a(26) = 37888, a(28) = 96, a(30) = 284, a(32) = 117, a(34) = 354, a(36) = 175, a(37) = 1089, a(38) = 84, a(40) = 153, a(42) = 176, a(44) = 3893, a(46) = 267, a(48) = 210. - Daniel Suteu, May 25 2018

a(5) > 3*10^12, if it exists. - Giovanni Resta, May 26 2018

			sigma(4+1) - sigma(4) = -1, so a(1) = 4, since this does not happen for x < 4.
sigma(8+3) - sigma(8) = -3, so a(3) = 8, since this does not happen for x < 8.

Cf. A000203, A015886.

a(n) = {my(k=1); while(sigma(k+n) != sigma(k) - n, k++); k;}

A255354 a(n) = smallest number k such that (k + n)' = k', or -1 if no such number exists, where k' is the arithmetic derivative of k.

Original entry on oeis.org

2, 3, 2, 3, 2, 5, 110, 3, 2, 3, 2, 5, 50145, 3, 2, 3, 2, 5, 53115, 3, 2, 7, 189, 5, 273, 3, 2, 3, 2, 7, 75, 5, 930642191642, 3, 2, 5, 165, 3, 2, 3, 2, 5, 12, 3, 2, 7, 99, 5, 182, 3, 2, 7, 706, 5, 1523965807, 3, 2, 3, 2, 7, 494, 5

Offset: 1

Paolo P. Lava, Feb 24 2015

The sequence begins (first 100 terms):
2, 3, 2, 3, 2, 5, 110, 3, 2, 3, 2, 5, 50145, 3, 2, 3, 2, 5, 53115, 3, 2, 7, 189, 5, 273, 3, 2, 3, 2, 7, 75, 5, 930642191642, 3, 2, 5, 165, 3, 2, 3, 2, 5, 12, 3, 2, 7, 99, 5, 182, 3, 2, 7, 706, 5, 1523965807, 3, 2, 3, 2, 7, 494, 5, -1, 3, 2, 5, 1151559, 3, 2, 3, 2, 7, 705, 5, 20, 3, 2, 5, 4526, 3, 2, 7, 1102, 5, 1509626, 3, 2, 13, 778, 7, 226429394, 5, -1, 3, 2, 5, 1910, 3, 2, 3 where the other missing terms (designated by -1: a(63), a(93)) are > 10^12, if they exist.
a(91) = 226429394. - Michel Marcus, Feb 28 2015
a(63), a(93) > 10^12. - Giovanni Resta, Jun 22 2018

			a(1) = 2 because (2 + 1)' = 2' = 1.
a(2) = 3 because (3 + 2)' = 3' = 1.
a(3) = 2 because (2 + 3)' = 2' = 1.
...
a(7) = 110 because (110 + 7)' = 110' = . Etc.

Cf. A003415, A007015, A007365, A015886, A065559.

with(numtheory): P:=proc(q) local a,b,k,n,p;
for n from 1 to q do for k from 1 to q do
a:=k*add(op(2,p)/op(1,p),p=ifactors(k)[2]); b:=(k+n)*add(op(2,p)/op(1,p),p=ifactors(k+n)[2]);
if a=b then print(k); break; fi; od;
od; end: P(10^20);

a(33)-a(62) from Giovanni Resta, Jun 22 2018

cp's OEIS Frontend

A206768 a(n) = smallest number k such that sigma(k-n) = sigma(k) - n, with k > n+1.

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A063680 Solutions to sigma(k) + 7 = sigma(k+7).

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A305196 a(n) is the smallest number k such that tau(k + n) = tau(k) + n where tau(n) is the number of divisors of n (A000005).

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A217259 Numbers n such that sigma(n+1) - sigma(n-1) = 2; sigma(n) = A000203(n) = sum of divisors of n.

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A290125 Square array read by antidiagonals T(n,k) = sigma(k + n) - sigma(k) - n, with n>=0 and k>=1.

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A304994 a(n) is the smallest number k such that sigma(k + n) = sigma(k) - n, or -1 if no such number exists.

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Author

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Examples

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Programs

PARI

A255354 a(n) = smallest number k such that (k + n)' = k', or -1 if no such number exists, where k' is the arithmetic derivative of k.

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Author

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Comments

Examples

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Programs

Maple

Extensions