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.

User: Metin Sariyar

Metin Sariyar's wiki page.

Metin Sariyar has authored 37 sequences. Here are the ten most recent ones:

A354603 Numbers k such that sum of distinct primes dividing k is equal to the sum of proper divisors of k+1.

Original entry on oeis.org

3, 7, 14, 31, 127, 206, 2974, 8191, 19358, 20490, 131071, 147454, 286122, 289650, 292332, 441276, 524287, 909498, 1207358, 1657968, 1782540, 2490042, 3368860, 9274806, 11367402, 14107852, 16776156, 18589386, 22910988, 24450316, 26867718, 28959606, 32674506, 33163372
Offset: 1

Views

Author

Metin Sariyar, Jul 08 2022

Keywords

Comments

Numbers k such that A008472(k) = A001065(k+1). All Mersenne primes are terms.

Examples

			Example:  14 is a term because A008472(14) = 2+7 = A001065(15) = 1+3+5.

Crossrefs

Cf. A000668, A001065, A008472, A006145.

Programs

  • Mathematica
    Select[Range[19358],Sum[f, {f, Select[Divisors[#], PrimeQ]}]==DivisorSigma[1,#+1]-(#+1)&]

Extensions

More terms from Amiram Eldar, Jul 08 2022

A354746 Non-repdigit numbers k such that every permutation of the digits of k has the same number of distinct prime divisors.

Original entry on oeis.org

12, 13, 15, 16, 17, 21, 23, 26, 28, 31, 32, 36, 37, 39, 45, 51, 54, 56, 57, 58, 61, 62, 63, 65, 68, 69, 71, 73, 75, 79, 82, 85, 86, 93, 96, 97, 113, 116, 117, 122, 131, 155, 156, 161, 165, 171, 177, 178, 187, 199, 212, 221, 224, 226, 228, 242, 245, 248, 254, 255, 258, 262, 282
Offset: 1

Views

Author

Metin Sariyar, Jun 05 2022

Keywords

Examples

			156 is a term because omega(156) = omega(165) = omega (516) = omega(561) = omega(615) = omega(651) = 3, where omega(n) is the number of distinct prime divisors of n.

Links

Crossrefs

Cf. A001221, A003459, A115662, A354745.

Programs

  • Mathematica
    Select[Range[10000],CountDistinct[PrimeNu[FromDigits /@ Permutations[IntegerDigits[#]]]]==1&&CountDistinct[IntegerDigits[#]]>1&]
  • Python
    from sympy import factorint
from itertools import permutations
def ok(n):
    s, pf = str(n), len(factorint(n))
    if len(set(s)) == 1: return False
    return all(pf==len(factorint(int("".join(p)))) for p in permutations(s))
print([k for k in range(500) if ok(k)]) # Michael S. Branicky, Jun 05 2022

A354745 Non-repdigit numbers k such that every permutation of the digits of k has the same number of divisors.

Original entry on oeis.org

13, 15, 17, 24, 26, 31, 37, 39, 42, 51, 58, 62, 71, 73, 79, 85, 93, 97, 113, 117, 131, 155, 171, 177, 178, 187, 199, 226, 262, 288, 311, 337, 339, 355, 373, 393, 515, 535, 551, 553, 558, 585, 622, 711, 717, 718, 733, 771, 781, 817, 828, 855, 871, 882, 899, 919, 933, 989, 991, 998
Offset: 1

Views

Author

Metin Sariyar, Jun 05 2022

Keywords

Comments

After a(93) = 84444, no further terms < 10^18. - Michael S. Branicky, Jun 08 2022

Examples

			871 is a term because d(871) = d(817) = d(178) = d(187) = d(718) = d(781) = 4, where d(n) is the number of divisors of n.

Crossrefs

Cf. A000005, A003459, A067012, A062895, A350867, A354746.

Programs

  • Mathematica
    Select[Range[10000],CountDistinct[DivisorSigma[0,FromDigits /@ Permutations[IntegerDigits[#]]]]==1&&CountDistinct[IntegerDigits[#]]>1&]
  • Python
    from sympy import divisor_count
from itertools import permutations
def ok(n):
    s, d = str(n), divisor_count(n)
    if len(set(s)) == 1: return False
    return all(d==divisor_count(int("".join(p))) for p in permutations(s))
print([k for k in range(5500) if ok(k)]) # Michael S. Branicky, Jun 05 2022

A354466 Numbers k such that the decimal expansion of the sum of the reciprocals of the digits of k starts with the digits of k in the same order.

Original entry on oeis.org

1, 13, 145, 153, 1825, 15789, 16666, 21583, 216666, 2416666, 28428571, 265833333, 3194444444, 3333333333, 9111111111, 35333333333, 3166666666666, 3819444444444, 26666666666666, 34166666666666, 527857142857142, 3944444444444444, 6135714285714285, 615833333333333333
Offset: 1

Views

Author

Metin Sariyar, Jun 01 2022

Keywords

Comments

The sequence is infinite because all numbers of the form 10^(10^n-6) + 6*(10^(10^n-6)-1)/9, (n>0) are terms.
All terms are zeroless since 1/0 is undefined.
If n gives a sum < 1 then that sum is taken as 0.xyz.. but n does not start with 0, so not a term.

Examples

			28428571 is a term because 1/2 + 1/8 + 1/4 + 1/2 + 1/8 + 1/5 + 1/7 + 1/1 = 2.8428571...
825 is not a term since 1/8 + 1/2 + 1/5 = 0.825.

Links

Crossrefs

Cf. A009994, A034708, A337904.

Programs

  • Mathematica
    Do[If[FreeQ[IntegerDigits[n], 0]&&Floor[Total[1/IntegerDigits[n]]*10^(IntegerLength[n]-IntegerLength[Floor[Total[1/IntegerDigits[n]]]])]==n&&Floor[Total[1/IntegerDigits[n]]]>0, Print[n]], {n, 1, 216666}]
  • PARI
    \\ See links.
  • Python
    # See links.

Extensions

a(12)-a(24) from Michael S. Branicky, Jun 03 2022

A348882 Numbers that are expressible as the product of the number of distinct prime factors of preceding integers.

Original entry on oeis.org

16, 48, 72, 96, 144, 432, 576, 1296, 2592, 5184, 20736, 32805, 221184, 1555200, 11197440, 55987200, 95551488, 268738560, 302330880, 382205952, 524880000, 671846400, 6718464000, 34012224000, 155520000000, 403107840000, 6856864358400, 107495424000000, 110075314176000
Offset: 1

Views

Author

Metin Sariyar, Nov 02 2021

Keywords

Examples

			The number of distinct prime factors of the numbers 15, 14, 13, 12, 11, 10 are respectively 2, 2, 1, 2, 1, 2 and 2*2*1*2*1*2 = 16, hence 16 is a term.

Crossrefs

Cf. A001221, A000961, A007774, A033992, A033993, A051270, A074969, A176655, A348266.

Programs

  • Mathematica
    om[n_] := om[n] = PrimeNu[n]; q[n_] := Module[{m = n, k = n - 1}, While[k > 1 && Divisible[m, om[k]], m /= om[k]; k--]; m == 1]; Select[Range[2, 10^6], q] (* Amiram Eldar, Nov 02 2021 *)

Extensions

a(13)-a(17) from Amiram Eldar, Nov 02 2021
More terms from David A. Corneth, Nov 02 2021

A348837 a(n) is the smallest k such that the sum of the number of divisors of the n numbers from k to k+n-1 equals tau(k+n).

Original entry on oeis.org

2, 10, 93, 236, 355, 2634, 2873, 5032, 11331, 20150, 18889, 80628, 55427, 207886, 205905, 371264, 369583, 617742, 166301, 1436380, 720699, 2227658, 1081057, 831576, 4633175, 3326374, 2633373, 5045012, 11850271, 6683010, 11642369, 9979168, 9424767, 8648606, 24418765
Offset: 1

Views

Author

Metin Sariyar, Nov 01 2021

Keywords

Examples

			a(1) = 2 because tau(2) = tau(3) = 2; a(1) = A005237(1).
a(2) = 10 because tau(10) + tau(11) = 4 + 2 =  6, the same as tau(12) = 6.
a(3) = 93 because tau(93) + tau(94) + tau(95) = 4 + 4 + 4 = 12, the same as tau(96) = 12.

Links

Crossrefs

Cf. A000005, A005237, A348335.

Programs

  • Mathematica
    a[n_] := Module[{div = DivisorSigma[0, Range[n]], k = n + 1}, While[(d = DivisorSigma[0, k]) != Plus @@ div, div = Join[Drop[div, 1], {d}]; k++]; k - n]; Array[a, 20] (* Amiram Eldar, Nov 01 2021 *)
  • PARI
    a(n) = my(k=1); while (sum(i=k, k+n-1, numdiv(i)) != numdiv(k+n), k++); k; \\ Michel Marcus, Nov 01 2021
  • PARI
    a(n)=my(v=vector(n,k,numdiv(k)),s=vecsum(v),t,i=n); forfactored(k=n+1,2^63-1, t=numdiv(k); if(s==t, return(k[1]-n)); if(i++>n,i=1); s+=t-v[i]; v[i]=t) \\ Charles R Greathouse IV, Nov 01 2021

Extensions

a(21)-a(26) from Michel Marcus, Nov 01 2021
a(27)-a(35) from Amiram Eldar, Nov 01 2021

A348335 a(n) = smallest k such that the sum of the divisors of the n numbers from k to k+n-1 equals sigma(k+n), or -1 if no such k exists.

Original entry on oeis.org

14, 1, 591357
Offset: 1

Views

Author

Metin Sariyar, Oct 13 2021

Keywords

Comments

a(4) > 10^9, if it exists. - Amiram Eldar, Oct 13 2021

Examples

			a(1) = 14 because sigma(14) = sigma(15) = 24; a(1) = A002961(1).
a(2) = 1 because sigma(1) + sigma(2) = 1 + 3 = 4, the same as sigma(3) = 4; a(2) = A104149(1).
a(3) = 591357 because sigma(591357) + sigma(591358) + sigma(591359) = 866880 + 890352 + 599760 = 2356992, the same as sigma(591360) = 2356992.

Crossrefs

Cf. A000203, A002961, A065900, A092403, A104149, A252922.

Programs

  • Mathematica
    a[n_] := Module[{sig = DivisorSigma[1, Range[n]], k = n + 1}, While[(s = DivisorSigma[1, k]) != Plus @@ sig, sig = Join[Drop[sig, 1], {s}]; k++]; k - n]; Array[a, 3] (* Amiram Eldar, Oct 29 2021 *)
  • PARI
    isok(m, nb) = sum(i=1, nb, sigma(m+i-1)) == sigma(m+nb);
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, Oct 28 2021

A348266 k-digit numbers whose digit(s) are the number of distinct prime factors in each of the preceding k integers.

Original entry on oeis.org

22, 313, 2232, 2323, 2333, 32215, 432152, 2434332, 4222423, 43332543, 332325334, 2535332433, 4532543535234, 5435433351423
Offset: 1

Views

Author

Metin Sariyar, Oct 09 2021

Keywords

Comments

a(12) <= 2535332433. - David A. Corneth, Oct 10 2021
a(12) >= 10^9. - Michel Marcus, Oct 11 2021

Examples

			22 is a term because omega(20) = 2 and omega(21) = 2, whose concatenation is 22.
313 is a term because preceding it omega(310) = 3, omega(311) = 1 and omega(312) = 3, and their concatenation is 313.
32215 is a term because, the number of distinct prime divisors of 32210, 32211, 32212, 32213 and 32214 are 3, 2, 2, 1, 5 and their ordered concatenation gives the next number 32215.

Crossrefs

Cf. A001221, A000961, A007774, A033992, A033993, A051270, A074969, A176655.
Cf. A323083.

Programs

  • Mathematica
    Select[Range[33000], FromDigits[PrimeNu /@ (# - Range[IntegerLength[#], 1, -1])] == # &] (* Amiram Eldar, Oct 09 2021 *)
  • PARI
    isok(m) = {my(s="", k=m, i=1); while(1, s = concat(s, Str(omega(k))); if (eval(s) == m+i, return (i)); if (eval(s) > m+i, return(0)); k++; i++;);}
lista(nn) = my(nb); for(n=1, nn, if (nb=isok(n), print1(n+nb, ", "))); \\ Michel Marcus, Oct 09 2021

Extensions

a(8)-a(9) from Amiram Eldar, Oct 09 2021
a(10)-a(11) from Michel Marcus, Oct 10 2021
a(12) confirmed by Martin Ehrenstein, Oct 28 2021
a(13)-a(14) from Martin Ehrenstein, Oct 30 2021

A346423 Numbers m such that Sum_{k=1..m} omega(k) = sigma(m).

Original entry on oeis.org

11, 230, 52830, 160908, 6134334960
Offset: 1

Views

Author

Metin Sariyar, Jul 16 2021

Keywords

Comments

Numbers k such that A013939(k) = sigma(k).

Examples

			The sum of number of distinct primes dividing numbers up to 11 is 1+1+1+1+2+1+1+1+2+1 = sigma(11), so 11 is a term.

Crossrefs

Cf. A000203, A013939.

Programs

  • Mathematica
    s=0;Do[s=s+PrimeNu[n];If[DivisorSigma[1,n]==s,Print[n]],{n,2,160908}]

Extensions

a(5) from Martin Ehrenstein, Aug 22 2021

A346386 Triangular numbers that are sum of squares of two distinct triangular numbers.

Original entry on oeis.org

1, 10, 36, 45, 136, 325, 666, 820, 1225, 2080, 3321, 5050, 7381, 10440, 11026, 14365, 18721, 19306, 25425, 32896, 41905, 52650, 65341, 80200, 90100, 97461, 101025, 117370, 140185, 166176, 195625, 197506, 228826, 266085, 307720, 314821, 354061, 405450, 454581, 462241
Offset: 1

Views

Author

Metin Sariyar, Jul 14 2021

Keywords

Examples

			136, 10 and 6 are triangular numbers and 136 = 10^2 + 6^2 so 136 is a term.
820, 28 and 6 are triangular numbers and 820 = 28^2 + 6^2 so 820 is a term.

Crossrefs

Cf. A000217, A004431, A037270, A073613.

Programs

  • Mathematica
    m = 50; t = Accumulate[Range[0, m]]; Select[Union[Plus @@@ Subsets[t^2, {2}]], # <= t[[-1]]^2 && IntegerQ @ Sqrt[8*# + 1] &] (* Amiram Eldar, Jul 16 2021 *)

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

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