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.

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A036925 Digit sum of composite odd number equals digit sum of juxtaposition of its prime factors (counted with multiplicity).

Original entry on oeis.org

27, 85, 121, 265, 319, 355, 391, 483, 517, 535, 627, 645, 663, 729, 825, 861, 895, 913, 915, 985, 1111, 1165, 1219, 1255, 1449, 1507, 1581, 1633, 1755, 1795, 1881, 1903, 1921, 1935, 2067, 2079, 2155, 2173, 2227, 2265, 2373, 2409, 2461, 2475, 2515, 2583
Offset: 1

Views

Author

Patrick De Geest, Jan 04 1999

Keywords

Comments

Odd Smith numbers. - Robert Israel, Aug 25 2024

Links

Crossrefs

Cf. A006753, A019506, A036924.

Programs

  • Maple
    filter:= proc(n) local F;
  if isprime(n) then return false fi;
  F:= ifactors(n)[2];
  convert(convert(n, base, 10), `+`) = convert(map(t -> t[2]*convert(convert(t[1], base, 10), `+`), F), `+`)
end proc:
select(filter, [seq(i,i=9..10000,2)]); # Robert Israel, Aug 25 2024

Extensions

Title made more precise by Sean A. Irvine, Nov 30 2020

A235766 Smallest first term of a sequence of exactly n consecutive hoax numbers.

Original entry on oeis.org

22, 84, 12955, 291090, 9538589, 3541285143, 136063250955, 253282144742
Offset: 1

Views

Author

Carlos Rivera, Jan 15 2014

Keywords

Comments

A hoax number is a composite number whose sum of digits is equal to the sum of the digits of its distinct prime factors.

Examples

			For n = 1, a(1) = 22 because 22 = 11*2 and the sum of the digits (SOD) in both sides is 4 but 23 is not composite.
For n = 2, a(2) = 84 because 84 = 2^2*3*7, SOD = 12; 85 = 5*17, SOD = 13 but 86 = 2*43 and SOD = 14 <> 9.

Links

  • Shyam Sunder Gupta, Smith Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 4, 127-157.
  • Giovanni Resta, Hoax numbers.
  • Eric Weisstein's World of Mathematics, Hoax Number.

Crossrefs

Cf. A019506.

Extensions

a(6)-a(8) from Giovanni Resta, Jan 15 2014

A331463 Numbers k such that k and k + 1 are both binary hoax numbers (A329936).

Original entry on oeis.org

8, 15, 49, 50, 252, 489, 699, 725, 755, 799, 951, 979, 980, 988, 989, 1023, 1134, 1350, 1351, 1370, 1390, 1599, 1629, 1630, 1660, 1690, 1694, 1763, 1854, 1908, 1929, 1939, 1940, 1960, 2006, 2015, 2166, 2312, 2358, 2645, 2700, 2779, 2787, 2862, 2923, 2930, 2988
Offset: 1

Views

Author

Amiram Eldar, Jan 17 2020

Keywords

Examples

			8 is a term since both 8 and 8 + 1 = 9 are binary hoax numbers: 8 = 2^3 in binary representation is 1000 = 10^3 and 1 + 0 + 0 + 0 = 1 + 0, and 9 = 3^2 in binary representation is 1001 = 11^2 and 1 + 0 + 0 + 1 = 1 + 1.

Links

Crossrefs

Cf. A050219, A019506, A329935, A329936, A331464.

Programs

  • Magma
    hoax:=func; [k:k in [2..3000]|hoax(k) and hoax(k+1)]; // Marius A. Burtea, Jan 17 2020
  • Mathematica
    binWt[n_] := Total @ IntegerDigits[n, 2]; binHoaxQ[n_] := CompositeQ[n] && Total[binWt /@ FactorInteger[n][[;; , 1]]] == binWt[n]; seq = {}; isHoax1 = binHoaxQ[1]; Do[isHoax2 = binHoaxQ[n]; If[isHoax1 && isHoax2, AppendTo[seq, n-1]]; isHoax1 = isHoax2, {n, 2, 3000}]; seq

A329942 a(n) begins the first run of exactly n consecutive binary hoax numbers (A329936).

Original entry on oeis.org

4, 8, 49, 3952, 117175, 2322232, 2437094, 15449349, 438134200, 1605609902, 85678432036, 132891678661, 8415592788756
Offset: 1

Views

Author

Amiram Eldar, Nov 24 2019

Keywords

Comments

a(14) > 10^13, a(15) = 6359937801959. - Giovanni Resta, Nov 28 2019

Examples

			a(2) = 8 since 8 and 9 are binary hoax numbers.
a(3) = 49 since 49, 50, and 51 are binary hoax numbers.

Crossrefs

Cf. A019506, A235766, A329936.

Programs

  • Mathematica
    binWt[n_] := Total@IntegerDigits[n, 2]; binHoaxQ[n_] := CompositeQ[n] && Total[binWt /@ FactorInteger[n][[;; , 1]]] == binWt[n]; n = 1; count = 0; max = 6; seq = Table[0, {max}]; While[count < max, n1 = n; If[binHoaxQ[n], While[binHoaxQ[++n1]]; d = n1 - n; If[d <= max && seq[[d]] == 0, count++; seq[[d]] = n]]; n = n1 + 1]; seq

Extensions

a(11)-a(13) from Giovanni Resta, Nov 28 2019

A036922 Even composite numbers whose digit sum equals the digit sum of (sum of prime factors, counted with multiplicity).

Original entry on oeis.org

4, 22, 94, 114, 150, 166, 202, 204, 222, 224, 274, 342, 346, 382, 438, 450, 454, 526, 540, 562, 612, 634, 640, 706, 852, 922, 1068, 1086, 1120, 1122, 1138, 1200, 1230, 1232, 1282, 1314, 1318, 1400, 1626, 1642, 1770, 1820, 1822, 1894, 1966, 2070, 2080
Offset: 1

Views

Author

Patrick De Geest, Jan 04 1999

Keywords

Crossrefs

Cf. A006753, A036920, A019506.

Programs

  • PARI
    isok(n)=my(f=factor(n)); n%2==0 && n>2 && sumdigits(sum(i=1, #f~, f[i,1]*f[i,2])) == sumdigits(n) \\ Andrew Howroyd, Jun 19 2021

Extensions

Title reworded by Sean A. Irvine, Nov 30 2020

A036923 Odd composite numbers n such that the digit sum of n equals digit sum of sum of its prime factors (counted with multiplicity).

Original entry on oeis.org

27, 105, 121, 265, 315, 355, 445, 517, 841, 913, 915, 1111, 1165, 1185, 1219, 1221, 1239, 1255, 1345, 1363, 1507, 1633, 1903, 2067, 2101, 2155, 2173, 2209, 2227, 2245, 2265, 2335, 2409, 2515, 2533, 2605, 2965, 3091, 3129, 3219, 3235, 3417, 3505, 3507
Offset: 1

Views

Author

Patrick De Geest, Jan 04 1999

Keywords

Crossrefs

Cf. A006753, A036920, A019506.

Programs

  • Mathematica
    ds[n_]:=Total[IntegerDigits[n]]; t={}; Do[If[!PrimeQ[n]&&ds[n]==ds[Total[ Times@@@FactorInteger[n]]],AppendTo[t,n]],{n,9,3508,2}]; t (* Jayanta Basu, Jun 04 2013 *)

Extensions

Title reworded by Sean A. Irvine, Nov 30 2020

A036926 Digit sum of 'even' number equals digit sum of 'sum' and 'juxtaposition' of its prime factors (counted with multiplicity).

Original entry on oeis.org

4, 22, 94, 166, 202, 274, 346, 382, 438, 454, 526, 562, 634, 706, 852, 922, 1086, 1282, 1626, 1642, 1822, 1894, 1966, 2182, 2326, 2362, 2434, 2614, 2722, 2902, 2974, 3046, 3138, 3226, 3246, 3258, 3442, 3622, 3694, 3802, 3946, 4054, 4126, 4162, 4306, 4414
Offset: 1

Views

Author

Patrick De Geest, Jan 04 1999

Keywords

Crossrefs

Cf. A006753, A036921, A019506.

A036927 Digit sum of 'odd' number equals digit sum of 'sum' and 'juxtaposition' of its prime factors (counted with multiplicity).

Original entry on oeis.org

27, 121, 265, 355, 517, 913, 915, 1111, 1165, 1219, 1255, 1507, 1633, 1903, 2067, 2155, 2173, 2227, 2265, 2409, 2515, 2605, 2965, 3091, 3505, 3615, 3865, 4209, 4765, 4855, 5071, 5305, 6115, 6315, 6457, 7051, 7447, 7465, 7915, 8005, 8023, 9015, 9031
Offset: 1

Views

Author

Patrick De Geest, Jan 04 1999

Keywords

Crossrefs

Cf. A006753, A036921, A019506.

A376157 Numbers k such that the sum of the digits of k equals the sum of its prime factors plus the sum of the multiplicities of each prime factor.

Original entry on oeis.org

4, 25, 36, 54, 125, 192, 289, 297, 343, 392, 448, 676, 756, 1089, 1536, 1764, 1936, 2646, 2888, 3872, 4802, 4860, 6174, 6250, 6776, 6860, 7290, 7488, 7680, 8750, 8775, 9408, 9747, 10648, 14739, 15309, 16848, 18432, 18865, 21296, 22869, 25725, 29988, 33750, 33957
Offset: 1

Views

Author

Jordan Brooks, Sep 12 2024

Keywords

Examples

			For k = 54, its prime factorization is 2^1*3^3: 5+4 = 2+1+3+3 = 9.
For k = 756, its prime factorization is 2^2*3^3*7^1: 7+5+6 = 2+2+3+3+7+1 = 18.

Crossrefs

Similar to A006753, A019506, A050689, A063737, A070275, A285494.
Cf. A007953, A008474.

Programs

  • Mathematica
    Select[Range[34000], DigitSum[#]==Total[Flatten[FactorInteger[#]]] &] (* Stefano Spezia, Sep 14 2024 *)
  • PARI
    isok(k)={my(f=factor(k)); vecsum(f[,1]) + vecsum(f[,2]) == sumdigits(k)} \\ Andrew Howroyd, Sep 26 2024
  • Python
    from sympy.ntheory import factorint
c = 2
while c < 10000:
    charsum = 0
    for char in str(c):
        charsum += int(char)
    pf = factorint(c)
    cand = 0
    for p in pf.keys():
        cand += p
        cand += pf[p]
    if charsum == cand:
        print(c)
        print(pf)
    c += 1

Formula

{ k : A007953(k) = A008474(k) }.
Previous Showing 11-19 of 19 results.

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