A007365 -id:A007365 - cp's OEIS frontend

A065932 Index values for new maxima in sequence A007365.

0, 1, 2, 3, 29, 43, 69, 1879, 2287, 3780, 4200, 4440, 4620, 5040, 6300, 7140, 7560, 8820, 9240, 10080, 12600, 13860, 15120, 17640, 18480, 20160, 22680, 25200, 27720, 28560, 30240, 32760, 37800, 42840, 50400, 55440, 57960, 60480, 64680, 65520, 73080, 75600

Offset: 1

Jason Earls, Nov 28 2001

nonn

RECORDS transform of A007365.

Donovan Johnson, Table of n, a(n) for n = 1..100
N. J. A. Sloane, Transforms

Cf. A007365, A065933.

sg(m) = {local(a,n,k); a = 0; for(k = 1,m,n = 1; while(sigma(n)! = sigma(n+k), n++); if(n>a,a = n; print(k,"\t",n)))} \\ Klaus Brockhaus

More terms from David Wasserman, Oct 10 2002

Offset corrected by Donovan Johnson, Nov 26 2013

A065933 Successive maxima in sequence A007365.

Original entry on oeis.org

1, 14, 33, 382, 406, 435, 8786, 14390, 16172, 16640, 16830, 17850, 21736, 25194, 29640, 30240, 37791, 41496, 46189, 50388, 62985, 65208, 75582, 80256, 92378, 100776, 113373, 125970, 138567, 140184, 151164, 184756, 188955, 230230, 251940, 277134, 289731, 302328

Offset: 1

Jason Earls, Nov 28 2001

nonn

RECORDS transform of A007365.

Donovan Johnson, Table of n, a(n) for n = 1..100
N. J. A. Sloane, Transforms

Cf. A007365, A065932.

More terms from David Wasserman, Oct 10 2002

Offset corrected by Donovan Johnson, Nov 26 2013

A276715 a(n) = the smallest number k such that k and k + n have the same number and sum of divisors (A000005 and A000203).

Original entry on oeis.org

1, 14, 33, 42677635, 51, 46, 155, 62, 69, 46, 174, 154, 285, 182, 141, 62, 138, 142, 235, 158, 123, 94, 213, 322, 295, 94, 177, 118, 159, 406, 376, 266, 177, 891528365, 321, 310, 355, 248, 249, 166, 213, 418, 376, 602, 426, 142, 570, 310, 445, 248, 249, 158

Offset: 0

Jaroslav Krizek, Sep 16 2016

nonn

If a(33) exists, it must be greater than 2*10^8.
a(n) for n >= 34: 321, 310, 355, 248, 249, 166, 213, 418, 376, 602, 426, 142, 570, 310, 445, 248, 249, 158, 267, 406, 632, 166, 267, ...
The records occur at indices 0, 1, 2, 3, 33, 207, 471, ... with values 1, 14, 33, 42677635, 891528365, 2944756815, 3659575815, ... - Amiram Eldar, Feb 17 2019

			a(2) = 33 because 33 is the smallest number such that tau(33) = tau(35) = 4 and simultaneously sigma(33) = sigma(35) = 48.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A065559 (smallest k such that tau(k) = tau(k+n)), A007365 (smallest k such that sigma(k) = sigma(k+n)).

Cf. Sequences with numbers n such that n and n+k have the same number and sum of divisors for k=1: A054004, for k=2: A229254, k=3: A276714.

A276715:=func; [A276715(n):n in[0..32]]

a[k_] := Module[{n=1}, While[DivisorSigma[0,n] != DivisorSigma[0,n+k] || DivisorSigma[1,n] != DivisorSigma[1,n+k], n++]; n]; Array[a, 50, 0] (* Amiram Eldar, Feb 17 2019 *)

from itertools import count
from sympy import divisor_sigma
def A276715(n): return next(k for k in count(1) if all(divisor_sigma(k,i)==divisor_sigma(n+k,i) for i in (0,1))) # Chai Wah Wu, Jul 25 2022

a(33) onwards from Amiram Eldar, Feb 17 2019

A333947 a(n) is the smallest k > 0 such that sigma(n+k) = sigma(n); if such k > 0 does not exist, then a(n) = 0.

Original entry on oeis.org

0, 0, 0, 0, 0, 5, 0, 0, 0, 7, 0, 0, 0, 1, 8, 9, 0, 0, 0, 6, 10, 0, 0, 14, 0, 15, 0, 11, 0, 16, 0, 0, 2, 19, 12, 0, 0, 21, 0, 18, 0, 20, 0, 21, 0, 5, 0, 27, 0, 0, 4, 45, 0, 2, 16, 31, 22, 31, 0, 18, 0, 7, 40, 0, 18, 4, 0, 14, 8, 24, 0, 0, 0, 39, 0, 63, 0, 14, 0

Offset: 1

Bernard Schott, Apr 11 2020

nonn

This sequence is inspired by A007365 where a(n) is the smallest k such that sigma(n+k) = sigma(k); indeed, n and k are switched between these two sequences.
There are three distinct cases for which a(n) = 0:
If n is prime then a(n) = 0,
If n is in A211658 then a(n) = 0,
If n is the largest number q_r of a sequence q_1 < q_2 < ... < q_r with q_r composite and sigma(q_1) = sigma(q_2) = ... =  sigma(q_r) then a(n) = 0. The first two such examples are a(25) = 0 and a(39) = 0 with sigma(16) = sigma(25) = 31, and sigma(28) = sigma(39) = 56.

			sigma(9) = 13 and there is no k>0 such that sigma(9+k) = 13, then a(9) = 0.
sigma(14) = sigma(15) = sigma(23) = 24, so a(14) = 1 and a(15) = 8, and as 23 is prime, a(23) = 0.

Robert Israel, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems

Cf. A000203, A007365, A054973.

Cf. A002961 (a(n)=1).

f:= proc(n) local s,k;
  s:= numtheory:-sigma(n);
for k from n+1 to s-1 do
  if numtheory:-sigma(k)=s then return k-n fi
od;
0
end proc:
map(f, [$1..100]); # Robert Israel, Apr 17 2020

a[n_] := Module[{k = n+1, s = DivisorSigma[1, n]}, While[k < s && DivisorSigma[1, k] != s, k++];If[k >= s, 0, k-n]]; Array[a, 70] (* Amiram Eldar, Apr 12 2020 *)

a(n) = {my(s=sigma(n)); for (k= n+1, s-1, if (sigma(k) == s, return (k-n));); return(0);} \\ Michel Marcus, Apr 11 2020

A094466 a[n] is the smallest integer with property that phi[n+a[n]]=sigma[n].

Original entry on oeis.org

1, 3, 5, 3, 19, 3, 7, 3, 11, 23, 5, 19, 6, 23, 7, 47, 6, 14, 6, 7, 15, 11, 43, 46, 10, 20, 6, 11, 33, 23, 45, 22, 6, 15, 14, 38, 51, 15, 14, 21, 10, 41, 12, 19, 27, 23, 69, 22, 24, 22, 15, 14, 15, 14, 21, 26, 23, 55, 15, 14, 30, 20, 23, 87, 15, 14, 31, 26, 27, 139, 15, 14, 35, 20, 33

Offset: 1

Labos Elemer, May 11 2004

nonn

			n=6:a[6]=19 because sigma[19]=20=phi[19+6]=phi[25]=20.
n=16:a[16]=23 because sigma[23]=24=phi[23+16]=phi[39]=24

Cf. A007015, A007365.

Least solution to Min{x; phi[x+n]=sigma[x]}.

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

A298654 Least number k such that the sum of the anti-divisors of k is equal to the sum of the anti-divisors of k+n.

Original entry on oeis.org

8, 55, 26, 15, 43, 10, 89, 22, 20, 129, 118, 430, 43, 32, 39, 88, 174, 179, 35, 31, 45, 161, 53, 27, 228, 407, 122, 86, 90, 149, 87, 288, 46, 177, 283, 28, 117, 130, 222, 158, 200, 82, 68, 62, 383, 932, 32, 63, 120, 375, 1107, 67, 298, 110, 119, 352, 122, 277

Offset: 1

Paolo P. Lava, Jan 24 2018

			a(1) = 8 because the sum of the anti-divisors of 8 is 8 and of 9 is 8 again;
a(2) = 55 because the sum of the anti-divisors of 55 is 74 and of 57 is 74 again.

Cf. A007365, A066417.

with(numtheory): P:=proc(q) local a,b,i,j,k,n; for i from 0 to q do for n from 1 to q do
k:=0; j:=n; while j mod 2 <> 1 do k:=k+1; j:=j/2; od;
a:=sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2;
k:=0; j:=n+i; while j mod 2 <> 1 do k:=k+1; j:=j/2; od;
b:=sigma(2*(n+i)+1)+sigma(2*(n+i)-1)+sigma((n+i)/2^k)*2^(k+1)-6*(n+i)-2;
if a=b then print(n); break; fi; od; od; end: P(10^5);

cp's OEIS Frontend

A065932 Index values for new maxima in sequence A007365.

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A065933 Successive maxima in sequence A007365.

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A276715 a(n) = the smallest number k such that k and k + n have the same number and sum of divisors (A000005 and A000203).

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A333947 a(n) is the smallest k > 0 such that sigma(n+k) = sigma(n); if such k > 0 does not exist, then a(n) = 0.

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Maple

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A094466 a[n] is the smallest integer with property that phi[n+a[n]]=sigma[n].

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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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Maple

Extensions

A298654 Least number k such that the sum of the anti-divisors of k is equal to the sum of the anti-divisors of k+n.

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Maple