A049084 -id:A049084 - cp's OEIS frontend

A062537 Nodes in riff (rooted index-functional forest) for n.

0, 1, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 3, 4, 4, 4, 5, 5, 5, 4, 5, 4, 5, 4, 5, 5, 6, 5, 4, 6, 5, 6, 5, 5, 5, 6, 6, 5, 6, 5, 6, 6, 5, 6, 5, 4, 5, 6, 6, 4, 5, 7, 6, 6, 6, 5, 7, 5, 6, 6, 4, 7, 7, 5, 6, 6, 7, 6, 6, 6, 6, 6, 6, 7, 7, 6, 6, 4, 6, 5, 7, 7, 6, 7, 7, 6, 7, 7, 6, 7, 7, 7, 6, 5, 5, 7, 6, 6, 7, 5, 7, 8

Offset: 1

David W. Wilson, Jun 25 2001

A061396(n) is the number of times n appears in this sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Jon Awbrey, Illustrations of riffs for small integers.
Jon Awbrey, Riffs and Rotes.

Cf. A061396.

Cf. A001221, A027748, A049084, A124010.

import Data.Function (on)
a062537 1 = 0
a062537 n = sum $ map (+ 1) $
   zipWith ((+) `on` a062537) (map a049084 $ a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Feb 26 2013

a[1] = 0; a[n_] := a[n] = Sum[{p, e} = pe; a[PrimePi[p]] + a[e] + 1, {pe, FactorInteger[n]}]; Array[a, 105] (* Jean-François Alcover, Jul 26 2019 *)

a(Product(p_i^e_i)) = Sum(a(i)+a(e_i)+1), product over nonzero e_i in prime factorization.

a(n) = Sum_{k=1..A001221(n)} (a(A049084(A027748(n,k))) + a(A124010(n,k)) + 1). - Reinhard Zumkeller, Feb 26 2013

A033548 Honaker primes: primes P(k) such that sum of digits of P(k) equals sum of digits of k.

Original entry on oeis.org

131, 263, 457, 1039, 1049, 1091, 1301, 1361, 1433, 1571, 1913, 1933, 2141, 2221, 2273, 2441, 2591, 2663, 2707, 2719, 2729, 2803, 3067, 3137, 3229, 3433, 3559, 3631, 4091, 4153, 4357, 4397, 4703, 4723, 4903, 5009, 5507, 5701, 5711, 5741, 5801, 5843

Offset: 1

Calculated by Jud McCranie

A090431(A049084(a(n))) = 0.

			131 is the 32nd prime and sum of digits of both is 5.

Proposed by G. L. Honaker, Jr.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A007605, A033549, A049084, A072439, A090431.

a033548 n = a033548_list !! (n-1)
a033548_list = filter ((== 0) . a090431 . a049084) a000040_list
-- Reinhard Zumkeller, Mar 16 2014

read("transforms") :
isA033548 := proc(n)
    if isprime(n) and digsum(n) = digsum(numtheory[pi](n)) then
        true ;
    else
        false;
    end if;
end proc:
A033548 := proc(n)
    local p, k;
    if n = 1 then
        131;
    else
        p := nextprime(procname(n-1)) ;
        while true  do
            if isA033548(p) then
                return p;
            end if;
            p := nextprime(p) ;
        end do:
    end if;
end proc:
seq(A033548(n),n=1..40) ; # R. J. Mathar, Jul 07 2021

Prime[ Select[ Range[ 2000 ], Apply[ Plus, IntegerDigits[ # ] ] == Apply[ Plus, IntegerDigits[ Prime[ # ] ] ] & ] ] (* Santi Spadaro, Oct 14 2001 *)
Select[ Prime@ Range@ 5927, Plus @@ IntegerDigits@ # == Plus @@ IntegerDigits@ PrimePi@ # &]  (* Robert G. Wilson v, Jun 07 2009 *)
nn=800;Transpose[Select[Thread[{Prime[Range[nn]],Range[nn]}],Total[IntegerDigits[First[#]]]== Total[ IntegerDigits[ Last[#]]]&]][[1]] (* Harvey P. Dale, Jun 13 2011 *)

is(n)=isprime(n) && sumdigits(n)==sumdigits(primepi(n)) \\ Charles R Greathouse IV, Jun 18 2015

from sympy.ntheory.factor_ import digits
from sympy import primepi, primerange
print([n for n in primerange(1, 5901) if (sum(digits(n)[1:])==sum(digits(primepi(n))[1:]))]) # Indranil Ghosh, Jun 27 2017, after Charles R Greathouse IV

a(n) = A000040(A033549(n)). - R. J. Mathar, Jul 07 2021

More terms from Robert G. Wilson v, Jun 07 2009

A054841 If n = 2^a * 3^b * 5^c * 7^d * ... then a(n) = a + 10 * b + 100 * c + 1000 * d + ... .

Original entry on oeis.org

0, 1, 10, 2, 100, 11, 1000, 3, 20, 101, 10000, 12, 100000, 1001, 110, 4, 1000000, 21, 10000000, 102, 1010, 10001, 100000000, 13, 200, 100001, 30, 1002, 1000000000, 111, 10000000000, 5, 10010, 1000001, 1100, 22, 100000000000, 10000001, 100010

Offset: 1

Henry Bottomley, Apr 11 2000

Are there any other numbers besides n=12 for which n=a(n) ? - Ctibor O. Zizka, Oct 08 2008
The sequence is a morphism from (N*,*) into (N,+), cf. formula. Up to n=1023, the digit sum A007953(a(n)) equals Omega(n) = A001222(n). This holds whenever A051903(n)<10. Restricted to these n, the sequence is also injective. However, when n is a multiple of 2^10, 3^10, 5^10 etc, then a(n) is equal to some a(m) with mM. F. Hasler, Nov 16 2008
This has been called the "Exponential Prime Power Representation" of n by W. Nissen in a post to the sci.math newsgroup (where probably some more sophisticated convention for representing digits > 10 would be used). - M. F. Hasler, Jul 03 2016

			a(25) = 200 because 25 = 5^2 * 3^0 * 2^0.
a(1024) = 10 = a(3), because 1024 = 2^10; but this two-digit multiplicity overflows into the 10^1 position, which encodes for powers of three.

Robert Israel, Table of n, a(n) for n = 1..10000 (n=1..1023 from Michael De Vlieger)
Evans A Criswell, A Sequence Puzzle (Posted to rec.puzzles Jan 01 1997)
Walter Nissen, Exponential Prime Power Representation, sci.math newsgroup, May 23 1995.

Row 10 of A104244.
Left inverse of A054842.
Cf. A001222, A048675, A090880, A090881, A090882, A276075, A276085 (analogous constructions for other bases), A090883, A090884, A049084, A027748, A124010, A056239.

a054841 1 = 0
a054841 n = sum $ zipWith (*)
                  (map ((10 ^) . subtract 1 . a049084) $ a027748_row n)
                  (map fromIntegral $ a124010_row n)
-- Reinhard Zumkeller, Aug 03 2015

A:= n -> add(t[2]*10^(numtheory:-pi(t[1])-1),t= ifactors(n)[2]);
seq(A(n), n=1..1000); # Robert Israel, Jul 24 2014

a054841[n_Integer] := Catch[FromDigits[IntegerDigits[Apply[Plus,
     Which[n == 0, Throw["undefined"],
        n == 1, 0,
        Max[Last /@ FactorInteger @ n] > 9, Throw["overflow"],
        True, Power[10, PrimePi[Abs[#]] - 1]] & /@
      Flatten[ConstantArray @@@ FactorInteger[n]]]]]] (* Michael De Vlieger, Jul 24 2014 *)

A054841(n)=sum(i=1,#n=factor(n)~,10^primepi(n[1,i])*n[2,i])/10 \\ M. F. Hasler, Nov 16 2008

from sympy import factorint, primepi
def a(n): return sum(e*10**(primepi(p)-1) for p, e in factorint(n).items())
print([a(n) for n in range(1, 41)]) # Michael S. Branicky, Mar 17 2024

a(m*n) = a(m) + a(n) for all m,n > 0. A007953(a(n))=A001222(n) for all n such that A051903(n) < 10. - M. F. Hasler, Nov 16 2008
a(n) = sum(10^(A049084(A027748(k))-1) * A124010(k): k = 1..A001221(n)). - Reinhard Zumkeller, Aug 03 2015
a(A054842(n)) = n for all n >= 0. - Antti Karttunen, Aug 29 2016
a(n) = Sum_{i>0} e_i*10^(i-1) when n = Product_{i>0} prime(i)^e_i. - M. F. Hasler, Mar 14 2018

A087794 Products of prime-indices of factors of semiprimes.

Original entry on oeis.org

1, 2, 4, 3, 4, 6, 8, 5, 9, 6, 10, 7, 12, 8, 12, 9, 16, 14, 15, 16, 10, 11, 18, 18, 12, 20, 13, 21, 14, 20, 24, 22, 15, 24, 16, 24, 27, 17, 28, 25, 18, 26, 28, 32, 19, 30, 20, 30, 30, 21, 33, 22, 32, 36, 23, 36, 34, 24, 36, 36, 35, 25, 38, 26, 40, 39, 27, 40, 40, 28, 42, 44, 29

Offset: 1

Reinhard Zumkeller, Oct 09 2003

nonn

A semiprime (A001358) is a product of any two prime numbers. A prime index of n is a number m such that the m-th prime number divides n. The multiset of prime indices of n is row n of A112798. - Gus Wiseman, Dec 04 2020

			A001358(20)=57=3*19=A000040(2)*A000040(8), therefore a(20)=2*8=16.
From _Gus Wiseman_, Dec 04 2020: (Start)
The sequence of all semiprimes together with the products of their prime indices begins:
   4: 1 * 1 = 1
   6: 1 * 2 = 2
   9: 2 * 2 = 4
  10: 1 * 3 = 3
  14: 1 * 4 = 4
  15: 2 * 3 = 6
  21: 2 * 4 = 8
  22: 1 * 5 = 5
  25: 3 * 3 = 9
  26: 1 * 6 = 6
(End)

A003963 is the version for not just semiprimes.
A176504 gives the sum of the same two indices.
A176506 gives the difference of the same two indices.
A339361 is the squarefree case.
A001358 lists semiprimes.
A006881 lists squarefree semiprimes.
A289182/A115392 list the positions of odd/even terms of A001358.
A338898/A338912/A338913 give the prime indices of semiprimes.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
A338904 groups semiprimes by weight.
Cf. A001222, A046315, A056239, A065516, A084126, A084127, A112798, A128301, A300912, A318990, A320655, A338909, A339362.

Table[If[SquareFreeQ[n],Times@@PrimePi/@First/@FactorInteger[n],PrimePi[Sqrt[n]]^2],{n,Select[Range[100],PrimeOmega[#]==2&]}] (* Gus Wiseman, Dec 04 2020 *)

a(n) = A049084(A084126(n))*A049084(A084127(n)) = A003963(A001358(n)).

a(n) = A003963(A001358(n)) = A338912(n) * A338913(n). - Gus Wiseman, Dec 04 2020

A073520 Smallest magic constant for any n X n magic square made from consecutive primes, or 0 if no such magic square exists.

Original entry on oeis.org

2, 0, 4440084513, 258, 313, 484, 797, 2016, 2211, 2862, 4507, 6188, 6325, 9660, 12669, 13016, 16857, 19530, 23069, 28184, 38761, 46302, 42515, 49846, 59087, 70260, 73385, 78960, 97267, 98316, 111023, 124454, 134641, 152952, 163043, 180596, 195975, 218432, 237623, 293182, 276243, 298868

Offset: 1

N. J. A. Sloane, Aug 29 2002

			A square of order 15 found by _Natalia Makarova_, communicated by Stefano Tognon, Sep 23 2009:
[  131  167  229  461  541  617  733  911  967 1091 1259 1279 1319 1471 1493
   547  907 1583 1613  149 1423  193 1601  941  137  233  389 1039 1283  631
  1019  181  751  163 1453 1301 1297 1277  271 1619 1327  691  277  281  761
  1307  719  359  919 1063  653 1237  269 1433  863 1439  313  191 1021  883
   503 1367  433 1013  829 1153  317  347 1109  491 1249  677 1451 1489  241
   421  311 1487  439 1049 1409 1123  463  409  983  449 1031 1163  373 1559
  1399 1193  419 1531  971  647  977 1051  709  479 1229  379  353 1093  239
   599  953 1213  587  499  727 1321  787  307 1151  157 1571 1033  773  991
   211 1291 1499  577 1087  349  947  467  739  613 1171 1609  173  839 1097
   563  139 1373 1459 1289  443  619 1201 1427  809  881 1303  331  263  569
   607 1607 1511  673 1181 1481 1217  523  661  857  223  743  197  431  757
   853  643  701  179 1483  571  769  859 1447  659  929  997 1223 1129  227
  1549  887  257  557  367 1061  601  337 1361  937 1231  811 1543  293  877
  1579 1187  397 1069  509  683  797 1567  401  383  641  283  823  827 1523
  1381 1117  457 1429  199  151  521 1009  487 1597  251  593 1553 1103 821]

Allan W. Johnson, Jr., Journal of Recreational Mathematics, vol. 14:2, 1981-82, pp. 152-153.
Allan W. Johnson, Jr., Journal of Recreational Mathematics, vol. 23:3, 1991, pp. 190-191.
H. L. Nelson, Journal of Recreational Mathematics, 1988, vol. 20:3, p. 214.
Clifford A. Pickover, The Zen of Magic Squares, Circles and Stars: An Exhibition of Surprising Structures across Dimensions, Princeton University Press, 2002.

M. F. Hasler, Table of n, a(n) for n = 1..63
Mutsumi Suzuki, Study of Magic Squares, 1957, in Japanese. Gives minimal squares of orders from 4 to 9 composed of consecutive primes.
Harvey Heinz, Prime Magic Squares
N. Makarova Squares 7x7, 8x8, 9x9, Squares 10x10, 11x11, 12x12, Square 14x14 (in Russian)
Stefano Tognon, Table for prime magic squares
Eric Weisstein's World of Mathematics, Prime Magic Square
Index entries for sequences related to magic squares

Cf. A104157: smallest element in an n X n magic squares of consecutive primes.
Cf. A073519 and A320873 (3 X 3 magic square of consecutive primes), A073521 (consecutive primes of a 4 X 4 magic square), A245721 and A320874 (4 X 4 pandigital magic square of consecutive primes), A073522 (consecutive primes of a 5 X 5 magic square, non-minimal and non-pandiagonal), A073523 and A320876 (6 X 6 pandigital magic square of consecutive primes).
Cf. A256234: magic sums of 4 X 4 pandiagonal magic squares of consecutive primes.

A073520(n,p=A104157[n])=sum(i=2,n^2,p=nextprime(p+1),p)/n \\ Assumes a pre-computed array A104157, but can be used to find a(n) and A104157(n) by calculating this for supplied primes p until the result satisfies the condition of the conjecture in FORMULA. - M. F. Hasler, Oct 29 2018

Conjecture: for n >= 5, a(n) equals the smallest integer of the form (A000040(s+1) + ... + A000040(s+n^2))/n = (A007504(s+n^2) - A007504(s))/n of the same parity as n.
a(2) = 0, otherwise a(n) = (1/n) * Sum_{m=k..n^2+k-1} A000040(m), where k = A049084(A104157(n)). - Arkadiusz Wesolowski, Nov 06 2015
In the above, A049084 could be replaced by A000720 = primepi. - M. F. Hasler, Oct 29 2018

a(5)-a(6) corrected and a(7)-a(14) added, from the work of Stefano Tognon and Natalia Makarova, by Max Alekseyev, Sep 23 2009
a(15) from Natalia Makarova, a(16) and further terms from Stefano Tognon
Edited by Max Alekseyev, Oct 13 2009
Edited and more terms (using A104157) from M. F. Hasler, Oct 29 2018

A002645 Quartan primes: primes of the form x^4 + y^4, x > 0, y > 0.

Original entry on oeis.org

2, 17, 97, 257, 337, 641, 881, 1297, 2417, 2657, 3697, 4177, 4721, 6577, 10657, 12401, 14657, 14897, 15937, 16561, 28817, 38561, 39041, 49297, 54721, 65537, 65617, 66161, 66977, 80177, 83537, 83777, 89041, 105601, 107377, 119617, 121937

Offset: 1

N. J. A. Sloane

The largest known quartan prime is currently the largest known generalized Fermat prime: The 1353265-digit 145310^262144 + 1 = (145310^65536)^4 + 1^4, found by Ricky L Hubbard. - Jens Kruse Andersen, Mar 20 2011

Primes of the form (a^2 + b^2)/2 such that |a^2 - b^2| is a square. - Thomas Ordowski, Feb 22 2017

			a(1) =   2 = 1^4 + 1^4.
a(2) =  17 = 1^4 + 2^4.
a(3) =  97 = 2^4 + 3^4.
a(4) = 257 = 1^4 + 4^4.

A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929; see Vol. 1, pp. 245-259.
N. D. Elkies, Primes of the form a^4 + b^4, Mathematical Buds, Ed. H. D. Ruderman Vol. 3 Chap. 3 pp. 22-8 Mu Alpha Theta 1984.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Zak Seidov, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
A. J. C. Cunningham, High quartan factorisations and primes, Messenger of Mathematics 36 (1907), pp. 145-174.
A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.

Subsequence of A002313 and of A028916.
Intersection of A004831 and A000040.
Cf. A002646, A000583, A003336, A000290, A256852.

a002645 n = a002645_list !! (n-1)
a002645_list = 2 : (map a000040 $ filter ((> 1) . a256852) [1..])
-- Reinhard Zumkeller, Apr 11 2015

nn = 100000; Sort[Reap[Do[n = a^4 + b^4; If[n <= nn && PrimeQ[n], Sow[n]], {a, nn^(1/4)}, {b, a}]][[2, 1]]]
With[{nn=20},Select[Union[Flatten[Table[x^4+y^4,{x,nn},{y,nn}]]],PrimeQ[ #] && #<=nn^4+1&]] (* Harvey P. Dale, Aug 10 2021 *)

upto(lim)=my(v=List(2),t);forstep(x=1,lim^.25,2,forstep(y=2,(lim-x^4)^.25,2,if(isprime(t=x^4+y^4),listput(v,t))));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 05 2011

list(lim)=my(v=List([2]),x4,t); for(x=1,sqrtnint(lim\=1,4), x4=x^4; forstep(y=1+x%2,min(sqrtnint(lim-x4,4), x-1),2, if(isprime(t=x4+y^4), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Aug 20 2017

A000040 INTERSECTION A003336. - Jonathan Vos Post, Sep 23 2006

A256852(A049084(a(n))) > 1 for n > 1. - Reinhard Zumkeller, Apr 11 2015

More terms from Victoria A Sapko (vsapko(AT)canes.gsw.edu), Nov 07 2002

A091205 Factorization and index-recursion preserving isomorphism from binary codes of GF(2) polynomials to integers.

Original entry on oeis.org

0, 1, 2, 3, 4, 9, 6, 5, 8, 15, 18, 7, 12, 23, 10, 27, 16, 81, 30, 13, 36, 25, 14, 69, 24, 11, 46, 45, 20, 21, 54, 19, 32, 57, 162, 115, 60, 47, 26, 63, 72, 61, 50, 33, 28, 135, 138, 17, 48, 35, 22, 243, 92, 39, 90, 37, 40, 207, 42, 83, 108, 29, 38, 75, 64, 225, 114, 103

Offset: 0

Antti Karttunen, Jan 03 2004

nonn

This "deeply multiplicative" bijection is one of the deep variants of A091203 which satisfy most of the same identities as the latter, but it additionally preserves also the structures where we recurse on irreducible polynomial's A014580-index. E.g., we have: A091238(n) = A061775(a(n)). The reason this holds is that when the permutation is restricted to the binary codes for irreducible polynomials over GF(2) (A014580), it induces itself: a(n) = A049084(a(A014580(n))).

On the other hand, when this permutation is restricted to the union of {1} and reducible polynomials over GF(2) (A091242), permutation A245813 is induced.

Antti Karttunen, Table of n, a(n) for n = 0..8192
A. Karttunen, Scheme-program for computing this sequence.
Index entries for sequences operating on GF(2)[X]-polynomials
Index entries for sequences that are permutations of the natural numbers

Inverse: A091204.
Similar or related permutations: A091203, A106443, A106445, A106447, A235042, A245704, A245813, A245821.
Cf. A000040, A007097, A010051, A014580, A049084, A061775, A091238, A091225, A091226, A091230, A091242.

allocatemem(123456789);
v091226 = vector(2^22);
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
n=2; while((n < 2^22), if(isA014580(n), v091226[n] = v091226[n-1]+1, v091226[n] = v091226[n-1]); n++)
A091226(n) = v091226[n];
A091205(n) = if(n<=1,n,if(isA014580(n),prime(A091205(A091226(n))),{my(irfs,t); irfs=subst(lift(factor(Mod(1,2)*Pol(binary(n)))),x,2); irfs[,1]=apply(t->A091205(t),irfs[,1]); factorback(irfs)}));
for(n=0, 8192, write("b091205.txt", n, " ", A091205(n)));
\\ Antti Karttunen, Aug 16 2014

a(0)=0, a(1)=1. For n that is coding an irreducible polynomial, that is if n = A014580(i), we have a(n) = A000040(a(i)) and for reducible polynomials a(ir_i X ir_j X ...) = a(ir_i) * a(ir_j) * ..., where ir_i = A014580(i), X stands for carryless multiplication of polynomials over GF(2) (A048720) and * for the ordinary multiplication of integers (A004247).
As a composition of related permutations:
a(n) = A245821(A245704(n)).
Other identities.
For all n >= 0, the following holds:
a(A091230(n)) = A007097(n). [Maps iterates of A014580 to the iterates of primes. Permutation A245704 has the same property.]
For all n >= 1, the following holds:
A010051(a(n)) = A091225(n). [After a(1)=1, maps binary representations of irreducible GF(2) polynomials, A014580, bijectively to primes and the binary representations of corresponding reducible polynomials, A091242, to composite numbers, in some order. The permutations A091203, A106443, A106445, A106447, A235042 and A245704 have the same property.]

Name changed by Antti Karttunen, Aug 16 2014

A006378 Prime self (or Colombian) numbers: primes not expressible as the sum of an integer and its digit sum.

Original entry on oeis.org

3, 5, 7, 31, 53, 97, 211, 233, 277, 367, 389, 457, 479, 547, 569, 613, 659, 727, 839, 883, 929, 1021, 1087, 1109, 1223, 1289, 1447, 1559, 1627, 1693, 1783, 1873, 2099, 2213, 2347, 2437, 2459, 2503, 2549, 2593, 2617, 2683, 2729, 2819, 2953, 3023, 3067

Offset: 1

N. J. A. Sloane

M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 116.
D. R. Kaprekar, Puzzles of the Self-Numbers. 311 Devlali Camp, Devlali, India, 1959.
D. R. Kaprekar, The Mathematics of the New Self Numbers, Privately Printed, 311 Devlali Camp, Devlali, India, 1963.
D. R. Kaprekar, The Mathematics of the New Self Numbers (Part V). 311 Devlali Camp, Devlali, India, 1967.
Jeffrey Shallit, personal communication c. 1999.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Donovan Johnson, Table of n, a(n) for n = 1..10000
Shyam Sunder Gupta, On Some Marvellous Numbers of Kaprekar, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
D. R. Kaprekar, The Mathematics of the New Self Numbers [annotated and scanned]
B. Recamán, Problem E2408, Amer. Math. Monthly, 81 (1974), p. 407.
T. Trotter, Charlene Numbers (Copy as of Jan. 2018 on web.archive.org; original page no longer available.) Originally published Nov. 2010.
Index entries for Colombian or self numbers and related sequences

Subsequence of A247104.

Cf. A003052, A007953, A047791, A048521, A062028, A000040, A107740, A049084.

a006378 n = a006378_list !! (n-1)
a006378_list = map a000040 $ filter ((== 0) . a107740) [1..]
-- Reinhard Zumkeller, Sep 27 2014

With[{nn=3200},Complement[Prime[Range[PrimePi[nn]]],Table[n+Total[ IntegerDigits[n]],{n,nn}]]] (* Harvey P. Dale, Dec 30 2011 *)

select( is_A006378(n)=is_A003052(n)&&isprime(n), primes([1,3000])) \\ M. F. Hasler, Nov 08 2018

A107740(A049084(a(n))) = 0. [Corrected by Reinhard Zumkeller, Sep 27 2014]

A091202 Factorization-preserving isomorphism from nonnegative integers to binary codes for polynomials over GF(2).

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 11, 8, 5, 14, 13, 12, 19, 22, 9, 16, 25, 10, 31, 28, 29, 26, 37, 24, 21, 38, 15, 44, 41, 18, 47, 32, 23, 50, 49, 20, 55, 62, 53, 56, 59, 58, 61, 52, 27, 74, 67, 48, 69, 42, 43, 76, 73, 30, 35, 88, 33, 82, 87, 36, 91, 94, 39, 64, 121, 46, 97, 100, 111, 98

Offset: 0

Antti Karttunen, Jan 03 2004

E.g. we have the following identities: A000005(n) = A091220(a(n)), A001221(n) = A091221(a(n)), A001222(n) = A091222(a(n)), A008683(n) = A091219(a(n)), A014580(n) = a(A000040(n)), A049084(n) = A091227(a(n)).

Antti Karttunen, Table of n, a(n) for n = 0..8192
A. Karttunen, Scheme-program for computing this sequence.
Index entries for sequences operating on GF(2)[X]-polynomials
Index entries for sequences that are permutations of the natural numbers

Inverse: A091203.
Several variants exist: A235041, A091204, A106442, A106444, A106446.
Cf. also A000005, A091220, A001221, A091221, A001222, A091222, A008683, A091219, A000040, A014580, A048720, A049084, A091227, A245703, A234741.
Cf. also A302023, A302025, A305417, A305427 for other similar permutations.

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2));
A305420(n) = { my(k=1+n); while(!A091225(k),k++); (k); };
A305421(n) = { my(f = subst(lift(factor(Pol(binary(n))*Mod(1, 2))),x,2)); for(i=1,#f~,f[i,1] = Pol(binary(A305420(f[i,1])))); fromdigits(Vec(factorback(f))%2,2); };
A091202(n) = if(n<=1,n,if(!(n%2),2*A091202(n/2),A305421(A091202(A064989(n))))); \\ Antti Karttunen, Jun 10 2018

a(0)=0, a(1)=1, a(p_i) = A014580(i) for primes p_i with index i and for composites a(p_i * p_j * ...) = a(p_i) X a(p_j) X ..., where X stands for carryless multiplication of GF(2)[X] polynomials (A048720).
Other identities. For all n >= 1, the following holds:
A091225(a(n)) = A010051(n). [Maps primes to binary representations of irreducible GF(2) polynomials, A014580, and nonprimes to union of {1} and the binary representations of corresponding reducible polynomials, A091242. The permutations A091204, A106442, A106444, A106446, A235041 and A245703 have the same property.]
From Antti Karttunen, Jun 10 2018: (Start)
For n <= 1, a(n) = n, for n > 1, a(n) = 2*a(n/2) if n is even, and if n is odd, then a(n) = A305421(a(A064989(n))).
a(n) = A305417(A156552(n)) = A305427(A243071(n)).
(End)

A091203 Factorization-preserving isomorphism from binary codes of GF(2) polynomials to integers.

Original entry on oeis.org

0, 1, 2, 3, 4, 9, 6, 5, 8, 15, 18, 7, 12, 11, 10, 27, 16, 81, 30, 13, 36, 25, 14, 33, 24, 17, 22, 45, 20, 21, 54, 19, 32, 57, 162, 55, 60, 23, 26, 63, 72, 29, 50, 51, 28, 135, 66, 31, 48, 35, 34, 243, 44, 39, 90, 37, 40, 99, 42, 41, 108, 43, 38, 75, 64, 225, 114, 47, 324

Offset: 0

Antti Karttunen, Jan 03 2004

nonn

E.g. we have the following identities: A000040(n) = a(A014580(n)), A091219(n) = A008683(a(n)), A091220(n) = A000005(a(n)), A091221(n) = A001221(a(n)), A091222(n) = A001222(a(n)), A091225(n) = A010051(a(n)), A091227(n) = A049084(a(n)), A091247(n) = A066247(a(n)).

Antti Karttunen, Table of n, a(n) for n = 0..8192
A. Karttunen, Scheme-program for computing this sequence.
Index entries for sequences operating on GF(2)[X]-polynomials
Index entries for sequences that are permutations of the natural numbers

Inverse: A091202.
Several variants exist: A235042, A091205, A106443, A106445, A106447.
Cf. also A000005, A000040, A001221, A001222, A004247, A008683, A010051, A014580, A048720, A049084, A066247, A091219, A091220, A091221, A091222, A091225, A091227, A091247, A234741, A234742, A245703, A245704, A278233.
Cf. also A302024, A302026, A305418, A305428 for other similar permutations.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2));
A305419(n) = if(n<3,1, my(k=n-1); while(k>1 && !A091225(k),k--); (k));
A305422(n) = { my(f = subst(lift(factor(Pol(binary(n))*Mod(1, 2))),x,2)); for(i=1,#f~,f[i,1] = Pol(binary(A305419(f[i,1])))); fromdigits(Vec(factorback(f))%2,2); };
A091203(n) = if(n<=1,n,if(!(n%2),2*A091203(n/2),A003961(A091203(A305422(n))))); \\ Antti Karttunen, Jun 10 2018

a(0)=0, a(1)=1. For n's coding an irreducible polynomial ir_i, that is if n=A014580(i), we have a(n) = A000040(i) and for composite polynomials a(ir_i X ir_j X ...) = p_i * p_j * ..., where p_i = A000040(i) and X stands for carryless multiplication of GF(2)[X] polynomials (A048720) and * for the ordinary multiplication of integers (A004247).
Other identities. For all n >= 1, the following holds:
A010051(a(n)) = A091225(n). [After a(1)=1, maps binary representations of irreducible GF(2) polynomials, A014580, to primes and the binary representations of corresponding reducible polynomials, A091242, to composite numbers. The permutations A091205, A106443, A106445, A106447, A235042 and A245704 have the same property.]
From Antti Karttunen, Jun 10 2018: (Start)
For n <= 1, a(n) = n, for n > 1, a(n) = 2*a(n/2) if n is even, and if n is odd, then a(n) = A003961(a(A305422(n))).
a(n) = A005940(1+A305418(n)) = A163511(A305428(n)).
A046523(a(n)) = A278233(n).
(End)

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