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 11-15 of 15 results.

A224252 Nonpalindromic n such that the factorizations of n and its digital reverse differ only for the exponents order.

Original entry on oeis.org

277816, 618772, 14339143, 34193341, 1125355221, 1225535211, 2613391326, 6231933162, 26157457326, 62375475162, 100504263021, 102407325111, 111523704201, 120362405001, 144326261443, 275603902756, 277816277816, 344162623441, 392739273927, 392875758639
Offset: 1

Views

Author

Giovanni Resta, Apr 02 2013

Keywords

Comments

Subset of A110751 and A110819.

Examples

			277816 and its reverse 618772 are in the sequence since 277816 = 2^3*7*11^2*41 and 618772 = 2^2*7^3*11*41 have the same prime divisors and the same exponents (1,1,2,3).

Links

Crossrefs

Cf. A110751, A110819, A224253.

Programs

  • Mathematica
    Do[fn = FactorInteger@n; fr = FactorInteger@ FromDigits@ Reverse@ IntegerDigits@n; If[fn != fr && First /@ fn == First /@ fr && Sort[Last /@ fn] == Sort[Last /@ fr], Print[n]], {n, 15*10^6}]
  • Python
    from sympy import primefactors, factorint
A224252 = [n for n in range(1,10**6) if n != int(str(n)[::-1]) and primefactors(n) == primefactors(int(str(n)[::-1])) and sorted(factorint(n).values()) == sorted(factorint(int(str(n)[::-1])).values())] # Chai Wah Wu, Aug 21 2014

A147882 Positive integers k that are balanced, meaning that if k has d digits, then its initial ceiling(d/2) digits have the same sum as its last ceiling(d/2) digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 222, 232, 242, 252, 262, 272, 282, 292, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 404, 414, 424, 434, 444, 454, 464, 474, 484, 494
Offset: 1

Views

Author

Jason Tarver (scottarver(AT)gmail.com), Nov 17 2008

Keywords

Comments

Differs from A110751 in cases like n=1010, 1089, 1102, 1120, 1203, 1212, 1230, etc. - R. J. Mathar, Dec 13 2008

Examples

			From _David A. Corneth_, Sep 28 2023: (Start)
353 is a term as it has k = 3 digits and so we see that the sum of the first ceiling(k/2) = ceiling(3/2) = 2 and the last ceiling(k/2) = ceiling(3/2) = 2 are equal and indeed 3 + 5 = 5 + 3.
13922 is a term as it has k = 5 digits and so we see that the sum of the first ceiling(k/2) = ceiling(5/2) = 2 and the last ceiling(k/2) = ceiling(5/2) = 2 are equal and indeed 1 + 3 + 9 = 9 + 2 + 2. (End)

Links

Crossrefs

Cf. A002113, A110751.

Programs

  • PARI
    is(n) = {my(d = digits(n), qdp1 = #d + 1); sum(i = 1, #d\2, d[i]-d[qdp1 - i]) == 0} \\ David A. Corneth, Sep 28 2023

Extensions

Definition clarified by N. J. A. Sloane, Oct 06 2023

A201009 Numbers m such that the set of distinct prime divisors of m is equal to the set of distinct prime divisors of the arithmetic derivative m'.

Original entry on oeis.org

1, 4, 16, 27, 108, 144, 256, 432, 500, 784, 972, 1323, 1728, 2700, 2916, 3125, 3456, 5292, 8788, 11664, 12500, 13068, 15376, 16875, 19683, 20736, 23328, 25000, 27648, 28125, 31212, 34300, 47916, 54000, 57132, 65536, 72000, 78732, 97556, 102400, 103788, 104544
Offset: 1

Views

Author

Paolo P. Lava, Jan 09 2013

Keywords

Comments

A027748(n,k) = A027748(A003415(n),k) for k=1..A001221(n). - Reinhard Zumkeller, Jan 16 2013
A051674 is a subsequence of this sequence.

Examples

			n = 1728 = 2^6*3^3, n' = 6912 = 2^8*3^3 have the same prime factors 2 and 3.

Links

Crossrefs

Cf. A001221, A003415, A027748, A051674, A055744, A081377, A110751, A110819.

Programs

  • Haskell
    a201009 = a201009_list
a201009_list = 1 : filter
   (\x -> a027748_row x == a027748_row (a003415 x)) [2..]
-- Reinhard Zumkeller, Jan 16 2013
  • Maple
    with(numtheory);
A201009:=proc(q)
local a,b,k,n;
for n from 1 to q do
  a:=ifactors(n)[2]; b:=ifactors(n*add(op(2,p)/op(1,p),p=ifactors(n)[2]))[2];
  if nops(a)=nops(b) then
    if product(a[k][1],k=1..nops(a))=product(b[k][1],k=1..nops(a)) then print(n);
fi; fi; od; end:
A201009(100000); # Paolo P. Lava, Jan 09 2013
  • Python
    from sympy import primefactors, factorint
A201009 = [n for n in range(1,10**5) if primefactors(n) == primefactors(sum([int(n*e/p) for p,e in factorint(n).items()]) if n > 1 else 0)] # Chai Wah Wu, Aug 21 2014

A242092 Numbers n such that n and the digital reversal of the n-th prime in base 10 have the same distinct prime factors.

Original entry on oeis.org

86, 1357, 24146, 1028736826, 33667786628, 2132089369082
Offset: 1

Views

Author

Chai Wah Wu, Aug 14 2014

Keywords

Comments

First 3 terms are all products of 2 primes.
a(4) > 10^8. - Chai Wah Wu, Aug 15 2014
a(7) > 10^13. - Giovanni Resta, Dec 09 2019

Examples

			86 = 2^1*43^1, the 86th prime is 443 and 344 = 2^3*43^1.
1357 = 59^1*23^1, the 1357th prime is 11213 and 31211 = 59^1*23^2.

Crossrefs

Cf. A110751.

Programs

  • PARI
    rev(n)=r="";d=digits(n);for(i=1,#d,r=concat(Str(d[i]),r));eval(r)
for(n=1,10^7,p=rev(prime(n));if(omega(n)==omega(p),if(gcd(n,p)==min(n,p),print1(n,", ")))) \\ Derek Orr, Aug 14 2014
  • Python
    from sympy import primefactors, prime
A242092 = [n for n in range(1,10**7) if primefactors(n) == primefactors(int(str(prime(n))[::-1]))]

Extensions

a(4)-a(6) from Giovanni Resta, Dec 09 2019

A110755 a(n) = Tau(N), where N = the number obtained as a concatenation of 9801 with itself n times. Tau(n) = number of divisors of n.

Original entry on oeis.org

15, 60, 288, 240, 480, 2304, 960, 7680, 5376, 7680, 5120, 18432, 1920, 15360, 2359296, 122880, 3840, 344064, 240, 122880, 7077888, 81920, 3840, 9437184, 491520, 30720, 786432, 491520, 30720, 150994944, 960, 983040, 12582912, 61440
Offset: 1

Views

Author

Amarnath Murthy, Aug 11 2005

Keywords

Comments

9801, has the property that any number of concatenation of it with self and the digit reversal have same prime divisors.

Examples

			a(2) =tau(98019801) = 60.

Crossrefs

Cf. A110751, A110752, A110753, A110754.

Extensions

More terms from Alexis Olson (AlexisOlson(AT)gmail.com), Nov 14 2008
More terms from David Wasserman, Dec 18 2008
Previous Showing 11-15 of 15 results.

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

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