A280928 -id:A280928 - 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

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

A280971 Composite numbers having the same bits as their prime factors (with multiplicity), including zero bits.

Original entry on oeis.org

159, 287, 303, 319, 591, 623, 679, 687, 699, 763, 1135, 1167, 1203, 1243, 1247, 1271, 1351, 1371, 1391, 1631, 2167, 2173, 2231, 2285, 2319, 2359, 2463, 2471, 2495, 2519, 2743, 2779, 2787, 2809, 2863, 2931, 2933, 2991, 3029, 3039, 3503, 4223, 4279, 4287, 4319, 4343, 4411, 4439, 4479, 4487

Offset: 1

Ely Golden, Jan 11 2017

Binary equivalent of A280928.

Subsequence of A278909 as well as A280967. The terms in A278909 and A280967 but not this sequence are given by A280972.

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

Cf. A278909, A280967, A280928.

from sympy import factorint
def ok(n):
    f = factorint(n)
    return sum(f.values()) > 1 and sorted(bin(n)[2:]) == sorted("".join(bin(p)[2:]*f[p] for p in f))
print([k for k in range(5000) if ok(k)]) # Michael S. Branicky, Apr 20 2025

def factorbits(x):
    if(x<2):
        return (0,0);
    s=0;t=0
    f=list(factor(x));
    #ensures inequality of numfactorbits(x) and bin(x).count("1") if x is prime
    if((len(f)==1)&(f[0][1]==1)):
        return (0,0);
    for c in range(len(f)):
        s+=bin(f[c][0]).count("1")*f[c][1]
        t+=(bin(f[c][0]).count("0")-1)*f[c][1]
    return (s,t);
counter=2
index=1
while(index<=10000):
    if(factorbits(counter)==(bin(counter).count("1"),bin(counter).count("0")-1)):
        print(str(index)+" "+str(counter))
        index+=1;
    counter+=1;

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.

A281336 a(n) is the smallest composite number having the same base-n digits (both type and quantity) as its prime factors (with multiplicity).

Original entry on oeis.org

159, 7847, 1135, 83494, 57, 30057, 85, 72646, 1255, 18193, 185, 101212405, 4119, 63791, 88357, 31054582, 489, 4196517, 451, 4598494, 13315, 1012985, 679, 7009758597, 26533, 2884373, 985, 646372334, 1057, 989775393, 1285, 1395750166, 179503, 73294351, 1387

Offset: 2

Hans Havermann, Jan 20 2017

A278981(n) <= A281336(n)

			In base 2: The digits of 159 (10011111) are the same type and quantity as the total of those in its prime factors, 3 (11) and 53 (110101); 2 zeros, 6 ones.
In base 26: The digits of 26533 (1,13,6,13) are the same type and quantity as the total of those in its prime factors, 13 (13), 13 (13), and 157 (6,1); 1 one, 1 six, 2 thirteens.

Cf. A278981, A280928, A281335.

A348306 List of Agathokakological Numbers "k": string of digits of the juxtaposition of the prime factors of k has the same length as k but these digits do not appear in k.

Original entry on oeis.org

10, 14, 21, 49, 106, 111, 118, 129, 134, 146, 158, 161, 166, 177, 201, 219, 249, 259, 267, 329, 343, 413, 511, 553, 623, 1011, 1029, 1046, 1077, 1081, 1101, 1106, 1114, 1119, 1138, 1149, 1167, 1186, 1227, 1299, 1318, 1354, 1358, 1363, 1418, 1454, 1466, 1538, 1541, 1546, 1561, 1589, 1591

Offset: 1

Samuel Harkness, Oct 11 2021

Theorem: (See PDF "PROOFS" in Links)
Of Agathokakological Numbers k,
No k have a leading 9.
No k end in 2 or 5.
10 is the only k to end in 0. It is also the only k with 5 as a prime factor.
Can only be square terms when k is of the order 10^m where m is odd.
For k written as a*10^m, k can only be even when 1<=a<1.888...
Empirical observation: When graphed with the log of the n-th term on x axis and the log of the n-th term's value on the y axis a pattern appears with a similar shape for each new power of ten (see figure "LogLogGraph" in Links)
Special cases 28651 = 7*4093 and 65821 = 7*9043 use all digits 0-9 once.
"Agathokakological" is a Greek word meaning "composed of both good and evil." (Merriam-Webster) The composition (prime factorization) of Agathokakological Numbers is both good (same length) and evil (no common digits).

			158 = 2 * 79 since {2,7,9} do not appear in {1,5,8} and both have 3 digits.

Samuel Harkness, Table of n, a(n) for n = 1..6388
Samuel Harkness, MATLAB
Samuel Harkness, LogLogGraph
Samuel Harkness, PROOFS

Cf. A055642, A076649, A280928.
Intersection of A035139 and A109608.
Subsequence of A047201 from n=2.

q[n_] := Module[{d = IntegerDigits[n], f = FactorInteger[n]}, Length[d] == Plus @@ ((Last[#]*IntegerLength[First[#]]) & /@ f ) && Intersection[d, Join @@ IntegerDigits[f[[;; , 1]]]] == {}]; Select[Range[1600], q] (* Amiram Eldar, Oct 12 2021 *)

digsf(n) = my(f=factor(n), list=List()); for (k=1, #f~, my(dk=digits(f[k,1])); for (i=1, f[k,2], for (j=1, #dk, listput(list, dk[j])))); Vec(list);
isokd(m) = my(df=digsf(m), d=digits(m)); (#df == #d) && (#setintersect(Set(df), Set(d)) == 0); \\ Michel Marcus, Oct 11 2021

from sympy import factorint
def ok(n):
    s, f = str(n), factorint(n)
    pfd = set("".join(str(p) for p in f))
    if set(s) & pfd != set(): return False
    return len(s) == sum(len(str(p))*f[p] for p in f)
print(list(filter(ok, range(1601)))) # Michael S. Branicky, Oct 11 2021

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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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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A280971 Composite numbers having the same bits as their prime factors (with multiplicity), including zero bits.

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SageMath

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

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