A080718 -id:A080718 - 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

A048936 Subset of vampire numbers A014575 having exactly two representations of the desired form.

Original entry on oeis.org

125460, 11930170, 12054060, 12417993, 12600324, 12827650, 13002462, 22569480, 23287176, 26198073, 26373600, 26839800, 46847920, 61360780, 1001795850, 1013265360, 1017509850, 1018172470, 1044022896, 1047395790

Offset: 1

Eric W. Weisstein

			125460 = 204*615 = 246*510.
11930170 = 1301*9170 = 1310*9107.
12054060 = 2004*6015 = 2406*5010.

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

Shyam Sunder Gupta, Cab and Vampire Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 20, 499-512.
Eric Weisstein's World of Mathematics, Vampire Number.

Closely related: A020342, A014575, A080718, A280928, A144563.

Cf. A014575, A048933, ..., A048939.

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

Name edited by M. F. Hasler, Mar 11 2021

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")

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

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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A048936 Subset of vampire numbers A014575 having exactly two representations of the desired form.

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

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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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