A033075 -id:A033075 - cp's OEIS frontend

A355078 a(n) is the smallest number k with exactly n divisors in A033075.

1, 2, 4, 6, 18, 12, 24, 60, 120, 168, 360, 1080, 840, 3360, 2520, 7560, 15120, 30240, 84840, 196560, 339360, 254520, 1102920, 763560, 1527120, 4581360, 3054240, 9162720, 9926280, 19852560, 59557680, 39705120, 119115360, 277935840, 674987040, 1151448480, 1469089440

Offset: 1

Marius A. Burtea, Jul 11 2022

			1 has a single divisor in A033075, so a(1) = 1.
2 has divisors 1 and 2 in A033075, so a(2) = 2;
3 has only divisors 1, 3 in A033075, 4 has divisors 1, 2, 4 in A033075, so a(3) = 4.
5 has only divisors 1, 5 in A033075, 6 has divisors 1, 2, 3, 6 in A033075, so a(4) = 6.

Cf. A033075.

alt:=func; a:=[]; for n in [1..37] do k:=1; while #[d:d in Divisors(k)|alt(d)] ne n do k:=k+1; end while; Append(~a,k); end for; a;

diff(v)=vector(#v-1, i, v[i+1]-v[i]);
is(n)=if(n>9, Set(abs(diff(digits(n))))==[1], n>0);
a(n) = my(k=1); while (sumdiv(k, d, is(d)) != n, k++); k; \\ Michel Marcus, Jul 11 2022

A048398 Primes with consecutive digits that differ exactly by 1.

Original entry on oeis.org

2, 3, 5, 7, 23, 43, 67, 89, 101, 787, 4567, 12101, 12323, 12343, 32321, 32323, 34543, 54323, 56543, 56767, 76543, 78787, 78989, 210101, 212123, 234323, 234343, 432121, 432323, 432343, 434323, 454543, 456767, 654323, 654343, 678767, 678989

Offset: 1

Patrick De Geest, Apr 15 1999

Or, primes in A033075. - Zak Seidov, Feb 01 2011

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 67, p. 23, Ellipses, Paris 2008.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms n = 1..1500 from Zak Seidov)

Cf. A033075, A048399-A048405, A052016, A052017, A006055.

Cf. A010051; intersection of A033075 and A000040.

a048398 n = a048398_list !! (n-1)
a048398_list = filter ((== 1) . a010051') a033075_list
-- Reinhard Zumkeller, Feb 21 2012, Nov 04 2010
(Python 3.2 or higher)
from itertools import product, accumulate
from sympy import isprime
A048398_list = [2,3,5,7]
for l in range(1,17):
    for d in [1,3,7,9]:
        dlist = [d]*l
        for elist in product([-1,1],repeat=l):
            flist = [str(d+e) for d,e in zip(dlist,accumulate(elist)) if 0 <= d+e < 10]
            if len(flist) == l and flist[-1] != '0':
                n = 10*int(''.join(flist[::-1]))+d
                if isprime(n):
                    A048398_list.append(n)
A048398_list = sorted(A048398_list) # Chai Wah Wu, May 31 2017

Select[Prime[Range[10000]], # < 10 || Union[Abs[Differences[IntegerDigits[#]]]] == {1} &]

A215014 Numbers where any two consecutive decimal digits differ by 1 after arranging the digits in decreasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 21, 23, 32, 34, 43, 45, 54, 56, 65, 67, 76, 78, 87, 89, 98, 102, 120, 123, 132, 201, 210, 213, 231, 234, 243, 312, 321, 324, 342, 345, 354, 423, 432, 435, 453, 456, 465, 534, 543, 546, 564, 567, 576, 645, 654, 657, 675, 678, 687, 756, 765, 768, 786, 789, 798, 867

Offset: 1

Tanya Khovanova and Charles R Greathouse IV, Jul 31 2012

a(4091131) = 9876543210 is the last term.
Numbers n such that A004186(n) is a term of A033075. - Felix Fröhlich, Dec 26 2017
Also 0 together with positive integers having k distinct digits and the difference between the largest and the smallest digit equal to k-1. - David A. Corneth, Dec 26 2017

Ely Golden, Table of n, a(n) for n = 1..10000

Cf. A004186, A033075.

lst = {}; Do[If[Times @@ Differences@Sort@IntegerDigits[n] == 1, AppendTo[lst, n]], {n, 0, 675}]; lst (* Arkadiusz Wesolowski, Aug 01 2012 *)
Join[Range[0,9],Select[Range[1000],Union[Differences[Sort[ IntegerDigits[ #]]]] == {1}&]] (* Harvey P. Dale, Jan 14 2015 *)

is(n)=my(v=vecsort(eval(Vec(Str(n)))));for(i=2,#v,if(v[i]!=1+v[i-1],return(0)));1

is(n) = if(!n, return(1)); my(d = digits(n), v = vecsort(d,,8)); #d == #v && v[#v] - v[1] == #v - 1

# Ely Golden, Dec 26 2017
def consecutive(li):
  for i in range(len(li)-1):
    if(li[i+1]!=1+li[i]): return False
  return True
def sorted_digits(n):
  lst=[]
  while(n>0):
    lst+=[n%10] ; n//=10
  lst.sort() ; return lst
j=0
for i in range(1,10001):
  while(not consecutive(sorted_digits(j))): j+=1
  print(str(i)+" "+str(j)) ; j+=1

# alternate for generating full sequence in seconds
from itertools import permutations as perms
frags = ["0123456789"[i:j] for i in range(10) for j in range(i+1, 11)]
afull = sorted(set(int("".join(s)) for f in frags for s in perms(f)))
print(afull[:70]) # Michael S. Branicky, Aug 04 2022

If zero is excluded, the number of terms with k digits, 1 <= k <= 10, is (11-k)*k! - (k-1)!. - Franklin T. Adams-Watters, Aug 01 2012

Name edited by Felix Fröhlich, Dec 26 2017

A090994 Number of meaningful differential operations of the n-th order on the space R^9.

Original entry on oeis.org

9, 17, 32, 61, 116, 222, 424, 813, 1556, 2986, 5721, 10982, 21053, 40416, 77505, 148785, 285380, 547810, 1050876, 2017126, 3869845, 7427671, 14250855, 27351502, 52479500, 100719775, 193258375, 370895324, 711682501, 1365808847, 2620797529

Offset: 1

Branko Malesevic, Feb 29 2004

nonn

Also number of meaningful compositions of the n-th order of the differential operations and Gateaux directional derivative on the space R^8. - Branko Malesevic and Ivana Jovovic (ivana121(AT)EUnet.yu), Jun 21 2007
Also (starting 5,9,...) the number of zig-zag paths from top to bottom of a rectangle of width 10, whose color is that of the top right corner. [From Joseph Myers, Dec 23 2008]
Also, number of n-digit terms in A033075 (stated without proof in A033075). - Zak Seidov, Feb 02 2011

G. C. Greubel, Table of n, a(n) for n = 1..1000
B. Malesevic, Some combinatorial aspects of differential operation composition on the space R^n, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 9 (1998), 29-33.
Branko Malesevic, Some combinatorial aspects of differential operation compositions on space R^n, arXiv:0704.0750 [math.DG], 2007.
B. Malesevic and I. Jovovic, The Compositions of the Differential Operations and Gateaux Directional Derivative, arxiv:0706.0249 [math.CO], 2007.
Joseph Myers, BMO 2008--2009 Round 1 Problem 1---Generalisation
Index entries for linear recurrences with constant coefficients, signature (1, 4, -3, -3, 1).

Cf. A090989-A090995, A000079, A007283, A020701, A020714, A129638, A033075.

a:=[9,17,32,61,116];; for n in [6..40] do a[n]:=a[n-1]+4*a[n-2] - 3*a[n-3]-3*a[n-4]+a[n-5]; od; a; # G. C. Greubel, Feb 02 2019

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(  x*(9+8*x-21*x^2-12*x^3+5*x^4)/(1-x-4*x^2+3*x^3+3*x^4-x^5) )); // G. C. Greubel, Feb 02 2019

NUM := proc(k :: integer) local i,j,n,Fun,Identity,v,A; n := 9; # <- DIMENSION Fun := (i,j)->piecewise(((j=i+1) or (i+j=n+1)),1,0); Identity := (i,j)->piecewise(i=j,1,0); v := matrix(1,n,1); A := piecewise(k>1,(matrix(n,n,Fun))^(k-1),k=1,matrix(n,n,Identity)); return(evalm(v&*A&*transpose(v))[1,1]); end:

LinearRecurrence[{1, 4, -3, -3, 1}, {9, 17, 32, 61, 116}, 31] (* Jean-François Alcover, Nov 20 2017 *)

my(x='x+O('x^40)); Vec(x*(9+8*x-21*x^2-12*x^3+5*x^4)/(1-x-4*x^2 +3*x^3+3*x^4-x^5)) \\ G. C. Greubel, Feb 02 2019

a=(x*(9+8*x-21*x^2-12*x^3+5*x^4)/(1-x-4*x^2+3*x^3+3*x^4-x^5)).series(x, 40).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 02 2019

a(k+5) = a(k+4) + 4*a(k+3) - 3*a(k+2) - 3*a(k+1) + a(k).

G.f.: x*(9+8*x-21*x^2-12*x^3+5*x^4)/(1-x-4*x^2+3*x^3+3*x^4-x^5). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009; corrected by R. J. Mathar, Sep 16 2009

More terms from Joseph Myers, Dec 23 2008

A228326 Start with 0. Successive digits in the sequence must differ by 1. Adjoin the smallest number not yet in the sequence.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 87, 65, 43, 21, 23, 45, 67, 89, 876, 54, 32, 10, 12, 34, 56, 76, 78, 98, 765, 432, 101, 210, 121, 212, 123, 232, 321, 234, 323, 434, 343, 454, 345, 456, 543, 2101, 2121, 2123, 2321, 2323, 2343, 2345, 654, 545, 656, 565, 676, 567

Offset: 1

N. J. A. Sloane, Aug 24 2013

Is the sequence infinite?

Yes, as for any digit d, there are infinitely many terms in A033075 with leading digit d+1 or d-1. - Paul Tek Aug 25 2013

Eric Angelini, Posting to the Sequence Fans Mailing List, Aug 23 2013.

Paul Tek, Table of n, a(n) for n = 1..10000
Paul Tek, PERL program for this sequence

Cf. A228327, A228328, A033075.

A134336 Nonnegative integers n containing each digit between n's smallest and largest decimal digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 21, 22, 23, 32, 33, 34, 43, 44, 45, 54, 55, 56, 65, 66, 67, 76, 77, 78, 87, 88, 89, 98, 99, 100, 101, 102, 110, 111, 112, 120, 121, 122, 123, 132, 201, 210, 211, 212, 213, 221, 222, 223, 231, 232, 233, 234, 243, 312, 321, 322, 323

Offset: 1

Rick L. Shepherd, Oct 21 2007

A032981 is a subsequence; the term 102 is the first positive integer not also in A032981. A171102 (pandigital numbers) and A033075 are subsequences. Union of A010785 (repdigits) and A108965.
a(n) = A178403(n) for n < 48. - Reinhard Zumkeller, May 27 2010
Equivalently, numbers with the property that the set of its decimal digits is a set of consecutive numbers. - Tanya Khovanova and Charles R Greathouse IV, Jul 31 2012

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A032981, A050278, A033075 (a subsequence), A010785, A108965.

is(n)=my(v=vecsort(eval(Vec(Str(n))),,8));for(i=2,#v,if(v[i]!=1+v[i-1],return(0)));1 \\ Tanya Khovanova and Charles R Greathouse IV, Jul 31 2012

is_A134336(n)={vecmax(n=Set(digits(n)))-vecmin(n)==#n-1} \\ M. F. Hasler, Dec 24 2014

def ok(n): d = sorted(set(map(int, str(n)))); return d[-1]-d[0]+1 == len(d)
print([k for k in range(324) if ok(k)]) # Michael S. Branicky, Dec 12 2023

a(n) ~ n. - Charles R Greathouse IV, Sep 09 2011

Edited by N. J. A. Sloane, Aug 06 2012

A048410 Numbers whose consecutive digits differ by 8.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 80, 91, 191, 808, 919, 1919, 8080, 9191, 19191, 80808, 91919, 191919, 808080, 919191, 1919191, 8080808, 9191919, 19191919, 80808080, 91919191, 191919191, 808080808, 919191919, 1919191919, 8080808080

Offset: 0

Patrick De Geest, Apr 15 1999

Cf. A033075, A033088, A033081, A048407-A048409.

With[{nn=10},Join[Range[0,9],Table[FromDigits[PadRight[{},n,{1,9}]],{n,nn}], Table[FromDigits[PadRight[{},m,{9,1}]],{m,nn}],Table[ FromDigits[ PadRight[ {},k,{8,0}]],{k,nn}]]]//Union (* Harvey P. Dale, Jul 17 2017 *)

A048407 Numbers whose consecutive digits differ by 5.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 27, 38, 49, 50, 61, 72, 83, 94, 161, 272, 383, 494, 505, 616, 727, 838, 949, 1616, 2727, 3838, 4949, 5050, 6161, 7272, 8383, 9494, 16161, 27272, 38383, 49494, 50505, 61616, 72727, 83838, 94949, 161616, 272727, 383838

Offset: 0

Patrick De Geest, Apr 15 1999

Cf. A033075, A033088, A033081, A048406-A048410.

Join[Range[0,9],Select[Range[400000],Union[Abs[Differences[ IntegerDigits[ #]]]] == {5}&]] (* Harvey P. Dale, Sep 23 2013 *)

A048406 Numbers whose consecutive digits differ by 4.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 26, 37, 40, 48, 51, 59, 62, 73, 84, 95, 151, 159, 262, 373, 404, 484, 515, 595, 626, 737, 840, 848, 951, 959, 1515, 1595, 2626, 3737, 4040, 4048, 4840, 4848, 5151, 5159, 5951, 5959, 6262, 7373, 8404, 8484, 9515, 9595, 15151

Offset: 0

Patrick De Geest, Apr 15 1999

Cf. A033075, A033088, A033081, A048407-A048410.

A048411 Squares whose consecutive digits differ by 1.

Original entry on oeis.org

0, 1, 4, 9, 121, 676, 12321, 1234321, 123454321, 12345654321, 1234567654321, 123456787654321, 12345678987654321

Offset: 1

Patrick De Geest, Apr 15 1999

a(14), if it exists, is > 10^34. - Lars Blomberg, Nov 25 2016
Is it true that all terms are palindromes? - Chai Wah Wu, Apr 06 2018
a(14), if it exists, is > 10^52. - Michael S. Branicky, Apr 22 2025

Cf. A002477, A048412.

Cf. A010052; intersection of A033075 and A000290.

a048411 n = a048411_list !! (n-1)
a048411_list = filter ((== 1) . a010052) a033075_list
-- Reinhard Zumkeller, Feb 21 2012

Select[Range[0, 10^7]^2, Or[# == 0, IntegerLength@ # == 1, Union@ Abs@ Differences@ IntegerDigits@ # == {1}] &] (* Michael De Vlieger, Nov 25 2016 *)

from sympy.ntheory.primetest import is_square
def gen(d, s=None):
    if d == 0: yield tuple(); return
    if s == None:
        yield from [(i, ) + g for i in range(1, 10) for g in gen(d-1, s=i)]
    else:
        if s > 0: yield from [(s-1, ) + g for g in gen(d-1, s=s-1)]
        if s < 9: yield from [(s+1, ) + g for g in gen(d-1, s=s+1)]
def afind(maxdigits):
    print(0, end=", ")
    for d in range(1, maxdigits+1):
        for g in gen(d, s=None):
            t = int("".join(map(str, g)))
            if is_square(t): print(t, end=", ")
afind(17) # Michael S. Branicky, Sep 26 2021

a(n) = A048412(n)^2.

cp's OEIS Frontend

A355078 a(n) is the smallest number k with exactly n divisors in A033075.

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A048398 Primes with consecutive digits that differ exactly by 1.

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A215014 Numbers where any two consecutive decimal digits differ by 1 after arranging the digits in decreasing order.

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A090994 Number of meaningful differential operations of the n-th order on the space R^9.

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A228326 Start with 0. Successive digits in the sequence must differ by 1. Adjoin the smallest number not yet in the sequence.

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A134336 Nonnegative integers n containing each digit between n's smallest and largest decimal digits.

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PARI

PARI

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A048410 Numbers whose consecutive digits differ by 8.

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Mathematica