A065559 -id:A065559 - cp's OEIS frontend

A113465 Rectangular array read by antidiagonals: a(n, d) is the smallest number that starts an arithmetic progression with common difference d of n numbers with the same number of divisors.

1, 2, 1, 33, 3, 1, 242, 3, 2, 1, 11605, 213, 119, 3, 1, 28374, 213, 3445, 3, 2, 1, 171893, 1383, 15026, 111, 77, 5, 1, 1043710445721, 3091, 74783, 201, 8718, 5, 8, 1

Offset: 1

David Wasserman, Jan 08 2006

First two columns are A006558 and A113466. First four rows are A000012, A065559, A113467 and A113468. a(9, 1) is unknown; the rest of the 9th antidiagonal is 8129,88015,201,8718,5,8,3,1.

			a(4, 3) = 3445 because 3445, 3448, 3451 and 3454 each have 8 divisors.

Cf. A006558, A065559, A113456, A113466, A113467, A113468.

A276713 Numbers n such that n and n+3 have the same number of divisors (A000005).

Original entry on oeis.org

2, 35, 55, 62, 74, 82, 91, 102, 115, 119, 122, 135, 142, 143, 155, 158, 172, 186, 202, 203, 206, 214, 215, 218, 242, 255, 259, 262, 282, 295, 298, 299, 302, 323, 326, 343, 351, 354, 355, 362, 391, 395, 399, 425, 426, 435, 451, 466, 478, 482, 492, 502, 511, 514

Offset: 1

Jaroslav Krizek, Sep 16 2016

nonn

			35 is in sequence because tau(35) = tau(38) = 4.

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

Cf. A065559 (smallest k such that tau(k) = tau(k+n)), A015861 (sigma(n) = sigma(n+3)), A276714.

Cf. Similar sequences with numbers n such that n and n+k have the same number of divisors for k = 1: A005237, for k = 2: A062832.

[n: n in [1..10000] | NumberOfDivisors(n) eq  NumberOfDivisors(n+3)]

with(numtheory): A276713:=n->`if`(tau(n) = tau(n+3), n, NULL): seq(A276713(n), n=1..10^3); # Wesley Ivan Hurt, May 02 2017

SequencePosition[DivisorSigma[0,Range[600]],{x_,,,x_}][[All,1]] (* Harvey P. Dale, Nov 12 2022 *)

isok(n) = numdiv(n) == numdiv(n+3); \\ Michel Marcus, May 03 2017

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

A347338 a(n) is the smallest number k such that tau(k) = tau(k+n), and there is no number m, k < m < k+n such that tau(m) = tau(k).

Original entry on oeis.org

2, 3, 35, 7, 4, 12, 39, 20, 146, 30, 52, 32, 175, 88, 693, 9, 99, 108, 188, 847, 1014, 392, 124, 25, 315, 234, 195, 416, 196, 477, 225, 48, 2262, 1327, 1330, 252, 368, 160, 1636, 640, 5067, 168, 441, 884, 1183, 1064, 1377, 120, 1328, 112, 4908, 3872, 891, 396, 512

Offset: 1

David James Sycamore, Aug 27 2021

nonn

The prohibition in the definition distinguishes this sequence from A065559. This sequence identifies the first occurrence of a gap between numbers with the same tau, where no intervening number has that tau. Each tau t > 1 has a corresponding sequence of gaps (e.g., for t = 2, A001223, for t = 3, A069482), and a(n) is the smallest index of terms in A000005 corresponding to the first occurrence of a gap of length n in all of these (same tau) gap sequences.

A number appearing in this sequence cannot appear again, and many numbers do not appear at all (1 is not in because it is the only number with 1 divisor; 5 6 and 8 are not in because 3 is already a term; 10 is not in because 7 is a term, etc.).

			a(1) = 2 because 2 is the smallest k such that tau(k) = tau(k+1); a(2) = 3 because it is the smallest k with tau(k) = tau(k+2) with no intervening same tau number; a(3) = 35 because d(35) = 4, d(36) = 9, d(37) = 2, d(38) = 4 = d(35) and this is the least case of a gap of 3. (Here d means tau, namely, A000005.)

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf: A000005, A001223, A005237, A069482, A276963, A005179, A065559.

Block[{a = {}, k, d = DivisorSigma[0, Range[2^14]]}, Do[k = 1; While[Or[#[[1]] != #[[-1]], Count[#, #[[1]]] > 2] &@ d[[k ;; k + n]], k++]; AppendTo[a, k], {n, 55}]; a] (* Michael De Vlieger, Aug 27 2021 *)

a(n) = {my(i); for(i = 1, oo, if(iscan(i, n), return(i) ) ) }
iscan(k, n) = { my(c); c = numdiv(k); if(numdiv(k + n) != c, return(0) ); for(i = 1, n-1, if(numdiv(k + i) == c, return(0) ) ); 1 } \\ David A. Corneth, Aug 27 2021

from sympy import divisor_count
from functools import lru_cache
@lru_cache(maxsize=None)
def tau(n): return divisor_count(n)
def a(n):
    k = 2
    while True:
        while tau(k) != tau(k+n): k += 1
        if not any(tau(m) == tau(k) for m in range(k+1, k+n)): return k
        k += 1
print([a(n) for n in range(1, 56)]) # Michael S. Branicky, Aug 27 2021

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

A113465 Rectangular array read by antidiagonals: a(n, d) is the smallest number that starts an arithmetic progression with common difference d of n numbers with the same number of divisors.

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A276713 Numbers n such that n and n+3 have the same number of divisors (A000005).

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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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A347338 a(n) is the smallest number k such that tau(k) = tau(k+n), and there is no number m, k < m < k+n such that tau(m) = tau(k).

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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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