cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A258813 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) + sigma (b) = sigma(k) - k.

Original entry on oeis.org

9, 15, 27, 39, 51, 77, 143, 207, 329, 377, 473, 611, 903, 1241, 1243, 1273, 1437, 1591, 2117, 2303, 2975, 4189, 8401, 8657, 11993, 13849, 15611, 16771, 18239, 18599, 19359, 25331, 28877, 37291, 41747, 41807, 61549, 67037, 72601, 82169, 83411, 83711, 87449, 99329
Offset: 1

Views

Author

Paolo P. Lava, Jun 11 2015

Keywords

Examples

			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 sigma(2) + sigma(1) = sigma(9) - 9 = 4.
180953 in base 2 is 101100001011011001. If we take 101100001011011001 = concat(101100001011,011001) then 101100001011 and 011001 converted to base 10 are 2827 and 25. Finally sigma(2827) + sigma(25) = sigma(180953) - 180953 = 3127.

Links

Crossrefs

Cf. A253824, A253825, A258843, A258844.

Programs

  • Maple
    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)+sigma(b)=sigma(n)-n then print(n);
break; fi; fi; od; od; end: P(10^9);
  • PARI
    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 && (sigma(da)+sigma(db) == sigma(n)-n), return(1));););} \\ Michel Marcus, Jun 13 2015

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).

Original entry on oeis.org

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

Views

Author

Paolo P. Lava, Jun 12 2015

Keywords

Comments

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

Examples

			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).

Links

Crossrefs

Cf. A232355, A258813, A258844.

Programs

  • Maple
    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);
  • Mathematica
    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 *)
  • PARI
    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

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

Views

Author

Paolo P. Lava, Jun 26 2015

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

Programs

  • Maple
    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

Views

Author

Paolo P. Lava, Jul 03 2015

Keywords

Examples

			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.

Crossrefs

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

Programs

  • Maple
    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);
  • Mathematica
    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 *)
Showing 1-4 of 4 results.

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