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
A244079
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)*sigma(b) = n.
Original entry on oeis.org
28, 120, 162, 196, 868, 1260, 1302, 1800, 2016, 2480, 3780, 4464, 6804, 7440, 8370, 9882, 22200, 32640, 34290, 35640, 40640, 73152, 127008, 187488, 213776, 489888, 572880, 602640, 674082, 1074528, 1077120, 1397088, 1536192, 1582560, 1662120, 1669164, 1781136, 1905120
Offset: 1
28 in base 2 is 11100. If we take 11100 = concat(11,100) then 11 and 100 converted to base 2 are 3 and 4. Finally sigma(3)*sigma(4) = 4 * 7 = 28;
120 in base 2 is 1111000. If we take 1111000 = concat(111,1000) then 111 and 1000 converted to base 10 are 7 and 8. Finally sigma(7)*sigma(8) = 8 * 15 = 120.
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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)*sigma(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] DivisorSigma[1, #2] & @@@ f@ #, #] &] (* Michael De Vlieger, Jul 07 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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