A020342 -id:A020342 - cp's OEIS frontend

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

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

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.

Original entry on oeis.org

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

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")

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

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

A370676 Number of unordered pairs of natural numbers k1, k2 such that their product is an n-digit number and has the same multiset of digits as in both k1 and k2.

Original entry on oeis.org

0, 0, 3, 15, 98, 596, 3626, 22704, 146834, 983476, 6846451, 49364315, 367660050

Offset: 1

Danila Potapov, Feb 26 2024

Since multiplication and multiset union are commutative operations, we count unordered pairs, i.e., we can assume that k1 <= k2.
The sequence could be redefined in terms of the number of distinct n-digit numbers that could be factorized into such pairs.
From David A. Corneth, Feb 27 2024: (Start)
a(n) >= 2*a(n-1).
As we need k1 + k2 == k1 * k2 (mod 9) there are two possible pairs of residues (k1, k2) mod 3, namely, (0, 0) and (2, 2), and six possible residues mod 9, namely, (0, 0), (2, 2), (3, 6), (5, 8), (6, 3), (8, 5). (End)

			For n=3 the a(3)=3 solutions are:
  3 * 51 = 153
  6 * 21 = 126
  8 * 86 = 688
For n=4 the a(4)=15 solutions are:
  3 * 501 = 1503
  3 * 510 = 1530
  5 * 251 = 1255
  6 * 201 = 1206
  6 * 210 = 1260
  8 * 473 = 3784
  8 * 860 = 6880
  9 * 351 = 3159
  15 * 93 = 1395
  21 * 60 = 1260
  21 * 87 = 1827
  27 * 81 = 2187
  30 * 51 = 1530
  35 * 41 = 1435
  80 * 86 = 6880

Cf. A370675 (number of such n-digit pairs), A020342.

Cf. A370678, A370679, A370680.

def a(n):
    count = 0
    for i in range(1, 10**(n-1)):
        for j in range(i, 10**n//i+1):
            if len(str(i*j)) == n and sorted(str(i)+str(j)) == sorted(str(i*j)):
                count += 1
    print(n, count)

a(9)-a(10) from Michael S. Branicky, Feb 26 2024
a(11) from Chai Wah Wu, Feb 27 2024
a(12)-a(13) from Martin Ehrenstein, Mar 02 2024

A179482 A subset of vampire numbers: n has a nontrivial factorization using n's digits in reverse order.

Original entry on oeis.org

126, 153, 688, 1395, 33579, 37668, 187029, 223524, 267034, 1008126, 1480368, 1514955, 1574253, 1766196, 1791495, 1831086, 1945944, 2784384, 10013323, 10353244, 18937617, 19437888, 23486976, 36528975, 38477586, 45334998, 48471696, 109019911, 116257833

Offset: 1

Adam Kertesz, Jul 16 2010

A subset of A020342.
Easy to prove that no vampire number has a factorization with n's digits in "normal" (left-to-right) order, so it was natural to search if any of the reverse order works.
A superset of A009944,permitting two or more(!) factors. [Adam Kertesz, Aug 07 2010]
Sequence is infinite, since it is a superset of A009944 which is infinite (see Comments at A009944). - Giovanni Resta, Mar 17 2013

			E.g. 126=6*21, 1395=5*9*31, 267034=4307*62.

Giovanni Resta, Table of n, a(n) for n = 1..155
Giovanni Resta, Decompositions of the first 155 terms

Cf. A020342, A009944.

a(10)-a(29) from Giovanni Resta, Mar 17 2013

A280967 Vampire numbers (definition 1) in binary: n has a nontrivial factorization using n's bits.

Original entry on oeis.org

159, 175, 287, 303, 315, 318, 319, 343, 350, 351, 375, 567, 574, 575, 591, 603, 606, 623, 627, 630, 636, 638, 679, 686, 687, 699, 700, 702, 735, 750, 763, 765, 1127, 1134, 1135, 1148, 1150, 1167, 1179, 1182, 1183, 1203, 1206, 1212, 1215, 1239, 1243, 1246, 1247, 1251, 1254, 1260, 1271, 1272

Offset: 1

Ely Golden, Jan 11 2017

Nontrivial means that there must be at least two factors.

			175 is a member as 175 = 7 * 25 = 10101111_2 = 111_2 * 11001_2
5887 is a member as 5887 = 7 * 29 * 29 = 1011011111111_2 = 111_2 * 11101_2 * 11101_2

Ely Golden, Table of n, a(n) for n = 1..10000
Ely Golden, General program for generating vampire number sequences

Cf. A020342

A280972 Numbers that appear in both A278909 and A280967 but not in A280971.

Original entry on oeis.org

765, 1275, 1467, 1503, 1515, 1695, 2910, 2975, 3066, 3423, 4335, 4539, 4605, 4862, 4923, 4947, 4975, 5110, 5295, 5335, 5375, 5559, 5787, 5790, 5835, 5885, 6069, 6123, 6495, 6735, 6783, 7035, 7134, 9195, 9567, 9583, 9645, 9819, 9915, 10087, 10155, 10218, 10234, 10491, 10686, 10959, 10983, 11211

Offset: 1

Ely Golden, Jan 11 2017

Binary equivalent of the sequence representing Numbers that appear in both A176670 and A020342 but not A280928 (currently no members are known).

			765 = A278909(41) = A280967(32) but is not present in A280971.

Ely Golden, Table of n, a(n) for n = 1..896

Cf. A278909, A280967, A280971.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Extensions

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

Views

Author

Keywords

Examples

References

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Extensions

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

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Extensions

A370676 Number of unordered pairs of natural numbers k1, k2 such that their product is an n-digit number and has the same multiset of digits as in both k1 and k2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Python