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.

Showing 1-6 of 6 results.

A275734 Prime-factorization representations of "factorial base slope polynomials": a(0) = 1; for n >= 1, a(n) = A275732(n) * a(A257684(n)).

Original entry on oeis.org

1, 2, 3, 6, 2, 4, 5, 10, 15, 30, 10, 20, 3, 6, 9, 18, 6, 12, 2, 4, 6, 12, 4, 8, 7, 14, 21, 42, 14, 28, 35, 70, 105, 210, 70, 140, 21, 42, 63, 126, 42, 84, 14, 28, 42, 84, 28, 56, 5, 10, 15, 30, 10, 20, 25, 50, 75, 150, 50, 100, 15, 30, 45, 90, 30, 60, 10, 20, 30, 60, 20, 40, 3, 6, 9, 18, 6, 12, 15, 30, 45, 90, 30, 60, 9, 18, 27
Offset: 0

Views

Author

Antti Karttunen, Aug 08 2016

Keywords

Comments

These are prime-factorization representations of single-variable polynomials where the coefficient of term x^(k-1) (encoded as the exponent of prime(k) in the factorization of n) is equal to the number of nonzero digits that occur on the slope (k-1) levels below the "maximal slope" in the factorial base representation of n. See A275811 for the definition of the "digit slopes" in this context.

Examples

			For n=23 ("321" in factorial base representation, A007623), all three nonzero digits are maximal for their positions (they all occur on "maximal slope"), thus a(23) = prime(1)^3 = 2^3 = 8.
For n=29 ("1021"), there are three nonzero digits, where both 2 and the rightmost 1 are on the "maximal slope", while the most significant 1 is on the "sub-sub-sub-maximal", thus a(29) = prime(1)^2 * prime(4)^1 = 2*7 = 28.
For n=37 ("1201"), there are three nonzero digits, where the rightmost 1 is on the maximal slope, 2 is on the sub-maximal, and the most significant 1 is on the "sub-sub-sub-maximal", thus a(37) = prime(1) * prime(2) * prime(4) = 2*3*7 = 42.
For n=55 ("2101"), the least significant 1 is on the maximal slope, and the digits "21" at the beginning are together on the sub-sub-maximal slope (as they are both two less than the maximal digit values 4 and 3 allowed in those positions), thus a(55) = prime(1)^1 * prime(3)^2 = 2*25 = 50.

Links

Crossrefs

Cf. A001221, A001222, A002110, A007489, A007814, A048675, A051903, A056169, A056170, A060130, A060502, A225901.
Cf. A257684, A275732.
Cf. A275811.
Cf. A260736, A275728, A275808, A275812, A275946, A275947, A275962.
Cf. A275804 (indices of squarefree terms), A275805 (of terms not squarefree).
Cf. also A275725, A275733, A275735, A276076 for other such prime factorization encodings of A060117/A060118-related polynomials.

Programs

  • Python
    from operator import mul
from sympy import prime, factorial as f
def a007623(n, p=2): return n if n
    0 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 a(n): return 1 if n==0 else a275732(n)*a(a257684(n))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 19 2017

Formula

a(0) = 1; for n >= 1, a(n) = A275732(n) * a(A257684(n)).
Other identities and observations. For all n >= 0:
a(n) = A275735(A225901(n)).
a(A007489(n)) = A002110(n).
A001221(a(n)) = A060502(n).
A001222(a(n)) = A060130(n).
A007814(a(n)) = A260736(n).
A051903(a(n)) = A275811(n).
A048675(a(n)) = A275728(n).
A248663(a(n)) = A275808(n).
A056169(a(n)) = A275946(n).
A056170(a(n)) = A275947(n).
A275812(a(n)) = A275962(n).

A275804 Numbers with at most one nonzero digit on each digit slope of the factorial base representation of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 13, 16, 18, 20, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 36, 37, 40, 42, 44, 48, 49, 50, 51, 52, 60, 61, 64, 66, 68, 72, 73, 76, 78, 79, 82, 90, 96, 98, 102, 104, 108, 120, 121, 122, 123, 124, 126, 127, 128, 129, 130, 132, 133, 136, 138, 140, 144, 145, 146, 147, 148, 150, 151, 152, 153, 154, 156, 157, 160
Offset: 0

Views

Author

Antti Karttunen, Aug 10 2016

Keywords

Comments

Indexing starts from zero, because a(0) = 0 is a special case in this sequence.
Numbers n for which A275947(n) = 0 or equally, for which A275811(n) <= 1.
Numbers n for which A008683(A275734(n)) <> 0, that is, indices of squarefree terms in A275734.
Numbers n for which A060130(n) = A060502(n).
Numbers with at most one nonzero digit on each digit slope of the factorial base representation of n (see A275811 and A060502 for the definition of slopes in this context). More exactly: numbers n in whose factorial base representation (A007623) there does not exist a pair of digit positions i_1 and i_2 with nonzero digits d_1 and d_2, such that (i_1 - d_1) = (i_2 - d_2).

Links

Crossrefs

Complement: A275805.
Indices of zeros in A275947 and A275962.
Intersection with A276005 gives A261220.
Cf. A059590 (a subsequence).
Cf. A007623, A008683, A060130, A060502, A275734, A275811.

Programs

  • Python
    from operator import mul
from sympy import prime, factorial as f
from sympy.ntheory.factor_ import core
def a007623(n, p=2): return n if n
    0 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 a(n): return 1 if n==0 else a275732(n)*a(a257684(n))
def ok(n): return 1 if n==0 else core(a(n))==a(n)
print([n for n in range(201) if ok(n)]) # Indranil Ghosh, Jun 19 2017

A275947 Number of distinct slopes with multiple nonzero digits in factorial base representation of n: a(n) = A056170(A275734(n)). (See comments for more exact definition).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 2, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 0
Offset: 0

Views

Author

Antti Karttunen, Aug 15 2016

Keywords

Comments

a(n) gives the number of distinct elements that have multiplicity > 1 in a multiset [(i_x - d_x) | where d_x ranges over each nonzero digit present and i_x is its position from the right].

Examples

			For n=525, in factorial base "41311", there are three occupied slopes. The maximal slope contains the nonzero digits "3.1", the sub-maximal digits "4..1.", and the sub-sub-sub-maximal just "1..." (the 1 in the position 4 from right is the sole occupier of its own slope). Thus there are two slopes with more than one nonzero digit, and a(525) = 2.
Equally, when we form a multiset of (digit-position - digit-value) differences for all nonzero digits present in "41311", we obtain a multiset [0, 0, 1, 1, 3], in which the distinct elements that occur multiple times are 0 and 1, thus a(525) = 2.

Links

Crossrefs

Cf. A056170, A275734.
Cf. A275804 (indices of zeros), A275805 (of nonzeros).
Cf. also A060502, A225901, A275946, A275949, A275962.

Programs

Formula

a(n) = A056170(A275734(n)).
Other identities and observations. For all n >= 0.
a(n) = A275949(A225901(n)).
A060502(n) = A275946(n) + a(n).
a(n) <= A275962(n).

A275962 Total number of nonzero digits that occur on the multiply occupied slopes of the factorial base representation of n: a(n) = A275812(A275734(n)). (See comments for more exact definition).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 2, 2, 0, 2, 0, 2, 0, 2, 2, 3, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 2, 2, 0, 2, 0, 2, 0, 2, 2, 3, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 4, 0, 0, 2, 2, 0, 2, 0, 2, 0, 2, 2, 3, 0, 0, 2, 2, 0, 2, 0, 0, 2, 2, 0, 2, 2, 2, 3, 3, 2, 4, 0, 2, 2, 4, 2, 3, 0, 2, 0, 2, 2, 3, 0, 2, 0, 2, 2, 3, 0, 2, 2, 4, 2, 3, 2, 3, 2, 3, 3, 4, 0
Offset: 0

Views

Author

Antti Karttunen, Aug 15 2016

Keywords

Comments

a(n) gives the total number of elements (counted with multiplicity) that have multiplicity > 1 in a multiset [(i_x - d_x) | where d_x ranges over each nonzero digit present and i_x is its position from the right].

Examples

			For n=525, in factorial base "41311", there are three occupied slopes. The maximal slope contains the nonzero digits "3.1", the sub-maximal the digits "4..1.", and the sub-sub-sub-maximal just "1..." (the 1 in the position 4 from right is the sole occupier of its own slope). There are two slopes with more than one nonzero digit, each having two such digits, and thus a(525) = 2+2 = 4.
Equally, when we form a multiset of (digit-position - digit-value) differences for all nonzero digits present in "41311", we obtain a multiset [0, 0, 1, 1, 3], in which the elements that occur multiple times are [0, 0, 1, 1], thus a(525) = 4.

Links

Crossrefs

Cf. A275734, A275812.
Cf. A275804 (indices of zeros), A275805 (of nonzeros).
Cf. also A060130, A225901, A275946, A275947, A275964.

Programs

Formula

a(n) = A275812(A275734(n)).
Other identities and observations. For all n >= 0.
a(n) = A275964(A225901(n)).
a(n) = A060130(n) - A275946(n).
a(n) >= A275947(n).

A275811 Number of nonzero digits on a maximally occupied slope of factorial base representation of n: a(n) = A051903(A275734(n)). See comments for the definition.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 2, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 2, 3, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 2, 3, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 3, 3, 2, 2, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 2, 3, 1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 4, 1
Offset: 0

Views

Author

Antti Karttunen, Aug 10 2016

Keywords

Comments

Digit slopes are called "maximal", "sub-maximal", "sub-sub-maximal", etc. For digit-positions we employ one-based indexing, thus we say that the least significant digit of factorial base expansion of n is in position 1, etc. The maximal digit slope is occupied when there is at least one digit-position k that contains digit k (maximal digit allowed in that position), so that A260736(n) > 0, and n is thus a term of A273670. The sub-maximal digit slope is occupied when there is at least one nonzero digit k in a digit-position k+1. The sub-sub-maximal slope is occupied when there is at least one nonzero digit k in a digit-position k+2, etc. This sequence gives the number of nonzero digits on a slope (of possibly several) for which there exists no other slopes with more nonzero digits. See the examples.
In other words: a(n) gives the number of occurrences of a most common element in the multiset [(i_x - d_x) | where d_x ranges over each nonzero digit present in factorial base representation of n and i_x is that digit's position from the right].
Involution A225901 maps this metric to another metric A264990 which gives the maximal number of equal nonzero digits occurring in factorial base representation (A007623) of n. See also A060502.

Examples

			For n=23 ("321" in factorial base representation, A007623), all three nonzero digits are maximal for their positions (they all occur on "maximal slope"), thus the "maximal slope" is also the "maximally occupied slope" (as there are no other occupied slopes present), and a(23) = 3.
For n=29 ("1021"), there are three nonzero digits, where both 2 and the rightmost 1 are on the "maximal slope", while the most significant 1 is on the "sub-sub-sub-maximal", thus here the "maximal slope" is also the "maximally occupied slope" (with 2 nonzero digits present), and a(29) = 2.
For n=37 ("1201"), there are three nonzero digits, where the rightmost 1 is on the maximal slope, 2 is on the sub-maximal, and the most significant 1 is on the "sub-sub-sub-maximal", thus there are three occupied slopes in total, all with just one nonzero digit present, and a(37) = 1.
For n=55 ("2101"), the least significant 1 is on the maximal slope, and the digits "21" at the beginning are together on the sub-sub-maximal slope (as they are both two less than the maximal digit values 4 and 3 allowed in those positions), thus here the sub-sub-maximal slope is the "maximally occupied slope" with its two occupiers, and a(55) = 2.

Links

Crossrefs

Cf. A051903, A060502, A225901, A264990, A275734.
Cf. A275804 (gives the indices of 0 and 1's), A275805 (gives the indices of terms > 1).

Programs

  • Python
    from sympy import prime, factorint
from operator import mul
from functools import reduce
from sympy import factorial as f
def a051903(n): return 0 if n==1 else max(factorint(n).values())
def a007623(n, p=2): return n if n
    0 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 a275734(n): return 1 if n==0 else a275732(n)*a275734(a257684(n))
def a(n): return 0 if n==0 else a051903(a275734(n))
print([a(n) for n in range(201)]) # Indranil Ghosh, Jun 20 2017

Formula

a(n) = A051903(A275734(n)).
a(n) = A264990(A225901(n)).

Extensions

Signs in comment corrected and clarification added by Antti Karttunen, Aug 16 2016

A275809 Indices of zeros in A275808.

Original entry on oeis.org

0, 5, 14, 19, 22, 54, 59, 74, 84, 89, 93, 97, 100, 111, 114, 119, 264, 269, 278, 283, 286, 366, 371, 408, 413, 422, 427, 430, 440, 463, 466, 482, 492, 497, 501, 536, 552, 557, 566, 571, 574, 579, 589, 592, 596, 601, 604, 615, 618, 623, 655, 658, 675, 685, 688, 692, 696, 701, 710, 715, 718, 1560, 1565, 1574, 1579, 1582, 1614, 1619, 1634, 1644
Offset: 0

Views

Author

Antti Karttunen, Aug 09 2016

Keywords

Comments

Indexing begins from zero, because a(0) = 0 is a special case in this sequence.
Terms A009445(n)-1, [0, 5, 119, 5039, 362879, 39916799, ...] form a subsequence, and also the terms of A010050(n)-2, [0, 22, 718, 40318, 3628798, ...] form a subsequence.

Links

Crossrefs

Cf. A275736, A275808.
Cf. A275810 (first differences).
Subsequence of A275813 and of A275805 (after the initial 0).
Cf. also A009445, A010050.
Showing 1-6 of 6 results.

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