A147808 -id:A147808 - cp's OEIS frontend

A145808 Non-palindromic balanced numbers: the first and the last half of digits have the same sum.

1010, 1102, 1120, 1203, 1212, 1230, 1304, 1313, 1322, 1340, 1405, 1414, 1423, 1432, 1450, 1506, 1515, 1524, 1533, 1542, 1560, 1607, 1616, 1625, 1634, 1643, 1652, 1670, 1708, 1717, 1726, 1735, 1744, 1753, 1762, 1780, 1809, 1818, 1827, 1836, 1845, 1854

Offset: 1

M. F. Hasler, Nov 17 2008

Numbers such that the first half of digits have the same sum than the last half of digits are called balanced in the linked "Problem 217". (Note that here the meaning of "balanced" is neither that of A020492, nor that of A031443.)

Clearly all palindromes (A002113) have this property. Since the first non-palindromic example, 1010, comes only after A002113(109)=1001, we list here only non-palindromic balanced numbers.

Zak Seidov, Table of n, a(n) for n=1..1000 [From _Zak Seidov_, Oct 20 2009]
Project Euler, Problem 217: Balanced Numbers.

Cf. A147808.

Reap[Do[id=IntegerDigits[n];m=Floor[Length[id]/2];If[Reverse[id]!=id&&Total[Take[id,m]]==Total[Take[id,-m]],Sow[n]],{n,1010,2000}]][[2,1]] (* Zak Seidov, Oct 20 2009 *)
npbnQ[n_]:=Module[{idn=IntegerDigits[n],len},len=Floor[Length[idn]/2];idn!=Reverse[idn]&&Total[Take[idn,len]]==Total[Take[idn,-len]]]; Select[ Range[1000,2000],npbnQ] (* Harvey P. Dale, Sep 25 2012 *)

is_A145808(n) = is_balanced(n) & !is_A002113(n)
is_balanced(n) = { local( d, t=1+#Str(n)); (n\10^(t\2)-n%10^((t-1)\2)) % 9 && return; d=Vecsmall(Str(n)); sum(i=1,(t-1)\2,d[i]-d[t-i])==0 }

A145808 = { m | A007953([m/10^A110654(A055642(m))]) = A007953(m mod 10^A004526(A055642(m))) } \ A002113

A147811 Alexandrian integers: numbers of the form n=pqr such that 1/n = 1/p - 1/q - 1/r for some integers p,q,r.

Original entry on oeis.org

6, 42, 120, 156, 420, 630, 930, 1428, 1806, 2016, 2184, 3192, 4950, 5256, 8190, 8364, 8970, 10296, 10998, 12210, 17556, 19110, 21114, 23994, 24492, 28050, 32640, 33306, 34362, 37506, 39270, 44310, 52326, 57684, 57840, 70686, 74256, 79800, 83076

Offset: 1

M. F. Hasler and Alexis Olson (AlexisOlson(AT)gmail.com), Dec 13 2008

nonn

The numbers are of the form p(p+d)(p+(p^2+1)/d), where d runs over divisors of p^2+1 and p runs over all positive integers. See also A147807..A147810. - M. F. Hasler, Jan 07 2009

			630 is an Alexandrian integer since 630 = 5(-7)(-18) and 1/630 = 1/5 - 1/7 - 1/18.

Robert Israel, Table of n, a(n) for n = 1..10000
Project Euler, Problem 221: Alexandrian integers

Cf. A147807, A147808, A147809, A147810.

N:= 10^5: # to get all terms <= N
A:= select(`<=`,{seq(seq(p*(p+d)*(p+(p^2+1)/d), d=numtheory:-divisors(p^2+1)),p=1..floor(N^(1/3)))},N):
sort(convert(A,list)); # Robert Israel, Dec 16 2018

is_A147811(n) = { my(d=divisors(n), c=#d+1); n<42 && return(n==6); for( i=2, c-3, d[i+1]^2>d[c-i] && return; d[c-i]%d[i]==1 | next; for( j=i+1, c-i,d[j]^2>d[c-i] && next(2); d[c-i]\d[j]*(d[j]-d[i]) == d[j]*d[i]+1 && return(1))) }

A147842 Sum of all balanced numbers less than 10^n.

Original entry on oeis.org

45, 540, 50040, 3364890, 334795890, 27671338200, 2761284321300, 241066406350560, 24071550717893160, 2164109364980520560, 216167870783354294560, 19804111510101766391920, 1978598460245429017365520, 183672136030518949796846320

Offset: 1

Alexis Olson (AlexisOlson(AT)gmail.com), Nov 14 2008

See reference for definition of balanced number.

Partial sums of A147808. Up to n=3, equal to the sum of palindromes A002113 < 10^n; then it differs from this sequence by the sum of terms < 10^n in A145808. - M. F. Hasler, Nov 23 2008

Wei Zhou, Table of n, a(n) for n = 1..46
Project Euler, Problem 217: Balanced Numbers

s=0; A147842 = vector(13,k,s+=A147808(k)) \\ M. F. Hasler, Nov 23 2008

a(n) = Sum_{k=1..n} A147808(k). - M. F. Hasler, Nov 23 2008

a(9)-a(12) from M. F. Hasler, Nov 23 2008

a(13)-a(46) added by Wei Zhou, Oct 10 2011

cp's OEIS Frontend

A145808 Non-palindromic balanced numbers: the first and the last half of digits have the same sum.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A147811 Alexandrian integers: numbers of the form n=pqr such that 1/n = 1/p - 1/q - 1/r for some integers p,q,r.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

A147842 Sum of all balanced numbers less than 10^n.

Views

Author

Keywords

Comments

Links

Programs

PARI

Formula

Extensions