A255411 -id:A255411 - cp's OEIS frontend

A257685 Left inverse for injection A255411: a(0) = 0, after which, if n = A255411(k) for some k, then a(n) = k, otherwise a(n) = 0.

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 4, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 9, 0, 10, 0, 0, 0, 11, 0, 12, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 14, 0, 0, 0, 15, 0, 16, 0, 0, 0, 17, 0, 18, 0, 0, 0, 19, 0, 0, 0, 0, 0, 0, 0, 20, 0, 0, 0, 21, 0, 22, 0, 0, 0, 23, 0, 0

Offset: 0

Antti Karttunen, May 04 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10080

Cf. A255411, A257680, A257681, A257684.

Position[Select[Range[0, 120], ! MemberQ[IntegerDigits[#, MixedRadix[Reverse@ Range@ 12]], 1] &], #] - 1 & /@ Range[0, 120] /. {} -> 0 // Flatten (* Michael De Vlieger, May 30 2016, Version 10.2 *)

from sympy import factorial as f
def a007623(n, p=2):
    return n if n0 else '0' for i in x)[::-1]
    return 0 if n==1 else sum([int(y[i])*f(i + 1) for i in range(len(y))])
def a257680(n): return 1 if '1' in str(a007623(n)) else 0
def a(n): return (1 - a257680(n))*a257684(n)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 21 2017

(define (A257685 n) (* (- 1 (A257680 n)) (A257684 n)))

a(0) = 0, after which, if n = A255411(k) for some k, then a(n) = k, otherwise a(n) = 0.
a(n) = (1-A257680(n)) * A257684(n).
Other identities:
For all n >= 0, a(A255411(n)) = n. [This sequence works as a left inverse of A255411.]

A275845 Permutation of natural numbers: a(0) = 0, a(A153880(n)) = A255411(n), a(A273670(n)) = A256450(a(n)).

Original entry on oeis.org

0, 1, 4, 2, 6, 3, 12, 8, 16, 5, 15, 10, 18, 21, 22, 7, 20, 13, 24, 27, 28, 9, 26, 17, 48, 30, 52, 33, 34, 11, 60, 32, 64, 23, 56, 36, 66, 61, 70, 39, 40, 14, 73, 38, 78, 29, 67, 42, 72, 80, 76, 74, 85, 45, 84, 46, 88, 19, 89, 44, 90, 97, 94, 35, 81, 49, 87, 99, 93, 91, 105, 53, 96, 104, 100, 54, 109, 25, 108, 110, 112, 51, 111, 121, 114, 117, 118, 41

Offset: 0

Antti Karttunen, Aug 13 2016

Inverse: A275846.
Cf. A000142, A052849, A153880, A225901, A255411, A256450, A257680, A266193, A273663, A273670.
Similar permutations: A273667 (a more recursed variant), A275847, A275848.

a(0) = 0; for n >= 1: if A257680(A225901(n)) = 0 [when n is one of the terms of A153880] then a(n) = A255411(A266193(n)), otherwise [when n is one of the terms of A273670], a(n) = A256450(a(A273663(n))).
Other identities:
a(A000142(n)) = A052849(n) for all n >= 2.

A275846 Permutation of natural numbers: a(0) = 0, a(A255411(n)) = A153880(n), a(A256450(n)) = A273670(a(n)).

Original entry on oeis.org

0, 1, 3, 5, 2, 9, 4, 15, 7, 21, 11, 29, 6, 17, 41, 10, 8, 23, 12, 57, 16, 13, 14, 33, 18, 77, 22, 19, 20, 45, 25, 101, 31, 27, 28, 63, 35, 129, 43, 39, 40, 87, 47, 165, 59, 53, 55, 111, 24, 65, 213, 81, 26, 71, 75, 141, 34, 89, 269, 105, 30, 37, 95, 99, 32, 183, 36, 46, 113, 341, 38, 135, 48, 42, 51, 119, 50, 125, 44, 231, 49, 64, 143, 431, 54, 52

Offset: 0

Antti Karttunen, Aug 13 2016

Inverse: A275845.
Cf. A153880, A255411, A256450, A257680, A257684, A273662, A273670.
Similar permutations: A273668 (a more recursed variant), A275847, A275848.

a(0) = 0; for n >= 1: if A257680(n) = 0 [when n is one of the terms of A255411] then a(n) = A153880(A257684(n)), otherwise [when n is one of the terms of A256450], a(n) = A273670(a(A273662(n))).

A276082 a(0) = 0, a(2n) = A153880(a(n)), a(2n+1) = 1+A255411(a(n)).

Original entry on oeis.org

0, 1, 2, 5, 6, 13, 14, 23, 24, 49, 50, 77, 54, 85, 86, 119, 120, 241, 242, 365, 246, 373, 374, 503, 264, 409, 410, 557, 414, 565, 566, 719, 720, 1441, 1442, 2165, 1446, 2173, 2174, 2903, 1464, 2209, 2210, 2957, 2214, 2965, 2966, 3719, 1560, 2401, 2402, 3245, 2406, 3253, 3254, 4103, 2424, 3289, 3290, 4157, 3294, 4165, 4166, 5039, 5040, 10081

Offset: 0

Antti Karttunen, Aug 21 2016

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to factorial base representation

Cf. A153880, A255411

Cf. also A059590, A275959, A276091, A225901, A276083.

from sympy import factorial as f
def a007623(n, p=2): return n if n0 else '0' for i in x)[::-1]
    return 0 if n==0 else sum([int(y[i])*f(i + 1) for i in range(len(y))])
def a153880(n):
    x=(str(a007623(n)) + '0')[::-1]
    return 0 if n==0 else sum([int(x[i])*f(i + 1) for i in range(len(x))])
def a(n): return 0 if n==0 else a153880(a(n//2)) if n%2==0 else 1 + a255411(a((n - 1)//2))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 20 2017

a(0) = 0, a(2n) = A153880(a(n)), a(2n+1) = 1+A255411(a(n)).
Other identities. For all n >= 0:
a(n) = A225901(A276083(n)).

A276083 a(0) = 0, a(2n) = A255411(a(n)), a(2n+1) = 1+A153880(a(n)).

Original entry on oeis.org

0, 1, 4, 3, 18, 13, 16, 9, 96, 73, 76, 51, 90, 61, 64, 33, 600, 481, 484, 363, 498, 373, 376, 249, 576, 433, 436, 291, 450, 301, 304, 153, 4320, 3601, 3604, 2883, 3618, 2893, 2896, 2169, 3696, 2953, 2956, 2211, 2970, 2221, 2224, 1473, 4200, 3361, 3364, 2523, 3378, 2533, 2536, 1689, 3456, 2593, 2596, 1731, 2610, 1741, 1744, 873, 35280, 30241

Offset: 0

Antti Karttunen, Aug 21 2016

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to factorial base representation

Cf. A153880, A255411.

Cf. also A059590, A275959, A276091, A225901, A276082.

from sympy import factorial as f
def a007623(n, p=2): return n if n0 else '0' for i in x)[::-1]
    return 0 if n==0 else sum([int(y[i])*f(i + 1) for i in range(len(y))])
def a153880(n):
    x=(str(a007623(n)) + '0')[::-1]
    return 0 if n==0 else sum([int(x[i])*f(i + 1) for i in range(len(x))])
def a(n): return 0 if n==0 else a255411(a(n//2)) if n%2==0 else 1 + a153880(a((n - 1)//2))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 20 2017

a(0) = 0, a(2n) = A255411(a(n)), a(2n+1) = 1+A153880(a(n)).
Other identities. For all n >= 0:
a(n) = A225901(A276082(n)).

A265908 a(n) = A255411(A265907(n)); also the first differences of A265907.

Original entry on oeis.org

4, 22, 256, 2500, 24598, 262192, 3005356, 36562174, 478487968, 12927533332, 280630789030, 5778343352464, 118583043104764, 2476606038823342, 53484469903211776, 1188931280602126420, 27430026590262346558, 653821165282804596712, 16172901278558141600116, 413537682797697142621894, 10959122779052635897843288, 300659352550430117464479652

Offset: 1

Antti Karttunen, Dec 20 2015

In factorial base (A007623) these numbers are almost just like those of A265906, but shifted once left, with an extra zero appended, and then each nonzero digit incremented by one:

20, 320, 20220, 324020, 4604320, 64004220, 824203020, , ...

Antti Karttunen, Table of n, a(n) for n = 1..120
Index entries for sequences related to factorial base representation

Row 2 of A275960.
Cf. A007623, A255411.
First differences of A265907.

a(n) = A255411(A265907(n))

a(n) = A265907(n+1) - A265907(n).

A377022 Numbers whose prime factorization has exponents that have no digit 1 in their factorial-base representation (A255411).

Original entry on oeis.org

1, 16, 81, 625, 1296, 2401, 4096, 10000, 14641, 28561, 38416, 50625, 65536, 83521, 130321, 194481, 234256, 262144, 279841, 331776, 456976, 531441, 707281, 810000, 923521, 1185921, 1336336, 1500625, 1874161, 2085136, 2313441, 2560000, 2825761, 3111696, 3418801

Offset: 1

Amiram Eldar, Oct 13 2024

Numbers that are "powerful" when they are factorized into factors of the form p^(k!), where p is a prime and k >= 1, a factorization that is done using the factorial-base representation of the exponents in the prime factorization (see A376885 for more details). Each factor p^(k!) has a multiplicity that is larger than 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A255411, A376885, A377019, A377020, A377021.
Analogous to A001694.
Subsequence of A036967.

expQ[n_] := expQ[n] = Module[{k = n, m = 2, r, s = 1}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, If[r == 1, s = 0; Break[]]; m++]; s == 1]; seq[lim_] := Module[{p = 2, s = {1}, emax, es}, While[(emax = Floor[Log[p, lim]]) > 3, es = Select[Range[0, emax], expQ]; s = Union[s, Select[Union[Flatten[Outer[Times, s, p^es]]], # <= lim &]]; p = NextPrime[p]]; s]; seq[4*10^6]

isexp(n) = {my(k = n, m = 2, r); while([k, r] = divrem(k, m); k != 0 || r != 0, if(r == 1, return(0)); m++); 1;}
is(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!isexp(e[i]), return(0))); 1;}

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=1} 1/p^A255411(k)) = 1.07819745085315583226... .

A276339 a(n) = A255411(n) - A276340(n).

Original entry on oeis.org

0, 0, 4, 4, 0, 0, 22, 22, 24, 24, 22, 22, 18, 18, 22, 22, 18, 18, 0, 0, 4, 4, 0, 0, 118, 118, 120, 120, 118, 118, 138, 138, 142, 142, 138, 138, 120, 120, 124, 124, 120, 120, 118, 118, 120, 120, 118, 118, 114, 114, 118, 118, 114, 114, 120, 120, 124, 124, 120, 120, 118, 118, 120, 120, 118, 118, 114, 114, 118, 118, 114, 114, 96

Offset: 0

Antti Karttunen, Sep 01 2016

Antti Karttunen, Table of n, a(n) for n = 0..5039
Index entries for sequences related to factorial base representation

Cf. A001563, A255411, A276340.

Cf. A276091 (seems to give the indices of zeros, checked empirically for the first 512 zeros).

(define (A276339 n) (- (A255411 n) (A276340 n)))

a(n) = A255411(n) - A276340(n).

A255567 a(1) = 1, a(2) = 2, after which, a(2n+1) = 1 + a(2n), a(2n) = A255411(a(n)).

Original entry on oeis.org

1, 2, 3, 12, 13, 16, 17, 72, 73, 76, 77, 90, 91, 94, 95, 480, 481, 484, 485, 498, 499, 502, 503, 576, 577, 580, 581, 594, 595, 598, 599, 3600, 3601, 3604, 3605, 3618, 3619, 3622, 3623, 3696, 3697, 3700, 3701, 3714, 3715, 3718, 3719, 4200, 4201, 4204, 4205, 4218, 4219, 4222, 4223, 4296, 4297, 4300, 4301, 4314, 4315, 4318, 4319

Offset: 1

Antti Karttunen, May 05 2015

nonn

From 2 onward, the sequence seems to give those n for which A256450(A255411(n))+1 = A255411(A256450(n)), i.e., grandparents for those cousins in tree A255566 where the cousin at the right side is one more than the cousin at the left side.

			This sequence can be represented as a binary tree. Apart from the 1 at root, and its children 2 and 3, from then on each left hand child is produced as A255411(n), and each right hand child as 1 + A255411(n) when parent contains n >= 2:
                   ..................1..................
                  2                                     3
        12......./ \.......13                 16......./ \.......17
       / \                 / \               / \                 / \
      /   \               /   \             /   \               /   \
     /     \             /     \           /     \             /     \
   72       73         76       77       90       91         94       95
480  481 484  485   498  499 502  503 576  577 580  581   594  595 598  599
etc.

Antti Karttunen, Table of n, a(n) for n = 1..1023

Cf. A256450, A255411. See also the tree illustration in A255566.

a(1) = 1, a(2) = 2, after which, a(2n+1) = 1 + a(2n), a(2n) = A255411(a(n)).

Edited because of the changed starting offset of A256450 - Antti Karttunen, May 30 2016

A276957 Permutation of natural numbers: a(A153880(n)) = A255411(n), a(A273670(n)) = A256450(n).

Original entry on oeis.org

1, 4, 2, 3, 5, 12, 6, 16, 7, 8, 9, 18, 10, 22, 11, 13, 14, 15, 17, 19, 20, 21, 23, 48, 24, 52, 25, 26, 27, 60, 28, 64, 29, 30, 31, 66, 32, 70, 33, 34, 35, 36, 37, 38, 39, 40, 41, 72, 42, 76, 43, 44, 45, 84, 46, 88, 47, 49, 50, 90, 51, 94, 53, 54, 55, 56, 57, 58, 59, 61, 62, 96, 63, 100, 65, 67, 68, 108, 69, 112, 71, 73

Offset: 1

Antti Karttunen, Sep 22 2016

Inverse: A276958.
Cf. A276950, A255411, A256450, A266193, A273663.
For more recursed variants see: A275845, A275847 & A273667.

(define (A276957 n) (cond ((zero? n) n) ((zero? (A276950 n)) (A255411 (A266193 n))) (else (A256450 (A273663 n)))))

If A276950(n) = 0, then a(n) = A255411(A266193(n)), otherwise a(n) = A256450(A273663(n)).

cp's OEIS Frontend

A257685 Left inverse for injection A255411: a(0) = 0, after which, if n = A255411(k) for some k, then a(n) = k, otherwise a(n) = 0.

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A275845 Permutation of natural numbers: a(0) = 0, a(A153880(n)) = A255411(n), a(A273670(n)) = A256450(a(n)).

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A275846 Permutation of natural numbers: a(0) = 0, a(A255411(n)) = A153880(n), a(A256450(n)) = A273670(a(n)).

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A276082 a(0) = 0, a(2n) = A153880(a(n)), a(2n+1) = 1+A255411(a(n)).

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A276083 a(0) = 0, a(2n) = A255411(a(n)), a(2n+1) = 1+A153880(a(n)).

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A265908 a(n) = A255411(A265907(n)); also the first differences of A265907.

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A377022 Numbers whose prime factorization has exponents that have no digit 1 in their factorial-base representation (A255411).

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A276339 a(n) = A255411(n) - A276340(n).

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A255567 a(1) = 1, a(2) = 2, after which, a(2n+1) = 1 + a(2n), a(2n) = A255411(a(n)).

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