A027751 -id:A027751 - cp's OEIS frontend

A006038 Odd primitive abundant numbers.

945, 1575, 2205, 3465, 4095, 5355, 5775, 5985, 6435, 6825, 7245, 7425, 8085, 8415, 8925, 9135, 9555, 9765, 11655, 12705, 12915, 13545, 14805, 15015, 16695, 18585, 19215, 19635, 21105, 21945, 22365, 22995, 23205, 24885, 25935, 26145, 26565, 28035, 28215

Offset: 1

N. J. A. Sloane

nonn

Dickson proves that there are only a finite number of odd primitive abundant numbers having n distinct prime factors. Sequence A188342 lists the smallest such numbers. - T. D. Noe, Mar 28 2011
Sequence A188439 sorts the numbers in this sequence by the number of distinct prime factors. Eight numbers have exactly three prime factors; 576 have exactly four prime factors. - T. D. Noe, Apr 04 2011
Any multiple of an abundant number (A005101) is again an abundant number. Primitive abundant numbers (A091191) are those not of this form, i.e., without an abundant proper divisor. We don't know any odd perfect number (A000396), so the (odd) terms here have only deficient proper divisors (A071395), and their prime factors p are less than sigma(n/p)/deficiency(n/p). See A005231 (odd abundant numbers) for an explanation why all terms have at least 3 distinct prime factors, and 5 prime factors when counted with multiplicity (A001222), whence a(1) = 3^3*5*7. All known terms are semiperfect (A005835, and thus in A006036): no odd weird number (A006037) is known, but if it exists, the smallest one is in this sequence. - M. F. Hasler, Jul 28 2016
So far, a(173) = 351351 is the only known term of A122036, i.e., which can't be written as sum of its proper divisors > 1. - M. F. Hasler, Jan 26 2020

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
L. E. Dickson, Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors, American Journal of Mathematics 35 (1913), pp. 413-422.
R. K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991
Jacob Liddy, An algorithm to determine all odd primitive abundant numbers with d prime divisors, Honors Research Projects (2018), 728.
Eric Weisstein's World of Mathematics, Primitive Abundant Number

Cf. A005101, A005231. Subsequence of A091191.

Cf. A000203, A027751, A379949 (subsequence of square terms).

a006038 n = a006038_list !! (n-1)
a006038_list = filter f [1, 3 ..] where
   f x = sum pdivs > x && all (<= 0) (map (\d -> a000203 d - 2 * d) pdivs)
         where pdivs = a027751_row x
-- Reinhard Zumkeller, Jan 31 2014

isA005101 := proc(n) is(numtheory[sigma](n) > 2*n ); end proc:
isA005100 := proc(n) is(numtheory[sigma](n) < 2*n ); end proc:
isA006038 := proc(n) local d; if type(n,'odd') and isA005101(n) then for d in numtheory[divisors](n) minus {1,n} do if not isA005100(d) then return false; end if; end do: return true;else false; end if; end proc:
n := 1 ; for a from 1 by 2 do if isA006038(a) then printf("%d %d\n",n,a) ; n := n+1 ; end if; end do: # R. J. Mathar, Mar 28 2011

t = {}; n = 1; While[Length[t] < 50, n = n + 2; If[DivisorSigma[1, n] > 2 n && Intersection[t, Divisors[n]] == {}, AppendTo[t, n]]]; t (* T. D. Noe, Mar 28 2011 *)

is(n)=n%2 && sumdiv(n,d,sigma(d,-1)>2)==1 \\ Charles R Greathouse IV, Jun 10 2013

is_A006038(n)=bittest(n,0) && sigma(n)>2*n && !for(i=1,#f=factor(n)[,1],sigma(n\f[i],-1)>2&&return) \\ More than 5 times faster. - M. F. Hasler, Jul 28 2016

A007422 Multiplicatively perfect numbers j: product of divisors of j is j^2.

Original entry on oeis.org

1, 6, 8, 10, 14, 15, 21, 22, 26, 27, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187

Offset: 1

N. J. A. Sloane

Or, numbers j such that product of proper divisors of j is j.
If M(j) denotes the product of the divisors of j, then j is said to be k-multiplicatively perfect if M(j) = j^k. All such numbers are of the form p q^(k-1) or p^(2k-1). This statement is in Sandor's paper. Therefore all 2-multiplicatively perfect numbers are semiprime p*q or cubes p^3. - Walter Kehowski, Sep 13 2005
All 2-multiplicatively perfect numbers except 1 have 4 divisors (as implied by Kehowski) and the converse is also true that all numbers with 4 divisors are 2-multiplicatively perfect. - Howard Berman (howard_berman(AT)hotmail.com), Oct 24 2008
Also 1 followed by numbers j such that A000005(j) = 4. - Nathaniel Johnston, May 03 2011
Fixed points of A007956. - Reinhard Zumkeller, Jan 26 2014

			The divisors of 10 are 1, 2, 5, 10 and 1 * 2 * 5 * 10 = 100 = 10^2.

Kenneth Ireland and Michael Ira Rosen, A Classical Introduction to Modern Number Theory. Springer-Verlag, NY, 1982, p. 19.
Edmund Landau, Elementary Number Theory, translation by Jacob E. Goodman of Elementare Zahlentheorie (Vol. I_1 (1927) of Vorlesungen ueber Zahlentheorie), by Edmund Landau, with added exercises by Paul T. Bateman and E. E. Kohlbecker, Chelsea Publishing Co., New York, 1958, pp. 31-32.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
William Chau, The tau, sigma, rho functions, and some related numbers, Pi Mu Epsilon Journal, Vol. 11, No. 10 (Spring 2004), pp. 519-534; entire issue.
József Sándor, Multiplicatively perfect numbers, J. Ineq. Pure Appl. Math., Vol. 2, No. 1 (2001), Article 3, 6 pp.
Eric Weisstein's World of Mathematics, Divisor Product.
Eric Weisstein's World of Mathematics, Multiplicative Perfect Number.

Cf. A030513 (same as this sequence but without the 1), A027751, A006881 (subsequence), A030078 (subsequence), A084110, A084116, A236473.

a007422 n = a007422_list !! (n-1)
a007422_list = [x | x <- [1..], a007956 x == x]
-- Reinhard Zumkeller, Jan 26 2014

IsA007422:=func< n | &*Divisors(n) eq n^2 >; [ n: n in [1..200] | IsA007422(n) ]; // Klaus Brockhaus, May 04 2011

k:=2: MPL:=[]: for z from 1 to 1 do for n from 1 to 5000 do if convert(divisors(n),`*`) = n^k then MPL:=[op(MPL),n] fi od; od; MPL; # Walter Kehowski, Sep 13 2005
# second Maple program:
q:= n-> n=1 or numtheory[tau](n)=4:
select(q, [$1..200])[];  # Alois P. Heinz, Dec 17 2021

Select[Range[200], Times@@Divisors[#] == #^2 &]  (* Harvey P. Dale, Mar 27 2011 *)

is(n)=n==1 || numdiv(n) == 4 \\ Charles R Greathouse IV, Oct 15 2015

from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A007422(n):
    def f(x): return int(n-1+x-primepi(integer_nthroot(x,3)[0])+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 16 2024

A084110(a(n)) = 1, see also A084116. - Reinhard Zumkeller, May 12 2003

The number of terms not exceeding x is N(x) ~ x * log(log(x))/log(x) (Chau, 2004). - Amiram Eldar, Jun 29 2022

Some numbers were omitted - thanks to Erich Friedman for pointing this out.

A065205 Number of subsets of proper divisors of n that sum to n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 5, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 7, 0, 0, 0, 3, 0, 2, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 3, 0, 2, 0, 0, 0, 34, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 31, 0, 0, 0, 0, 0, 1, 0, 6, 0, 0, 0, 25, 0, 0, 0, 1, 0, 23, 0, 0, 0, 0, 0, 21, 0, 0, 0, 2

Offset: 1

Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 19 2001

nonn

Deficient and weird numbers have a(n) = 0, perfect numbers and others (see A064771) have a(n) = 1.

Number of partitions of n into distinct proper divisors of n; a(A136447(n)) = 0; a(A005835(n)) > 0; a(A064771(n)) = 1. - Reinhard Zumkeller, Jan 21 2013

			a(20) = 1 because {1, 4, 5, 10} is the only subset of proper divisors of 20 that sum to 20.
a(24) = 5 because there are five different subsets we can use to sum up to 24: {1, 2, 3, 4, 6, 8}, {1, 2, 3, 6, 12}, {1, 3, 8, 12}, {2, 4, 6, 12}, {4, 8, 12}.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A064771, A005835.
Cf. A065218 for records.
Cf. A027751, A210442, A211110, A033630.

a065205 n = p (a027751_row n) n where
   p _      0 = 1
   p []     _ = 0
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
-- Reinhard Zumkeller, Jan 21 2013

a[n_] := (dd = Most[ Divisors[n] ]; cc = Array[c, Length[dd]]; Length[ {ToRules[ Reduce[ And @@ (0 <= # <= 1 &) /@ cc && dd . cc == n, cc, Integers]]}]); Table[ a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 23 2012 *)

a(n,s,d)={s || (s=sigma(n)-n) || return; d||d=vecextract(divisors(n),"^-1"); while(d[#d]>n, s-=d[#d]; d=d[1..-2]); s<=n && return(s==n); if( n>d[#d], a(n-d[#d],s-d[#d],d[1..-2]), 1)+a(n,s-d[#d],d[1..-2])} \\ M. F. Hasler, May 11 2015

a(n) = A033630(n) - 1.

More terms and additional comments from Jud McCranie, Oct 21 2001

A064771 Let S(n) = set of divisors of n, excluding n; sequence gives n such that there is a unique subset of S(n) that sums to n.

Original entry on oeis.org

6, 20, 28, 78, 88, 102, 104, 114, 138, 174, 186, 222, 246, 258, 272, 282, 304, 318, 354, 366, 368, 402, 426, 438, 464, 474, 490, 496, 498, 534, 572, 582, 606, 618, 642, 650, 654, 678, 748, 762, 786, 822, 834, 860, 894, 906, 940, 942, 978, 1002, 1014, 1038

Offset: 1

Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 19 2001

Perfect numbers (A000396) are a proper subset of this sequence. Weird numbers (A006037) are numbers whose proper divisors sum to more than the number, but no subset sums to the number.
Odd elements are rare: the first few are 8925, 32445, 351351, 442365; there are no more below 100 million. See A065235 for more details.
A065205(a(n)) = 1. - Reinhard Zumkeller, Jan 21 2013

			Proper divisors of 20 are 1, 2, 4, 5 and 10. {1,4,5,10} is the only subset that sums to 20, so 20 is in the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (terms 1..200 from T. D. Noe, terms 201..5000 from Amiram Eldar)

A005835 gives n such that some subset of S(n) sums to n. Cf. A065205.
Cf. A006037, A065205, A378448 (characteristic function).
Subsequences: A000396, A065235 (odd terms), A378519, A378530.
Cf. A027751.

a064771 n = a064771_list !! (n-1)
a064771_list = map (+ 1) $ elemIndices 1 a065205_list
-- Reinhard Zumkeller, Jan 21 2013

filter:= proc(n)
  local P,x,d;
  P:= mul(x^d+1, d = numtheory:-divisors(n) minus {n});
  coeff(P,x,n) = 1
end proc:
select(filter, [$1..2000]); # Robert Israel, Sep 25 2024

okQ[n_]:= Module[{d=Most[Divisors[n]]}, SeriesCoefficient[Series[ Product[ 1+x^i, {i, d}], {x, 0, n}], n] == 1];Select[ Range[ 1100],okQ] (* Harvey P. Dale, Dec 13 2010 *)

from sympy import divisors
def isok(n):
    dp = {0: 1}
    for d in divisors(n)[:-1]:
        u = {}
        for k in dp.keys():
            if (s := (d + k)) <= n:
                u[s] = dp.get(s, 0) + dp[k]
                if s == n and u[s] > 1:
                    return False
        for k,v in u.items():
            dp[k] = v
    return dp.get(n, 0) == 1
print([n for n in range(1, 1039) if isok(n)]) # Darío Clavijo, Sep 17 2024

More terms from Don Reble, Jud McCranie and Naohiro Nomoto, Oct 22 2001

A318990 Numbers of the form prime(x) * prime(y) where x divides y.

Original entry on oeis.org

4, 6, 9, 10, 14, 21, 22, 25, 26, 34, 38, 39, 46, 49, 57, 58, 62, 65, 74, 82, 86, 87, 94, 106, 111, 115, 118, 121, 122, 129, 133, 134, 142, 146, 158, 159, 166, 169, 178, 183, 185, 194, 202, 206, 213, 214, 218, 226, 235, 237, 254, 259, 262, 267, 274, 278, 289

Offset: 1

Gus Wiseman, Sep 06 2018

nonn

			The sequence of all dividing pairs (columns) begins:
  1  1  2  1  1  2  1  3  1  1  1  2  1  4  2  1  1  3  1  1  1  2  1  1
  1  2  2  3  4  4  5  3  6  7  8  6  9  4  8 10 11  6 12 13 14 10 15 16

Andrew Howroyd, Table of n, a(n) for n = 1..10000

A subset of A001358 (semiprimes), squarefree A006881.
The squarefree version is A339005.
The quotient is A358103 = A358104 / A358105.
A000040 lists the primes.
A001222 counts prime indices, distinct A001221.
A003963 multiplies together prime indices.
A056239 adds up prime indices.
A358192/A358193 gives quotients of semiprime indices.
Cf. A000837, A300912, A318991, A318992, A318993.
Cf. A000720, A027751, A032741, A215366, A296150.
Cf. A289508, A289509, A358106.

Select[Range[100],And[PrimeOmega[#]==2,Or[PrimePowerQ[#],Divisible@@Reverse[PrimePi/@FactorInteger[#][[All,1]]]]]&]

ok(n)={my(f=factor(n)); bigomega(f)==2 && (#f~==1 || primepi(f[2,1]) % primepi(f[1,1]) == 0)} \\ Andrew Howroyd, Oct 26 2018

A163870 Triangle read by rows: row n lists the nontrivial divisors of the n-th composite.

Original entry on oeis.org

2, 2, 3, 2, 4, 3, 2, 5, 2, 3, 4, 6, 2, 7, 3, 5, 2, 4, 8, 2, 3, 6, 9, 2, 4, 5, 10, 3, 7, 2, 11, 2, 3, 4, 6, 8, 12, 5, 2, 13, 3, 9, 2, 4, 7, 14, 2, 3, 5, 6, 10, 15, 2, 4, 8, 16, 3, 11, 2, 17, 5, 7, 2, 3, 4, 6, 9, 12, 18, 2, 19, 3, 13, 2, 4, 5, 8, 10, 20, 2, 3, 6, 7, 14, 21, 2, 4, 11, 22, 3, 5, 9, 15, 2, 23

Offset: 1

Juri-Stepan Gerasimov, Aug 06 2009

Row n contains row A002808(n) of table A027750.

T(n,k) = A027751(A002808(n),k+1), k = 1..A144925(n). - Reinhard Zumkeller, Mar 29 2014

			The table starts in row n=1 (with the composite 4) as
  2;
  2,3;
  2,4;
  3;
  2,5;
  2,3,4,6;
  2,7;
  3,5;
  2,4,8;
  2,3,6,9;
  2,4,5,10.

Reinhard Zumkeller, Rows n = 1..1000 of table, flattened

Cf. A002808, A027750.

Cf. A144925 (row lengths), A062825 (row sums), A056608 (left edge), A160180 (right edge).

a163870 n k = a163870_tabf !! (n-1) !! (k-1)
a163870_row n = a163870_tabf !! (n-1)
a163870_tabf = filter (not . null) $ map tail a027751_tabf
-- Reinhard Zumkeller, Mar 29 2014

Divisors[Select[Range[50], CompositeQ]][[All, 2 ;; -2]] (* Paolo Xausa, Dec 26 2024 *)

from itertools import islice
def g():
    n, j = 1, 2
    while True:
        n = (n << 1) | 1
        p = 1
        for k in range(2, (j >> 1) + 1):
            p = (p << 1) | 1
            if n % p == 0: yield k
        j+=1
print(list(islice(g(),95))) # Darío Clavijo, Dec 16 2024

Entries checked by R. J. Mathar, Sep 22 2009

A006036 Primitive pseudoperfect numbers.

Original entry on oeis.org

6, 20, 28, 88, 104, 272, 304, 350, 368, 464, 490, 496, 550, 572, 650, 748, 770, 910, 945, 1184, 1190, 1312, 1330, 1376, 1430, 1504, 1575, 1610, 1696, 1870, 1888, 1952, 2002, 2030, 2090, 2170, 2205, 2210, 2470, 2530, 2584, 2590, 2870, 2990, 3010, 3128, 3190, 3230, 3290, 3410, 3465, 3496, 3710, 3770, 3944, 4070, 4095, 4130, 4216, 4270, 4288, 4408, 4510, 4544, 4672, 4690, 4712, 4730, 4970

Offset: 1

N. J. A. Sloane, R. K. Guy

A primitive pseudoperfect number is a pseudoperfect number that is not a multiple of any other pseudoperfect number.
The odd entries so far are identical to the odd primitive abundant A006038. - Walter Kehowski, Aug 12 2005
Zachariou and Zachariou (1972) called these numbers "irreducible semiperfect numbers". - Amiram Eldar, Dec 04 2020

Richard K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, Section B2, pp. 74-75.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Donovan Johnson, Table of n, a(n) for n = 1..10000
Richard K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991
Eric Weisstein's World of Mathematics, Primitive Pseudoperfect Number.
Andreas Zachariou and Eleni Zachariou, Perfect, Semi-Perfect and Ore Numbers, Bull. Soc. Math. Grèce (New Ser.), Vol. 13, No. 13A (1972), pp. 12-22; alternative link.

Cf. A005835.

Cf. A210455, A027751.

a006036 n = a006036_list !! (n-1)
a006036_list = filter (all (== 0) . map a210455 . a027751_row) a005835_list
-- Reinhard Zumkeller, Jan 21 2013

with(numtheory): with(combinat): issemiperfect := proc(n) local b, S;
b:=false; S:=subsets(divisors(n) minus {n}); while not S[finished] do if
convert(S[nextvalue](),`+`)=n then b:=true; break fi od; return b end:
L:=remove(proc(z) isprime(z) end,[$1..5000]): PP:=[]: for zz from 1 to 1 do
for n in L do if issemiperfect(n) then PP:=[op(PP),n] fi od od;
sr := proc(l::list) local x, R, S, P, L; S:=sort(l); R:=[]; P:=S;
for x in S do
if not(x in R) then
L:=selectremove(proc(z) z>x and z mod x = 0 end, P);
R:=[op(R),op(L[1])]; P:=L[2];
fi; od; return P; end:
PPP:=sr(PP); # primitive pseudoperfect numbers less than 5000 # Walter Kehowski, Aug 12 2005

(* First run one of the programs for A005835 *) A006036 = A005835; curr = 1; max = A005835[[-1]]; While[curr < Length[A006036], currMult = A006036[[curr]]; A006036 = Complement[A006036, Range[2currMult, Ceiling[max/currMult] currMult, currMult]]; curr++]; A006036 (* Alonso del Arte, Sep 08 2012 *)

More terms from Walter Kehowski, Aug 12 2005

A080940 Smallest proper divisor of n which is a suffix of n in binary representation; a(n) = 0 if no such divisor exists.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 4, 1, 2, 1, 0, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 0, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 16, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 0, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 16, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 32, 1, 2, 1, 4, 1, 2, 1, 8

Offset: 1

Reinhard Zumkeller, Feb 25 2003

By definition, identical to A006519 except that a(2^k) = 0 for all k.
a(3*2^k)=2^k and a(m)<2^k for m<3*2^k (see A007283).
Also, the first repeating value of the periodic sequences created by 2^k mod n. - Alison J. McCrea, Apr 13 2025

			n=6='110', divisors<6: 1='1', 2='10' and 3='11', therefore a(6)=2='10';
n=7='111', divisors<7: 1='1', therefore a(7)=1;
n=8='1000', divisors<8: 1='1', 2='10' and 4='100', therefore a(8)=0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007088, A000079, A080941, A080942, A006519.

Cf. A030308, A027751.

import Data.List (isPrefixOf); import Data.Function (on)
a080940 n = if null ds then 0 else head ds  where
            ds = filter ((flip isPrefixOf `on` a030308_row) n) $
                        a027751_row n
-- Reinhard Zumkeller, Mar 27 2014

def A080940(n): return (m:=n&-n)*(m!=n) # Chai Wah Wu, Jun 20 2023

Definition improved by Reinhard Zumkeller, Mar 27 2014

A080941 Greatest proper divisor of n which is a suffix of n in binary representation; a(n) = 0 if no such divisor exists.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 4, 1, 2, 3, 0, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 3, 4, 1, 6, 1, 0, 1, 2, 1, 4, 1, 2, 3, 8, 1, 2, 1, 4, 5, 2, 1, 16, 1, 2, 3, 4, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 7, 0, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 3, 4, 1, 6, 1, 16, 1, 2, 1, 4, 5, 2, 3, 8, 1, 10, 1, 4, 1, 2, 1, 32, 1, 2, 3, 4, 1, 6, 1

Offset: 1

Reinhard Zumkeller, Feb 25 2003

a(n)=0 iff n=2^k (A000079);

			n=30='11110', divisors<30: 1='1', 2='10', 3='11', 5='101', 6='110', 10='1010' and 15='1111', therefore a(30)=2='10';
n=31='11111', divisors<31: 1='1', therefore a(31)=1;
n=32='100000', divisors<32: 1='1', 2='10', 4='100', 8='1000' and 16='10000', therefore a(32)=0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007088, A080940, A080942.

Cf. A030308, A027751.

import Data.List (isPrefixOf); import Data.Function (on)
a080941 n = if null ds then 0 else head ds  where
            ds = filter ((flip isPrefixOf `on` a030308_row) n) $
                        reverse $ a027751_row n
-- Reinhard Zumkeller, Mar 27 2014

Definition improved by Reinhard Zumkeller, Mar 27 2014

A072513 Product of all n - d, where d < n and d is a divisor of n.

Original entry on oeis.org

1, 1, 2, 6, 4, 60, 6, 168, 48, 360, 10, 47520, 12, 1092, 1680, 20160, 16, 440640, 18, 820800, 5040, 4620, 22, 734469120, 480, 7800, 11232, 4953312, 28, 3946320000, 30, 9999360, 21120, 17952, 28560, 439723468800, 36, 25308, 35568, 35852544000

Offset: 1

Amarnath Murthy, Jul 28 2002

nonn

			a(6) = (6-1)(6-2)(6-3) = 60.
For n = 16 the divisors d < n are 1,2,4 and 8, so a(16) = (16-1)*(16-2)*(16-4)*(16-8) = 15*14*12*8 = 20160.

Ivan Neretin, Table of n, a(n) for n = 1..1000

Cf. A072512, A080497, A080498, A080500 (similar products), A258324 (LCM instead of product).

Cf. A027751.

a072513 n = product $ map (n -) $ a027751_row n
-- Reinhard Zumkeller, May 27 2015

Table[Times @@ (n - Most[Divisors[n]]), {n, 1, 40}] (* Ivan Neretin, May 26 2015 *)

for(n=1,40,d=divisors(n); print1(prod(j=1,matsize(d)[2]-1,n-d[j]),","))

a(n)=factorback(apply(d->if(dCharles R Greathouse IV, May 26 2015

a(n) = (n-d_1)(n-d_2)...(n-d_k) where d_k is the largest divisor of n less than n (k = tau(n) - 1).
a(p) = p-1, a(pq) = pq(p-1)(q-1)(pq-1), p and q prime.
If n is not a prime or the square of a prime then n divides a(n).

Edited and extended by Klaus Brockhaus, Jul 31 2002

cp's OEIS Frontend

A006038 Odd primitive abundant numbers.

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A007422 Multiplicatively perfect numbers j: product of divisors of j is j^2.

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A065205 Number of subsets of proper divisors of n that sum to n.

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A064771 Let S(n) = set of divisors of n, excluding n; sequence gives n such that there is a unique subset of S(n) that sums to n.

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A318990 Numbers of the form prime(x) * prime(y) where x divides y.

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A163870 Triangle read by rows: row n lists the nontrivial divisors of the n-th composite.

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