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.

Previous Showing 11-19 of 19 results.

A119896 Numbers k such that 2^k, 3^k, 5^k, 7^k, 11^k, 13^k and 17^k have even digit sum.

Original entry on oeis.org

90, 185, 447, 470, 503, 552, 657, 758, 804, 811, 1182, 1412, 1546, 1566, 1638, 1655, 2275, 2390, 2504, 2670, 2721, 2803, 2814, 2902, 2928, 3002, 3060, 3087, 3135, 3393, 3660, 3751
Offset: 1

Views

Author

Zak Seidov, May 26 2006

Keywords

Crossrefs

Cf. A054683, A118734, A118867, A119894, A119895, A119897.

A270264 The cumulative sum of the digits of successive terms reproduces the prime number sequence; this is the lexicographically earliest sequence with this property.

Original entry on oeis.org

2, 1, 11, 20, 4, 101, 13, 110, 22, 6, 200, 15, 31, 1001, 40, 24, 33, 1010, 42, 103, 1100, 51, 112, 60, 8, 121, 2000, 130, 10001, 202, 59, 211, 105, 10010, 19, 10100, 114, 123, 220, 132, 141, 11000, 28, 20000, 301, 100001, 39, 48, 310, 100010, 400, 150, 100100, 37, 204, 213, 222, 101000, 231, 1003, 110000, 46, 68, 1012, 200000, 1021, 77, 240, 55, 1000001, 1030, 303, 17, 312, 321, 1102, 330, 26, 1111, 35, 64, 1000010, 73, 1000100, 402, 1120, 411, 44
Offset: 1

Views

Author

Eric Angelini and Jean-Marc Falcoz, Mar 14 2016

Keywords

Comments

Add the digits of (say) the first 4 terms of the sequence: you'll get 7 and 7 is the 4th prime number.
Add the digits of the first 5 terms of the sequence: you'll get 11 and 11 is the 5th prime number.
Add the digits of the first 6 terms of the sequence: you'll get 13 and 13 is the 6th prime number. Etc.
Presumably this is a permutation of the numbers {1} union A054683 (cf. A269740). - N. J. A. Sloane, Mar 15 2016
The conjecture that the sequence is equal to {1} union A054683 is equivalent to Polignac's conjecture (a generalization of the twin prime conjecture) which is still open. - Chai Wah Wu, Mar 15 2016

Links

Crossrefs

Cf. A000040, A054683.
A269740 says where n-th term of A054683 appears.

A268620 Numbers whose digital sum is a multiple of 4.

Original entry on oeis.org

0, 4, 8, 13, 17, 22, 26, 31, 35, 39, 40, 44, 48, 53, 57, 62, 66, 71, 75, 79, 80, 84, 88, 93, 97, 103, 107, 112, 116, 121, 125, 129, 130, 134, 138, 143, 147, 152, 156, 161, 165, 169, 170, 174, 178, 183, 187, 192, 196, 202, 206, 211, 215, 219, 220, 224, 228, 233, 237, 242, 246
Offset: 1

Views

Author

Bruno Berselli, Feb 09 2016

Keywords

Comments

a(1498) = 5999 is the smallest term that is congruent to 5 modulo 9.

Links

Crossrefs

Cf. A007953, A061383 (supersequence).
Cf. numbers whose digital sum is a multiple of k: A054683 (k=2), A008585 (k=3), this sequence (k=4), A227793 (k=5).
Subsequences: A002278, A038446, A052218, A052222, A061387, A169967, A235151, A235227, A235229.

Programs

  • Magma
    [n: n in [0..250] | IsIntegral(&+Intseq(n)/4)];
  • Maple
    select(t -> convert(convert(t,base,10),`+`) mod 4 = 0, [$1..1000]); # Robert Israel, Feb 09 2016
  • Mathematica
    Select[Range[0, 250], IntegerQ[Total[IntegerDigits[#]]/4] &]

A084682 Even evil numbers with an even digital sum.

Original entry on oeis.org

0, 6, 20, 24, 40, 46, 48, 60, 66, 68, 80, 86, 114, 116, 130, 132, 136, 150, 154, 156, 170, 172, 178, 190, 192, 198, 202, 204, 222, 226, 228, 240, 246, 260, 264, 282, 284, 288, 312, 318, 330, 332, 338, 350, 354, 356, 374, 378, 390, 394, 396, 402, 404, 408, 420
Offset: 1

Views

Author

Jason Earls, Jun 30 2003

Keywords

Links

Crossrefs

Cf. A001969.
Intersection of A054683 and A125592.

Programs

  • Maple
    filter:= n -> convert(convert(n,base,2),`+`)::even and convert(convert(n,base,10),`+`)::even:
select(filter, [seq(i,i=2..10000,2)]); # Robert Israel, Dec 31 2024
  • Mathematica
    eee[n_] :=  And @@ EvenQ /@ {n, Count[IntegerDigits[n, 2], 1], Total[IntegerDigits[n]]};
Select[Range[0, 420], eee] (* Jake L Lande, Jun 30 2024 *)
  • PARI
    is(n)={ bitand(n,1)==0 && bitand(sumdigits(n),1)==0 && bitand(hammingweight(n),1)==0 }
select(is, [0..500]) \\ Joerg Arndt, Jun 30 2024

Extensions

Offset changed by Andrew Howroyd, Sep 18 2024

A154387 Composite numbers with even sum of digits.

Original entry on oeis.org

4, 6, 8, 15, 20, 22, 24, 26, 28, 33, 35, 39, 40, 42, 44, 46, 48, 51, 55, 57, 60, 62, 64, 66, 68, 75, 77, 80, 82, 84, 86, 88, 91, 93, 95, 99, 105, 110, 112, 114, 116, 118, 121, 123, 125, 129, 130, 132, 134, 136, 138, 141, 143, 145, 147, 150, 152, 154, 156, 158, 161
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Jan 08 2009

Keywords

Examples

			15 is composite and 1 + 5 = 6 (an even number), so 15 is a term;
20 is composite and 2 + 0 = 2 (an even number), so 20 is a term.

Links

Crossrefs

Intersection of A002808 and A054683.

Programs

  • Maple
    sd := proc (n) options operator, arrow: add(convert(n, base, 10)[j], j = 1 .. nops(convert(n, base, 10))) end proc: a := proc (n) if isprime(n) = false and `mod`(sd(n), 2) = 0 then n else end if end proc: seq(a(n), n = 2 .. 180); # Emeric Deutsch, Jan 17 2009
  • Mathematica
    Select[Range[200],CompositeQ[#]&&EvenQ[Total[IntegerDigits[#]]]&] (* Harvey P. Dale, Aug 25 2016 *)

Extensions

Replaced 45 by 46 - R. J. Mathar, Jan 12 2009
More terms from Emeric Deutsch, Jan 17 2009

A251853 Nonnegative numbers n with all even digits such that the digital sum of the digits' sum is even.

Original entry on oeis.org

0, 2, 4, 6, 8, 20, 22, 24, 26, 40, 42, 44, 60, 62, 80, 200, 202, 204, 206, 220, 222, 224, 240, 242, 260, 400, 402, 404, 420, 422, 440, 488, 600, 602, 620, 668, 686, 688, 800, 848, 866, 868, 884, 886, 888, 2000, 2002, 2004, 2006, 2020, 2022, 2024, 2040, 2042, 2060, 2200
Offset: 1

Views

Author

Chase Fortier, Dec 09 2014

Keywords

Examples

			2288 is in the sequence because it is even, 2 and 8 are even, 2 + 2 + 8 + 8 = 20 is even, and 2 + 0 = 2 is even.

Links

Crossrefs

Cf. A007953, A014263, A054683.

Programs

  • Mathematica
    a251853[n_Integer] := Module[{digitSum}, digitSum[x_] := Plus @@ IntegerDigits[x]; Select[Range[n], And[And @@ EvenQ@IntegerDigits[#], EvenQ@digitSum[#], EvenQ@Nest[digitSum, #, 2]] &]]; a251853[2200] (* Michael De Vlieger, Dec 11 2014 *)
  • PARI
    isevend(v) = for (i=1, #v, if (v[i] % 2, return (0))); return (1);
isok(n) = isevend(digits(n)) && ((sumdigits(sumdigits(n)) % 2) == 0); \\ Michel Marcus, Dec 11 2014
  • Python
    A251853_list = [int(''.join(d)) for d in product('02468',repeat=4) if not sum(int(y) for y in str(sum(int(x) for x in d))) % 2] # Chai Wah Wu, Dec 20 2014
  • Sage
    [x for x in [0..2200] if prod([is_even(i) for i in x.digits()]) and sum(Integer(sum(x.digits())).digits())%2==0] # Tom Edgar, Dec 10 2014

Formula

Each digit in n is divisible by two, n is divisible by 2, the sum S of the digits of n is divisible by 2, and the sum of the digits of S is also divisible by 2.

A256785 Numbers n such that digitsum(n) is a whole number when n is represented in the fractional base 1.5 = 3/2.

Original entry on oeis.org

1, 5, 11, 14, 20, 21, 22, 23, 26, 29, 30, 31, 33, 34, 38, 39, 40, 41, 45, 46, 51, 52, 53, 56, 57, 58, 60, 61, 65, 69, 70, 71, 74, 78, 79, 83, 84, 85, 87, 88, 89, 90, 91, 95, 101, 105, 106, 110, 111, 112, 113, 116, 117, 118, 122, 126, 127, 132, 133, 135, 136, 140, 146, 149, 155, 159, 160, 161, 164, 165, 166, 168, 169, 173, 174, 175
Offset: 1

Views

Author

Anthony Sand, Apr 10 2015

Keywords

Comments

Base 1.5 requires three digits: 1, 0 and H = 0.5. For example:
1 = 1 = 1 * 1.5^0
2 = 1H = 1 * 1.5^1 + 0.5 * 1.5^0 = 1.5 + 0.5
3 = 1H0 = 1 * 1.5^2 + 0.5 * 1.5^1 = 2.25 + 0.75
4 = 1H1 = 1 * 1.5^2 + 0.5 * 1.5^1 + 1 * 1.5^0 = 2.25 + 0.75 + 1
5 = 1H0H = 1 * 1.5^3 + 0.5 * 1.5^2 + 0.5 * 1.5^0 = 3.375 + 1.125 + 0.5
6 = 1H10 = 1 * 1.5^3 + 0.5 * 1.5^2 + 1 * 1.5^1 = 3.375 + 1.125 + 1.5
7 = 1H11 = 1 * 1.5^3 + 0.5 * 1.5^2 + 1 * 1.5^1 + 1 * 1.5^0 = 3.375 + 1.125 + 1.5 + 1
The sequence above lists the n for which digsum(n,base=1.5) is a whole number.

Examples

			The sequence begins with 1, 5 and 11, because:
digsum(1,b=1.5) = 1
digsum(5,b=1.5) = 2 = digsum(1H0H) = 1 + 0.5 + 0.5
digsum(11,b=1.5) = 4 = digsum(1H11H) = 1 + 0.5 + 1 + 1 + 0.5
The digsums are all whole numbers. However, 2, 3 and 4 are excluded because:
digsum(2,b=1.5) = 1.5 = digsum(1H) = 1 + 0.5
digsum(3,b=1.5) = 1.5 = digsum(1H0) = 1 + 0.5 + 0
digsum(4,b=1.5) = 2.5 = digsum(1H1) = 1 + 0.5 + 1
The digsums are not whole numbers.

Links

Crossrefs

Cf. A007953, A054683, A054684.

Programs

  • PARI
    { b=3/2; dmx=30; d=vector(dmx); nmx=1000; n=0; ni=0; while(ni0, di++; d[di]=nn-floor(nn/b)*b; nn\=b; ); s=0; for(i=1,di,s+=d[i]); if(floor(s)==s, ni++; write("digsum.txt",ni," ",n)); ); }

A302438 Numbers with digits in nondecreasing order and even digital sum (in base 10) whose digits can be partitioned in two multisets with equal digital sum.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 112, 123, 134, 145, 156, 167, 178, 189, 224, 235, 246, 257, 268, 279, 336, 347, 358, 369, 448, 459, 1111, 1113, 1122, 1124, 1133, 1135, 1144, 1146, 1155, 1157, 1166, 1168, 1177, 1179, 1188, 1199, 1223, 1225, 1234, 1236, 1245, 1247
Offset: 1

Views

Author

David A. Corneth, Apr 08 2018

Keywords

Examples

			a(5000) = 11222699 is in this sequence as it has digits in nondecreasing order and an even digital sum, which is 32. The digits can be partitioned in two multisets with equal sum, for example as {1, 6, 9} and {1, 2, 2, 2, 9}, each having sum 16.

Links

Crossrefs

Cf. A009994, A054683.

Programs

  • PARI
    \\ See PARI link.

A358270 Numbers whose sum of digits is even and that have an even number of even digits.

Original entry on oeis.org

11, 13, 15, 17, 19, 20, 22, 24, 26, 28, 31, 33, 35, 37, 39, 40, 42, 44, 46, 48, 51, 53, 55, 57, 59, 60, 62, 64, 66, 68, 71, 73, 75, 77, 79, 80, 82, 84, 86, 88, 91, 93, 95, 97, 99, 1001, 1003, 1005, 1007, 1009, 1010, 1012, 1014, 1016, 1018, 1021, 1023, 1025, 1027, 1029, 1030
Offset: 1

Views

Author

Bernard Schott, Nov 06 2022

Keywords

Comments

There are only terms with an even number of digits, and precisely, there exist A137233(2*k) terms with 2*k digits.
The conditions separately are A054683 for even sum of digits, and A356929 for even number of even digits, so that this sequence is their intersection.
The opposite conditions, an odd sum of digits, and an odd number of odd digits, are the same and are A054684.

Examples

			26 is a term since 2+6 = 8 (even) and 26 has two even digits.
39 is a term since 3+9 = 12 (even) and 39 has zero even digits.
1012 is a term since 1+0+1+2 = 4 (even) and 1012 has two even digits.

Crossrefs

Intersection of A054683 and A356929.
Cf. A001637 (even length), A179081 (digit sum mod 2).
Cf. A014263, A054684, A137233.

Programs

  • Mathematica
    Select[Range[1000], EvenQ[Plus @@ IntegerDigits[#]] && EvenQ[Plus @@ DigitCount[#, 10, Range[0, 8, 2]]] &] (* Amiram Eldar, Nov 06 2022 *)
  • PARI
    a(n) = n*=2; n += 100^logint(110*n,100) \ 11; n - sumdigits(n)%2; \\ Kevin Ryde, Nov 10 2022
  • Python
    def ok(n): s = str(n); return sum(map(int, s))%2 == sum(1 for d in s if d in "02468")%2 == 0
print([k for k in range(1031) if ok(k)]) # Michael S. Branicky, Nov 06 2022
  • Python
    from itertools import count, islice, chain
def A358270_gen(): # generator of terms
    return filter(lambda n:not (len(s:=str(n))&1 or sum(int(d) for d in s)&1), chain.from_iterable((range(10**l,10**(l+1)) for l in count(1,2))))
A358270_list = list(islice(A358270_gen(),61)) # Chai Wah Wu, Nov 11 2022

Formula

a(n) = t - A179081(t) where t = A001637(2*n). - Kevin Ryde, Nov 10 2022
Previous Showing 11-19 of 19 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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