A258813 -id:A258813 - cp's OEIS frontend

A258843 Numbers k with the property that it is possible to write the base 2 expansion of k 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 + b) = sigma(k).

11, 123, 695, 991, 1919, 2839, 3707, 3841, 7615, 8047, 8055, 9347, 10703, 12847, 16195, 26743, 27089, 32127, 42251, 56419, 59027, 59179, 59389, 59815, 62749, 65113, 74671, 115289, 119211, 122847, 126895, 129495, 168739, 191051, 219295, 224281, 232315, 233729

Offset: 1

Paolo P. Lava, Jun 12 2015

It appears that a or b is equal to 1.
The terms that have b=1 are 11, 695, 991, 2839, 3707, 9347, ...; see A232355. - Michel Marcus, Jun 12 2015
If b=1, the number k can be expressed as 2a+b=2a+1. We are looking for numbers that satisfy the relation sigma(a+1)=sigma(k), namely sigma(a+1)=sigma(2a+1). In A232355 we have the numbers such that sigma(k)=sigma((k+1)/2) that match sigma(2a+1)=sigma((2a+1+1)/2)=sigma(a+1). That's why the two "subsequences" are the same thing. - Paolo P. Lava and Michel Marcus, Jun 16 2015

			11 in base 2 is 1011. If we take 1011 = concat(101,1) then 101 and 1 converted to base 10 are 5 and 1. Finally sigma(5 + 1) = sigma(6) = 12 = sigma(11).
123 in base 2 is 1111011. If we take 1111011 = concat(1,111011) then 1 and 111011 converted to base 10 are 1 and 59. Finally sigma(1 + 59) = sigma(60) = 168 = sigma(123).

Paolo P. Lava, Table of n, a(n) for n = 1..110

Cf. A232355, A258813, A258844.

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+b)=sigma(n) then print(n);
break; fi; fi; od; od; end: P(10^6);

f[n_] := Block[{d = IntegerDigits[n, 2], len, s}, len = Length@ d; s = FromDigits[#, 2] & /@ {Take[d, #], Take[d, -len + #]} & /@ Range[len - 1]; DeleteDuplicates[DivisorSigma[1, #1 + #2] == DivisorSigma[1, n] & @@@ s]]; Select[Range@ 250000, Length@ f@ # > 1 &] (* Michael De Vlieger, Jun 12 2015 *)

isok(n) = {b = binary(n); if (#b > 1, for (k=1, #b-1, vba = Vecrev(vector(k, i, b[i])); vbb = Vecrev(vector(#b-k, i, b[k+i])); da = sum(i=1, #vba, vba[i]*2^(i-1)); db = sum(i=1, #vbb, vbb[i]*2^(i-1)); if (sigma(da+db) == sigma(n), return(1));););} \\ Michel Marcus, Jun 12 2015

A258844 Numbers k with the property that it is possible to write the base 2 expansion of k 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)^2 = k.

Original entry on oeis.org

9, 36, 49, 100, 225, 784, 961, 1296, 2601, 3969, 7225, 8281, 14400, 16129, 18496, 21609, 29241, 34969, 42025, 65025, 116964, 123201, 133225, 246016, 261121, 278784, 465124, 508369, 672400, 700569, 828100, 1046529, 1368900, 1590121, 1782225, 4064256, 4190209, 4326400

Offset: 1

Paolo P. Lava, Jun 12 2015

Obviously all terms are squares.

The terms that have b=1 are: 9, 49, 225, 961, 3969, 16129, 65025, ...; see A060867 ((2^n-1)^2). - Michel Marcus, Jun 13 2015

			9 in base 2 is 1001. If we take 1001 = concat(10,01) then 10 and 01 converted to base 10 are 2 and 1. Finally (2 + 1)^2 = 3^2 = 9;
36 in base 2 is 100100. If we take 100100 = concat(10,0100) then 10 and 0100 converted to base 10 are 2 and 4. Finally (2 + 4)^2 = 6^2 = 36.

Cf. A258813, A258843.

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 (a+b)^2=n then print(n); break;
fi; fi; od; od; end: P(10^6);

isok(n) = {b = binary(n); if (#b > 1, for (k=1, #b-1, vba = Vecrev(vector(k, i, b[i])); vbb = Vecrev(vector(#b-k, i, b[k+i])); da = sum(i=1, #vba, vba[i]*2^(i-1)); db = sum(i=1, #vbb, vbb[i]*2^(i-1)); if ((da+ db)^2 == n, return(1));););} \\ Michel Marcus, Jun 13 2015

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

Paolo P. Lava, Jul 06 2015

a(31) > 5*10^8. - Hiroaki Yamanouchi, Sep 24 2015

			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.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..30

Cf. A001065, A244079, A258813, A258843, A259832.

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);

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 *)

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

a(13)-a(14) from Chai Wah Wu, Jul 17 2015

a(15)-a(28) from Hiroaki Yamanouchi, Sep 24 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

Paolo P. Lava, Jul 06 2015

a(62) > 10^8. - Hiroaki Yamanouchi, Sep 24 2015

			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).

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..61

Cf. A000203, A244079, A258813, A258843, A259831.

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);

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 *)

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

a(16)-a(21) from Chai Wah Wu, Jul 17 2015

a(22)-a(30) from Hiroaki Yamanouchi, Sep 24 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

Paolo P. Lava, Jul 06 2015

			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.

Cf. A000203, A258813, A258843, A259831, A259832.

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);

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 *)

A259397 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 phi(a + b) = phi(n), where phi(n) is the Euler totient function of n.

Original entry on oeis.org

6, 12, 14, 28, 30, 48, 62, 124, 126, 222, 224, 254, 448, 476, 496, 510, 768, 876, 1022, 1792, 1806, 2032, 2034, 2046, 2625, 2850, 2898, 3204, 3246, 3560, 3705, 3850, 4064, 4094, 7722, 7744, 7920, 7980, 7992, 8060, 8094, 8136, 8148, 8150, 8164, 8190, 11880, 13365

Offset: 1

Paolo P. Lava, Jun 26 2015

It appears that a or b is equal to 1. In particular, if b=1 we have 2625, 3705, 13365, 25545, 57645, ... that are a subset of A001274.

			6 in base 2 is 110. If we take 110 = concat(1,10) then 1 and 10 converted to base 10 are 1 and 2. Finally phi(1 + 2) = 2 = phi(6).
12 in base 2 is 1100. If we take 1100 = concat(1,100) then 1 and 100 converted to base 10 are 1 and 4. Finally phi(1 + 4) = 4 = phi(12);
2625 in base 2 is 101001000001. If we take 101001000001 = concat(10100100000,1) then 10100100000 and 1 converted to base 10 are 1312 and 1. Finally phi(1312 + 1) = 1200 = phi(2625); etc.

Paolo P. Lava, Table of n, a(n) for n = 1..150

Cf. A000010, A001274, A258813, A258843, A258844.

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 phi(a+b)=phi(n) then print(n); break;
fi; fi; od; od; end: P(10^8);

A259670 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 antisigma(a) + antisigma(b) = n.

Original entry on oeis.org

50, 77, 179, 346, 347, 550, 1758, 1909, 9205, 27884, 30660, 37354, 52019, 88052, 107974, 131590, 164413, 232447, 295682, 326133, 328491, 1494561, 1541005, 1541851

Offset: 1

Paolo P. Lava, Jul 03 2015

			50 in base 2 is 110010. If we take 110010 = concat(1100,10) then 1100 and 10 converted to base 10 are 12 and 2. Finally 12*13/2 - sigma(12) + 2*3/2 - sigma(2) = 78 - 28 + 3 - 3 = 50.
179 in base 2 is 1001101. If we take 1001101 = concat(11100,1) then 11100 and 1 converted to base 10 are 5 and 19. Finally 5*6/2 - sigma(5) + 19*20/2 - sigma(19) = 15 - 6 + 190 - 20 = 179.

Cf. A024816, A253824, A253825, A258813, A258843, A258844.

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 a*(a+1)/2-sigma(a)+b*(b+1)/2-sigma(b)=n then print(n);
break; fi; fi; od; od; end: P(10^9);

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[Total /@ (# (# + 1)/2 - DivisorSigma[1, #] &) /@ f@ #, #] &] (* Michael De Vlieger, Jul 03 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

Paolo P. Lava, Jul 07 2015

			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.

Cf. A003415, A244079, A258813, A258843, A259831, A259832.

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);

cp's OEIS Frontend

A258843 Numbers k with the property that it is possible to write the base 2 expansion of k 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 + b) = sigma(k).

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A258844 Numbers k with the property that it is possible to write the base 2 expansion of k 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)^2 = k.

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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.

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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).

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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.

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A259397 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 phi(a + b) = phi(n), where phi(n) is the Euler totient function of n.

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A259670 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 antisigma(a) + antisigma(b) = n.

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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.

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