A048936 -id:A048936 - cp's OEIS frontend

A014575 Vampire numbers (definition 2): numbers n with an even number of digits which have a factorization n = i*j where i and j have the same number of digits and the multiset of the digits of n coincides with the multiset of the digits of i and j.

1260, 1395, 1435, 1530, 1827, 2187, 6880, 102510, 104260, 105210, 105264, 105750, 108135, 110758, 115672, 116725, 117067, 118440, 120600, 123354, 124483, 125248, 125433, 125460, 125500, 126027, 126846, 129640

Offset: 1

Eric W. Weisstein

The numbers i and j may not both have trailing zeros. Numbers may have more than one such factorization. However, each n is listed only once. [Comment modified by Rick L. Shepherd, Nov 02 2009]

			1260 = 21*60, 1395 = 15*93, 1435 = 35*41, 1530 = 30*51, etc.

C. A. Pickover, "Vampire Numbers." Ch. 30 in Keys to Infinity. New York: Wiley, pp. 227-231, 1995.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (terms a(1)-a(87) by R. J. Mathar and a(88)-a(1006) by Manfred Scheucher)
Ely Golden, Sympy program for generating vampire numbers (definition 2)
Shyam Sunder Gupta, Cab and Vampire Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 20, 499-512.
Manfred Scheucher, Sage Script
Eric Weisstein's World of Mathematics, Vampire Number

The following sequences are all closely related: A020342, A014575, A080718, A280928, A048936, A144563.

Cf. A048933, A048934, A048935, A048936, A048937, A048938, A048939.

n := 1 :
for dgs from 4 to 10 by 2 do
    for a from 10^(dgs-1) to 10^dgs-1 do
        amset := sort(convert(a,base,10)) ;
        isv := false ;
        for d in numtheory[divisors](a) do
            m := a/d ;
            if ( m >= d ) then
                dset := convert(d,base,10) ;
                mset := convert(m,base,10) ;
                fset := sort([op(dset),op(mset)]) ;
                if fset = amset and nops(dset) = nops(mset) then
                    if (m mod 10 <> 0 ) or (d mod 10 <> 0 ) then
                    printf("%d %d\n",n,a) ;
                    isv := true ;
                    n := n+1 ;
                    end if;
                end if;
            end if;
            if isv then
                break;
            end if;
        end do:
    end do:
end do: # R. J. Mathar, Jan 10 2013

fQ[n_] := If[OddQ@ IntegerLength@ n, False, MemberQ[Map[Sort@ Flatten@ IntegerDigits@ # &, Select[Map[{#, n/#} &, TakeWhile[Divisors@ n, # <= Sqrt@ n &]], SameQ @@ Map[IntegerLength, #] &]], Sort@ IntegerDigits@ n]]; Select[Range[10^6], fQ] (* Michael De Vlieger, Jan 27 2017 *)

is(n)=my(v=digits(n));if(#v%2,return(0));fordiv(n,d,if(#Str(d)==#v/2 && #Str(n/d)==#v/2 && vecsort(v)==vecsort(digits(eval(Str(d,n/d)))) && (d%10 || (n/d)%10), return(1)));0 \\ Charles R Greathouse IV, Apr 19 2013

is_A014575(n)={my(v=vecsort(Vecsmall(Str(n)))); #v%2 && return; my( M=10^(#v\2), L=M\10); fordiv(n,d, dA048933) if vampire number, or false (empty, 0) else. - M. F. Hasler, Mar 11 2021

Edited by N. J. A. Sloane, Jan 03 2009

A020342 Vampire numbers: (definition 1): n has a nontrivial factorization using n's digits.

Original entry on oeis.org

126, 153, 688, 1206, 1255, 1260, 1395, 1435, 1503, 1530, 1827, 2187, 3159, 3784, 6880, 10251, 10255, 10426, 10521, 10525, 10575, 11259, 11439, 11844, 11848, 12006, 12060, 12384, 12505, 12546, 12550, 12595, 12600, 12762, 12768, 12798, 12843, 12955, 12964

Offset: 1

David W. Wilson

Nontrivial means that there must be at least two factors.

For any a(n), 10*a(n) is also in the sequence, and also in A144563. - M. F. Hasler, Nov 01 2021

			1395 = 31*9*5, so 1395 is a term.
179739 = 7 * 9 * 9 * 317 so 179739 is a term. - _Sean A. Irvine_, Feb 28 2023

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 65.

Ely Golden, Table of n, a(n) for n = 1..1000 (corrected by _Sean A. Irvine_)
Ely Golden, Sympy program for generating vampire numbers (definition 1)
Shyam Sunder Gupta, Cab and Vampire Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 20, 499-512.
Gordon Hamilton, Three integer sequences from recreational mathematics, Video (2013).
Sean A. Irvine, Java program (github)

Closely related: A014575, A080718, A280928, A048936, A144563.

is_A020342(n, m=0, D=vecsort(digits(n)))={ if(m && n >= m && vecsort(digits(n))==D, 1, #D<3, m && (D[1]>=m && vecprod(D)==n), n >= m^2, my(S=Set(D), i, C); fordiv(n,f, f < m && next; f*f > n && break; setminus(Set(digits(f)),S) && next; C=D; foreach(digits(f),d, if(i=vecsearch(C,d), C=C[^i], next(2))); C && is_A020342(n\f, f, C) && return(1)))} \\ See A144563 for a function counting the multiplicity of the representation. - M. F. Hasler, Nov 01 2021

Edited by N. J. A. Sloane, Jan 03 2009

A280928 Composite numbers having the same digits as their prime factors (with multiplicity), including zero digits.

Original entry on oeis.org

1255, 12955, 17482, 25105, 100255, 101299, 105295, 107329, 117067, 124483, 127417, 129595, 132565, 145273, 146137, 149782, 163797, 174082, 174298, 174793, 174982, 250105, 256315, 263155, 295105, 297463, 307183, 325615, 371893, 536539, 687919, 1002955, 1004251, 1012099, 1025095, 1029955

Offset: 1

Ely Golden, Jan 11 2017

Subsequence of A176670 as well as A020342.
Is this sequence the intersection of A176670 and A020342?
Excluding 1, all members of A080718 are members of this sequence. The first member of this sequence that is not a member of A080718 is a(17)=163797.

			100255 is a member of this sequence as 100255 = 5*20051, which is exactly the same set of digits as 100255.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..3643 from Ely Golden)

The following sequences are all closely related: A020342, A014575, A080718, A280928, A048936, A144563.

Cf. A176670

from sympy import factorint
def ok(n):
    f = factorint(n)
    return sum(f.values()) > 1 and sorted(str(n)) == sorted("".join(str(p)*f[p] for p in f))
print([k for k in range(700000) if ok(k)]) # Michael S. Branicky, Apr 20 2025

def digits(x, n):
    if((x<=0)|(n<2)):
        return []
    li=[]
    while(x>0):
        d=divmod(x, n)
        li.append(d[1])
        x=d[0]
    li.sort()
    return li;
def factorDigits(x, n):
    if((x<=0)|(n<2)):
        return []
    li=[]
    f=list(factor(x))
    #ensures inequality of digits(x, n) and factorDigits(x, n) if x is prime
    if((len(f)==1)&(f[0][1]==1)):
        return [];
    for c in range(len(f)):
        for d in range(f[c][1]):
            ld=digits(f[c][0], n)
            li+=ld
    li.sort()
    return li;
#this variable affects the radix
radix=10
c=2
index=1
while(index<=100):
    if(digits(c,radix)==factorDigits(c,radix)):
        print(str(index)+" "+str(c))
        index+=1
    c+=1
print("complete")

A080718 1, together with numbers n that are the product of two primes p and q such that the multiset of the digits of n coincides with the multiset of the digits of p and q.

Original entry on oeis.org

1, 1255, 12955, 17482, 25105, 100255, 101299, 105295, 107329, 117067, 124483, 127417, 129595, 132565, 145273, 146137, 149782, 174082, 174298, 174793, 174982, 250105, 256315, 263155, 295105, 297463, 307183, 325615, 371893, 536539

Offset: 1

Jeff Heleen, Mar 06 2003

Except for 1, this sequence is a subsequence of A280928. More specifically, members of A280928 are also members of this sequence if and only if they are semiprime. - Ely Golden, Jan 11 2017
This sequence has no equivalent in odd bases. This is because any equivalent of A280928 in an odd base must have all terms having at least 3 prime factors. - Ely Golden, Jan 11 2017
All entries other than 1 are congruent to 4 mod 9, because p*q == p + q mod 9 (with p and q not both divisible by 3) implies p*q == 4 mod 9. - Robert Israel, May 05 2014

			1255 = 5*251, 12955 = 5*2591, 17482 = 2*8741, 100255 = 5*20051, 146137=317*461, etc.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..145 from Robert Israel)

The following sequences are all closely related: A020342, A014575, A080718, A280928, A048936, A144563.

filter:= proc(n) local F,p,q,Ln,Lpq;
  F:= ifactors(n)[2];
  if nops(F) > 2 or convert(F,`+`)[2]<>2 then return false fi;
  p:= F[1][1];
  if nops(F) = 2 then q:= F[2][1] else q:= F[1][1] fi;
  Ln:= sort(convert(n,base,10));
  Lpq:= sort([op(convert(p,base,10)),op(convert(q,base,10))]);
  evalb(Ln = Lpq);
end proc:
filter(1):= true:
A080718:= select(filter,[1, seq(4+9*i,i=1..10^6)]); # Robert Israel, May 04 2014

ptpQ[n_]:=Module[{sidn=Sort[IntegerDigits[n]],fi=Transpose[ FactorInteger[ n]]}, fi[[2]]=={1,1}&&Sort[Flatten[ IntegerDigits/@ fi[[1]]]]==sidn]; Join[{1}, Select[Range[4,550000,9],ptpQ]] (* Harvey P. Dale, Jun 22 2014 *)

from sympy import factorint
from itertools import count, islice
def agen(): # generator
    yield 1
    for k in count(4, 9):
        t = sorted(str(k))
        f = factorint(k)
        if sum(f.values()) == 2:
            p, q = min(f), max(f)
            if t == sorted(str(p)+str(q)):
                yield k
print(list(islice(agen(), 30))) # Michael S. Branicky, Apr 20 2025

Edited by N. J. A. Sloane, Jan 03 2009

Incorrect entry 163797 removed by Robert Israel, May 04 2014

A144563 Subset of A020342 (vampire numbers, definition 1) listing numbers which have more than one such representation of the desired form.

Original entry on oeis.org

1260, 1395, 1530, 6880, 12060, 12550, 12600, 13950, 14350, 15030, 15300, 18270, 21870, 31590, 37840, 68800, 102510, 102550, 104260, 105210, 105250, 105264, 105750, 112590, 114390, 116928, 118440, 118480, 119682, 120060, 120600, 123840, 125050, 125460, 125500, 125950

Offset: 1

N. J. A. Sloane, Jan 03 2009, based on email from Zak Seidov

From M. F. Hasler, Nov 01 2021: (Start)
If x is in A020342, then 10*x is in this sequence, and this makes up most of the terms. Exceptions are the terms not ending in 0, {1395, 105264, 116928, 119682, 192375, 258795, 263736, 268398, 289674, 1008126, 1133484, 1173939, ...}. There are terms of the form 10*x in A020342 with x not in A020342, like {25510, 45760, 67950, 136590, 146520, 168520, 175560, 246150, 250510, 255010, ...}. Is any such term in this sequence A144563, or can it be proved there isn't?
All terms have at least 3 distinct prime factors (omega, A001221), and 4 prime factors counted with multiplicity (bigomega, A001222). The squarefree terms are {132430, 174370, 1012990, 1073290, 1094730, 1156990, 1170670, 117393, ...}. (End)

			1260 = 21*60 = 6*210. 1395 = 5*9*31 = 15*93. 1530 = 30*51 = 3*510.

M. F. Hasler, Table of n, a(n) for n = 1..184, Nov 01 2021

Closely related: A020342, A014575, A080718, A280928, A048936.

/* Helper function: count number of representations of n as product of numbers >= m whose multiset of digits is D, excluding the trivial representation if m = 0. */
VampRepCount(n, m=0, D=vecsort(digits(n)))={ if(#D<3, m && (D[1]>=m && vecprod(D)==n || vecsort(digits(n))==D), n >= m^2, my(S=Set(D), c=m && vecsort(digits(n))==D, i, C); fordiv(n, f, f < m && next; f*f > n && break; setminus(Set(digits(f)), S) && next; C=D; foreach(digits(f), d, if(i = vecsearch(C,d), C=C[^i], next(2))); C && c += VampRepCount(n\f, f, C) ); c, n >= m && vecsort(digits(n))==D)}
is_A144563(n)=VampRepCount(n)>1 \\ M. F. Hasler, Nov 01 2021

Subsequence of A020342; contains 10*A020342 as a subsequence. - M. F. Hasler, Nov 01 2021

Corrected A-number in definition. More terms and examples R. J. Mathar, Jan 05 2009

Terms beyond a(15) by M. F. Hasler, Nov 01 2021

A048933 Smallest factor i of any factorization used in the definition of A014575(n).

Original entry on oeis.org

21, 15, 35, 30, 21, 27, 80, 201, 260, 210, 204, 150, 135, 158, 152, 161, 167, 141, 201, 231, 281, 152, 231, 204, 251, 201, 261, 140, 179, 311, 323, 315, 317, 231, 351, 215, 146, 350, 351, 317, 156, 300, 251, 261, 356, 240, 269, 165, 176, 396, 221, 231, 371, 231, 225, 201, 225, 281, 216, 210, 210, 327, 395, 275, 252, 255

Offset: 1

Eric W. Weisstein

C. A. Pickover, "Vampire Numbers." Ch. 30 in Keys to Infinity. New York: Wiley, pp. 227-231, 1995.

Eric Weisstein's World of Mathematics, Vampire Number.

The following sequences are all closely related: A020342, A014575, A080718, A048936, A144563.

Cf. A048934, ..., A048939.

# print d, not a, in the program of A014575 - R. J. Mathar, Jul 15 2016

Edited by N. J. A. Sloane, Jan 03 2009

More terms from R. J. Mathar, Jul 15 2016

A262743 Predestined numbers: every n number is generated by at least one pair of products, such as n = ab = cd, and the multiset of the digits of a and b coincides with the multiset of the digits of c and d.

Original entry on oeis.org

64, 95, 130, 242, 325, 326, 392, 396, 435, 504, 544, 552, 572, 585, 632, 644, 664, 693, 740, 748, 756, 762, 770, 784, 806, 868, 952, 968, 973, 986, 990, 995, 1008

Offset: 1

Francesco Di Matteo, Sep 29 2015

For each pair of products, no more than 1 of the 4 numbers involved may have a trailing zero. E.g., 20 = 1*20 = 2*10 is trivial. Even 3920 = 49*80 = 4*980 is not valid (but note that 392 = 4*98 = 8*49 is a valid term). This rule is similar to that of the vampire numbers (A014575), and prevents trivial proliferations.
The name recalls the "genetic" metaphor of such numbers, that even if genes/digits are recombined, they remain with the same "destiny".
This sequence looks similar to A048936 (a subset of A014575, vampire numbers) but it's not the same because the factors' digits are exclusively the same as those of n, and also see, e.g., 11930170 = 1301*9170 = 1310*9107, which is not a valid predestined number.

			64  = 1*64 = 4*16;
95  = 1*95 = 5*19;
130 = 2*65 = 5*26;
242 = 2*121 = 11*22, etc.

Francesco Di Matteo, Sequenze ludiche, Game Edizioni (2015), pages 28-37.

Francesco Di Matteo, Table of n, a(n) for n = 1..2815
F. Di Matteo and A. Marchini, All the first 2815 terms calculated.

Cf. A245386, A262873.

good[w_] := Block[{L = {}}, Do[If[Length[Select[Join[w[[i]], w[[j]]], Mod[#, 10] == 0 &]] <= 1, AppendTo[L, {w[[i]], w[[j]]}]], {i, Length@w}, {j, i - 1}]; L]; prQ[n_] := Block[{t, d = Select[Divisors@n, #^2 <= n &]}, t = (Last /@ #) & /@ Select[SplitBy[ Sort@ Table[{ Sort@ Join[ IntegerDigits@ e, IntegerDigits [n/e]], {e, n/e}}, {e, d}], First], Length[#] > 1 &]; g = Select[good /@ t, # != {} &]; g != {}]; (* then *) Select[Range[1000], prQ] (* or *) Do[If[prQ@ n, Print[n," ", Flatten[g, 1]]], {n, 10^5}] (* Giovanni Resta, Oct 07 2015 *)

A048938 8-digit vampire numbers (definition 2).

Original entry on oeis.org

10025010, 10042510, 10052010, 10052064, 10081260, 10102252, 10124352, 10124757, 10127475, 10128235, 10129900, 10133788, 10134985, 10149750, 10165680, 10165815, 10166350, 10173820, 10185934, 10192248, 10195794

Offset: 1

Eric W. Weisstein

C. A. Pickover, "Vampire Numbers." Ch. 30 in Keys to Infinity. New York: Wiley, pp. 227-231, 1995.

Sean A. Irvine, Table of n, a(n) for n = 1..3228
Eric Weisstein's World of Mathematics, Vampire Number.

Cf. A014575, A048933, ..., A048936.

A094128 Numbers n that can be factorized n=x*y and containing in their ternary representation the same digits the same number of times as x and y together.

Original entry on oeis.org

44, 116, 128, 132, 152, 296, 320, 332, 344, 348, 380, 384, 396, 440, 452, 456, 464, 476, 800, 872, 880, 888, 944, 960, 980, 992, 996, 1024, 1028, 1032, 1044, 1136, 1140, 1152, 1184, 1188, 1196, 1232, 1280, 1304, 1316, 1320, 1352, 1356, 1368

Offset: 1

Reinhard Zumkeller, Jun 01 2004

			320=16*20, 320->'102212' with 1x'0', 2x'1' and 3x'2': 16->'121' and 20->'202' have together the same ternary digits the same number of times, therefore 320 is a term.

J. K. Andersen, Vampire Numbers
Eric Weisstein's World of Mathematics, Vampire Number
Eric Weisstein's World of Mathematics, Ternary

Cf. A020342, A048936, A007089.

A094208 Least vampire number with n fang pairs.

Original entry on oeis.org

1260, 125460, 13078260, 16758243290880, 24959017348650

Offset: 1

Lekraj Beedassy, May 27 2004

			a(2) is the smallest number which can be written in 2 ways as the product of two numbers with half as many digits and not both ending in 0: 125460 = 204*615 = 246*510. - _Jens Kruse Andersen_, Jun 14 2014

J. K. Andersen, Vampire Numbers

Cf. A020342, A014575, A048936, A048937.

Typo in Cf. to A014575 fixed by Jens Kruse Andersen, Jun 14 2014

cp's OEIS Frontend

A014575 Vampire numbers (definition 2): numbers n with an even number of digits which have a factorization n = i*j where i and j have the same number of digits and the multiset of the digits of n coincides with the multiset of the digits of i and j.

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A020342 Vampire numbers: (definition 1): n has a nontrivial factorization using n's digits.

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A280928 Composite numbers having the same digits as their prime factors (with multiplicity), including zero digits.

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SageMath

A080718 1, together with numbers n that are the product of two primes p and q such that the multiset of the digits of n coincides with the multiset of the digits of p and q.

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A144563 Subset of A020342 (vampire numbers, definition 1) listing numbers which have more than one such representation of the desired form.

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A048933 Smallest factor i of any factorization used in the definition of A014575(n).

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A262743 Predestined numbers: every n number is generated by at least one pair of products, such as n = a*b = c*d, and the multiset of the digits of a and b coincides with the multiset of the digits of c and d.

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A262743 Predestined numbers: every n number is generated by at least one pair of products, such as n = ab = cd, and the multiset of the digits of a and b coincides with the multiset of the digits of c and d.