A240927 -id:A240927 - cp's OEIS frontend

A068896 Primes containing 2k digits in which the sum of the first k digits is that of the last k digits.

11, 1423, 1607, 1753, 1973, 2011, 2213, 2341, 2543, 2617, 2671, 2819, 2837, 3407, 3461, 3517, 3571, 3719, 3847, 4013, 4637, 4673, 4691, 4729, 4783, 4967, 5023, 5261, 5519, 5573, 5591, 5647, 5683, 5849, 5867, 6143, 6217, 6271, 6473, 6491, 6529, 6547, 7043, 7649, 7759, 8017, 8053, 8219, 8237, 8273, 8291, 8329, 8677, 9137, 9173, 9283, 9467

Offset: 1

Amarnath Murthy, Mar 21 2002

			2341 is a member with 2+3 = 4+1.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A240927.

Select[Prime[Range[169,1229]],Length[Union[Total/@TakeDrop[ IntegerDigits[ #],2]]] == 1&] (* The program generates all 56 4-digit terms. To generate all 3669 of the 6-digit terms, change the Range constants to (9593, 78498) and change the 2 to 3. *) (* Harvey P. Dale, Aug 15 2021 *)

from sympy import primerange
def sd(s): return sum(map(int, s))
def auptod(digits):
    alst = []
    for d in range(2, digits+1, 2):
        for p in primerange(10**(d-1), 10**d):
            s = str(p)
            if sd(s[:len(s)//2]) == sd(s[len(s)//2:]): alst.append(p)
    return alst
print(auptod(4)) # Michael S. Branicky, Aug 15 2021

Corrected and extended by Harvey P. Dale, Aug 15 2021

11 prepended by David A. Corneth, Aug 15 2021

A068898 Triangular numbers containing 2k digits in which the sum of the first k digits = that of the rest.

Original entry on oeis.org

55, 66, 2415, 3003, 5050, 5151, 5995, 8778, 9045, 113050, 138075, 171405, 174345, 177906, 183921, 198765, 203203, 216153, 219453, 234270, 237705, 239086, 252405, 255255, 266815, 267546, 275653, 279378, 284635, 293761, 294528, 306153, 309291, 329266, 348195

Offset: 1

Amarnath Murthy, Mar 21 2002

			2415 is a term with 2+4 = 1+5.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Intersection of A000217 and A240927.

Cf. A068896, A068897.

dsQ[n_]:=Module[{idn=IntegerDigits[n],len=IntegerLength[n]/2}, Total[Take[ idn,len]] ==Total[ Take[idn,-len]]]; Select[Flatten[ Table[Table[(n(n+1))/2,{n,Ceiling[(Sqrt[8 10^i+1]-1)/2],Floor[ (Sqrt[8 10^(i+1)+1]-1)/2]}],{i,1,5,2}]],dsQ] (* Harvey P. Dale, Sep 29 2011 *)

Corrected and extended by Harvey P. Dale, Sep 29 2011

Offset changed by Andrew Howroyd, Sep 21 2024

A240929 Number of 10-digit positive integers in base n where the sum of the first k digits equals the sum of the last k digits.

Original entry on oeis.org

126, 6046, 88428, 694360, 3705741, 15192604, 51418473, 150420187, 392406145, 933294637, 2056947827, 4253047045, 8329101326, 15566783605, 27934647638, 48371293570, 81155221112, 132379936520, 210555362990, 327359243694, 498565022483, 745175639274, 1094795785319

Offset: 2

Martin Renner, Aug 03 2014

These integers are sometimes called balanced numbers.

Cambridge Colleges Sixth Term Examination Papers (STEP) 2007, Paper I, Section A (Pure Mathematics), Nr. 1.

Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A016061, A197083, A240927, A240928.

def A240929(n): return n*(n*(n*(n*(n*(n*(n*(n*(156190*n-140571)+29400)-30870)+3990)-8379)-3100)-1620)-5040)//362880 # Chai Wah Wu, May 08 2024

a(n) = n*(n-1)*(156190*n^7 + 15619*n^6 + 45019*n^5 + 14149*n^4 + 18139*n^3 + 9760*n^2 + 6660*n + 5040)/362880
From Chai Wah Wu, May 08 2024: (Start)
a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10) for n > 11.
G.f.: x^2*(x^7 + 326*x^6 + 7942*x^5 + 42341*x^4 + 67030*x^3 + 33638*x^2 + 4786*x + 126)/(x - 1)^10. (End)

A068897 Squares containing 2k digits in which the sum of the first k digits = that of the rest.

Original entry on oeis.org

5041, 108900, 122500, 128164, 137641, 155236, 173056, 185761, 203401, 206116, 216225, 287296, 288369, 302500, 324900, 342225, 368449, 423801, 434281, 459684, 485809, 515524, 531441, 540225, 675684, 698896, 720801, 737881, 749956, 779689

Offset: 1

Amarnath Murthy, Mar 21 2002

			5041 is a member with 5+0 = 4+1.

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

Intersection of A000290 and A240927.

Cf. A068896, A068898.

isok(n)={my(d=digits(n)); my(k=#d); k%2==0 && vecsum(d[1..k/2]) == vecsum(d[k/2+1..k])}
lista(n)={my(L=List(), k=0); while(#LAndrew Howroyd, Sep 19 2024

Corrected and extended by Harvey P. Dale, Mar 31 2002

Offset changed by Andrew Howroyd, Sep 19 2024

A240928 Number of 8-digit positive integers in base n where the sum of the first k digits equals the sum of the last k digits.

Original entry on oeis.org

35, 750, 6174, 31025, 114961, 346193, 896876, 2072694, 4379055, 8606312, 15936426, 28073487, 47400509, 77164915, 121695128, 186650684, 279308283, 408886194, 586909430, 827618109, 1148421417, 1570399589, 2118856324, 2823924050, 3721224455, 4852586700

Offset: 2

Martin Renner, Aug 03 2014

These integers are sometimes called balanced numbers.

Cambridge Colleges Sixth Term Examination Papers (STEP) 2007, Paper I, Section A (Pure Mathematics), Nr. 1.

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A016061, A197083, A240927, A240929.

Table[n(n-1)(1208n^5+151n^4+291n^3+116n^2+88n+60)/2520,{n,2,40}] (* Harvey P. Dale, Mar 18 2022 *)

a(n) = n*(n-1)*(1208*n^5+151*n^4+291*n^3+116*n^2+88*n+60)/2520.

G.f.: x^2*(x^5+83*x^4+673*x^3+1154*x^2+470*x+35)/(x-1)^8. - Alois P. Heinz, Mar 24 2022

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A068896 Primes containing 2k digits in which the sum of the first k digits is that of the last k digits.

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A068898 Triangular numbers containing 2k digits in which the sum of the first k digits = that of the rest.

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A240929 Number of 10-digit positive integers in base n where the sum of the first k digits equals the sum of the last k digits.

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A068897 Squares containing 2k digits in which the sum of the first k digits = that of the rest.

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A240928 Number of 8-digit positive integers in base n where the sum of the first k digits equals the sum of the last k digits.

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Keywords

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Programs

Mathematica

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