A259831
Numbers n with the property that it is possible to write the base 2 expansion of n as concat(a_2,b_2), with a_2>0 and b_2>0 such that, converting a_2 and b_2 to base 10 as a and b, we have (sigma(a)-a)*(sigma(b)-b) = n.
Original entry on oeis.org
216, 13296, 13464, 14416, 51480, 235200, 575484, 578592, 585000, 1032656, 1121400, 1599552, 4190364, 4786110, 8365968, 11268688, 13010634, 13253436, 21835624, 22108784, 23896320, 136311840, 152820243, 160380496, 170073324, 295999900, 421686580, 445421664
Offset: 1
216 in base 2 is 11011000. If we take 11011000 = concat(110,11000) then 110 and 11000 converted to base 10 are 6 and 24. Finally (sigma(6) - 6)*(sigma(24) - 24) = (12 - 6)*(60 - 24) = 6 * 36 = 216;
13296 in base 2 is 11001111110000. If we take 11001111110000 = concat(110,01111110000) then 110 and 01111110000 converted to base 10 are 6 and 1008. Finally (sigma(6) - 6)*(sigma(1008) - 1008) = (12 - 6)*(3224 - 1008)= 6 * 2216 = 13296.
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with(numtheory): P:=proc(q) local a,b,c,j,k,n;
for n from 1 to q do c:=convert(n, binary, decimal);
j:=0; for k from 1 to ilog10(c) do
a:=convert(trunc(c/10^k), decimal, binary);
b:=convert((c mod 10^k), decimal, binary);
if a*b>0 then if (sigma(a)-a)*(sigma(b)-b)=n then print(n);
break; fi; fi; od; od; end: P(10^9);
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f[n_] := Block[{d = IntegerDigits[n, 2], len = IntegerLength[n, 2], k}, ReplaceAll[Reap[Do[k = {FromDigits[Take[d, i], 2], FromDigits[Take[d, -(len - i)], 2]}; If[! MemberQ[k, 0], Sow@ k], {i, 1, len - 1}]], {} -> {1}][[-1, 1]]]; Select[Range@ 100000, MemberQ[(DivisorSigma[1, #1] - #1) (DivisorSigma[1, #2] - #2) & @@@ f@ #, #] &] (* Michael De Vlieger, Jul 07 2015 *)
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from sympy import divisor_sigma
A259831_list= []
for n in range(2,10**6):
s = format(n,'0b')
for l in range(1,len(s)):
n1, n2 = int(s[:l],2), int(s[l:],2)
if n2 > 0 and n == (divisor_sigma(n1)-n1)*(divisor_sigma(n2)-n2):
A259831_list.append(n)
break # Chai Wah Wu, Jul 17 2015
A259832
Numbers n with the property that it is possible to write the base 2 expansion of n as concat(a_2,b_2), with a_2>0 and b_2>0 such that, converting a_2 and b_2 to base 10 as a and b, we have (sigma(a)-a)*(sigma(b)-b) = sigma(n).
Original entry on oeis.org
7708, 9020, 86934, 92128, 120228, 325180, 372000, 491630, 565724, 739032, 862780, 1120024, 1344090, 1419304, 1440858, 1678232, 2752626, 2980515, 3684344, 4154418, 4860476, 7539610, 7565257, 9527064, 11025372, 12277728, 17002336, 20256672, 22528536, 24597984
Offset: 1
7708 in base 2 is 1111000011100. If we take 1111000011100 = concat(11110000, 11100) then 11110000 and 11100 converted to base 10 are 240 and 28. Finally (sigma(240) - 240)*(sigma(28) - 28) = (744 - 240)*(56 - 28) = 504 * 28 = 14112 = sigma(7708); 9020 in base 2 is 10001100111100. If we take 10001100111100= concat(10001100, 111100) then 110 and 01111110000 converted to base 10 are 140 and 60. Finally (sigma(140) - 140)*(sigma(60) - 60) = (336 - 140)*(168 - 60)= 196 * 108 = 21160 = sigma(9020).
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with(numtheory): P:=proc(q) local a,b,c,k,n;
for n from 1 to q do c:=convert(n, binary, decimal);
for k from 1 to ilog10(c) do
a:=convert(trunc(c/10^k), decimal, binary);
b:=convert((c mod 10^k), decimal, binary);
if a*b>0 then if (sigma(a)-a)*(sigma(b)-b)=sigma(n) then print(n);
break; fi; fi; od; od; end: P(10^9);
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f[n_] := Block[{d = IntegerDigits[n, 2], len = IntegerLength[n, 2], k}, ReplaceAll[Reap[Do[k = {FromDigits[Take[d, i], 2], FromDigits[Take[d, -(len - i)], 2]}; If[! MemberQ[k, 0], Sow@ k], {i, 1, len - 1}]], {} -> {1}][[-1, 1]]]; Select[Range@ 125000, MemberQ[(DivisorSigma[1, #1] - #1) (DivisorSigma[1, #2] - #2) & @@@ f@ #, DivisorSigma[1, #]] &] (* Michael De Vlieger, Jul 07 2015 *)
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from sympy import divisor_sigma
A259832_list= []
for n in range(2,10**6):
s, k = format(n,'0b'), divisor_sigma(n)
for l in range(1,len(s)):
n1, n2 = int(s[:l],2), int(s[l:],2)
if n2 > 0 and k == (divisor_sigma(n1)-n1)*(divisor_sigma(n2)-n2):
A259832_list.append(n)
break # Chai Wah Wu, Jul 17 2015
A259675
Numbers n with the property that it is possible to write the base 2 expansion of n as concat(a_2,b_2), with a_2>0 and b_2>0 such that, converting a_2 and b_2 to base 10 as a and b, we have a’ * b’ = n, where a’ and b’ are the arithmetic derivatives of a and b.
Original entry on oeis.org
1344, 1456, 2352, 5120, 5376, 6000, 9680, 25600, 36672, 38220, 73536, 76752, 77824, 86592, 96250, 110160, 114688, 122360, 141056, 161544, 249600, 314352, 382976, 471040, 486400, 553056, 822224, 1411536, 1525056, 1570800, 1612288, 1720320, 1886720, 2143220, 2359296
Offset: 1
1344 in base 2 is 10101000000. If we take 10101000000 = concat(1010, 1000000) then 1010 and 1000000 converted to base 10 are 10 and 64. Their arithmetic derivatives are 7 and 192. Finally 7 * 192 = 1344.
1456 in base 2 is 10110110000. If we take 10110110000 = concat(10110, 110000) then 10110 and 110000 converted to base 10 are 22 and 48. Their arithmetic derivatives are 13 and 112. Finally 13 * 112 = 1456.
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with(numtheory): P:=proc(q) local a,b,c,k,n,p;
for n from 1 to q do c:=convert(n,binary,decimal);
for k from 1 to ilog10(c) do
a:=convert(trunc(c/10^k),decimal,binary);
b:=convert((c mod 10^k),decimal,binary);
a:=a*add(op(2,p)/op(1,p),p=ifactors(a)[2]); b:=b*add(op(2,p)/op(1,p),p=ifactors(b)[2]);
if a*b>0 then if a*b=n then print(n);
break; fi; fi; od; od; end: P(10^9);
Showing 1-3 of 3 results.
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