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Search: a161904 -id:a161904
Displaying 1-5 of 5 results found. page 1
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A207375 Irregular array read by rows in which row n lists the (one or two) central divisors of n in increasing order. +10
43
1, 1, 2, 1, 3, 2, 1, 5, 2, 3, 1, 7, 2, 4, 3, 2, 5, 1, 11, 3, 4, 1, 13, 2, 7, 3, 5, 4, 1, 17, 3, 6, 1, 19, 4, 5, 3, 7, 2, 11, 1, 23, 4, 6, 5, 2, 13, 3, 9, 4, 7, 1, 29, 5, 6, 1, 31, 4, 8, 3, 11, 2, 17, 5, 7, 6, 1, 37, 2, 19, 3, 13, 5, 8, 1, 41, 6, 7, 1, 43 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
If n is a square then row n lists only the square root of n because the squares (A000290) have only one central divisor.
If n is not a square then row n lists the pair (j, k) of divisors of n, nearest to the square root of n, such that j*k = n.
Conjecture 1: It appears that the n-th record in this sequence is the last member of row A008578(n).
Column 1 gives A033676. Right border gives A033677. - Omar E. Pol, Feb 26 2019
The conjecture 1 follows from Bertrand's Postulate. - Charles R Greathouse IV, Feb 11 2022
Row products give A097448. - Omar E. Pol, Feb 17 2022
LINKS
Alois P. Heinz, Rows n = 1..5000
Omar E. Pol, Illustration of the divisors of the first 12 positive integers
EXAMPLE
Array begins:
1;
1, 2;
1, 3;
2;
1, 5;
2, 3;
1, 7;
2, 4;
3;
2, 5;
1, 11;
3, 4;
1, 13;
...
CROSSREFS
Row n has length A169695(n).
Row sums give A207376.
Cf. A000005, A000290, A008578, A027750, A033676, A033677, A097448, A161901, A161904.
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Feb 23 2012
STATUS
approved
A228813 Triangle read by rows T(n,k) in which column k lists 1's interleaved with A004526(k-1) zeros starting from the row A002620(k+1), with n>=1, k>=1. +10
13
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
1
COMMENTS
The sum of row n equals the number of divisors of n.
The number of zeros in row n equals A078152(n).
It appears that there are only eight rows that do not contain zeros. The indices of these rows are 1, 2, 3, 4, 6, 8, 12, 24, the divisors of 24, see A018253.
It appears that A066522 gives the indices of the rows in which the elements are in nonincreasing order.
LINKS
Table of n, a(n) for n=1..90.
EXAMPLE
For n = 10, row 10 is [1, 1, 1, 1, 0], and the sum of row 10 is 1+1+1+1+0 = 4. On the other hand, 10 has four divisors: 1, 2, 5, and 10. Note that the sum of row 10 is also A000005(10) = 4, the number of divisors of 10.
Triangle begins:
1;
1, 1;
1, 1;
1, 1, 1;
1, 1, 0;
1, 1, 1, 1;
1, 1, 0, 0;
1, 1, 1, 1;
1, 1, 0, 0, 1;
1, 1, 1, 1, 0;
1, 1, 0, 0, 0;
1, 1, 1, 1, 1, 1;
1, 1, 0, 0, 0, 0;
1, 1, 1, 1, 0, 0;
1, 1, 0, 0, 1, 1;
1, 1, 1, 1, 0, 0, 1;
1, 1, 0, 0, 0, 0, 0;
1, 1, 1, 1, 1, 1, 0;
1, 1, 0, 0, 0, 0, 0;
1, 1, 1, 1, 0, 0, 1, 1;
1, 1, 0, 0, 1, 1, 0, 0;
1, 1, 1, 1, 0, 0, 0, 0;
1, 1, 0, 0, 0, 0, 0, 0;
1, 1, 1, 1, 1, 1, 1, 1;
...
CROSSREFS
Row sums give A000005.
Row n has length A055086(n).
Columns 1 and 2: A000012. Columns 3 and 4: A059841.
Columns 5 and 6: A079978. Columns 7 and 8: A121262.
Columns 9 and 10: A079998. Columns 11 and 12: A079979.
Columns 13 and 14: A082784.
Cf. A002620, A004526, A018253, A027750, A066522, A078152, A147861, A161904, A196020, A210959, A212119, A212120, A221645, A228812, A228814.
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Sep 29 2013
STATUS
approved
A228814 Triangle read by rows T(n,k), n>=1, k>=1, in which column k starts in row A002620(k+1). If k is odd the column k lists j's interleaved with (k-1)/2 zeros, where j = (k+1)/2. Otherwise, if k is even the column k lists the positive integers but starting from k/2+1, interleaved with (k-2)/2 zeros. +10
6
1, 1, 2, 1, 3, 1, 4, 2, 1, 5, 0, 1, 6, 2, 3, 1, 7, 0, 0, 1, 8, 2, 4, 1, 9, 0, 0, 3, 1, 10, 2, 5, 0, 1, 11, 0, 0, 0, 1, 12, 2, 6, 3, 4, 1, 13, 0, 0, 0, 0, 1, 14, 2, 7, 0, 0, 1, 15, 0, 0, 3, 5, 1, 16, 2, 8, 0, 0, 4, 1, 17, 0, 0, 0, 0, 0, 1, 18, 2, 9, 3, 6, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
The number of positive terms of row n is A000005(n).
The positive terms of row n are the divisors of n.
The number of zeros in row n equals A078152(n).
Row n has length A055086(n).
The sum of row n is A000203(n).
Positive terms give A210959.
It appears that there are only eight rows that do not contain zeros. The indices of these rows are 1, 2, 3, 4, 6, 8, 12, 24, the divisors of 24, see A018253.
For another version see A228812.
LINKS
Table of n, a(n) for n=1..83.
EXAMPLE
For n = 60 the 60th row of triangle is [1, 60, 2, 30, 3, 20, 4, 15, 5, 12, 6, 10, 0, 0]. The row length is A055086(60) = 14. The number of zeros is A078152(60) = 2. The number of positive terms is A000005(60) = 12. The row sum is A000203(60) = 168.
Triangle begins:
1;
1, 2;
1, 3;
1, 4, 2;
1, 5, 0;
1, 6, 2, 3;
1, 7, 0, 0;
1, 8, 2, 4;
1, 9, 0, 0, 3;
1, 10, 2, 5, 0;
1, 11, 0, 0, 0;
1, 12, 2, 6, 3, 4;
1, 13, 0, 0, 0, 0;
1, 14, 2, 7, 0, 0;
1, 15, 0, 0, 3, 5;
1, 16, 2, 8, 0, 0, 4;
1, 17, 0, 0, 0, 0, 0;
1, 18, 2, 9, 3, 6, 0;
1, 19, 0, 0, 0, 0, 0;
1, 20, 2, 10, 0, 0, 4, 5;
1, 21, 0, 0, 3, 7, 0, 0;
1, 22, 2, 11, 0, 0, 0, 0;
1, 23, 0, 0, 0, 0, 0, 0;
1, 24, 2, 12, 3, 8, 4, 6;
...
CROSSREFS
Column 1 is A000012.
Cf. A000005, A000203, A002620, A004526, A018253, A027750, A055086, A078152, A147861, A161904, A196020, A210959, A212119, A212120, A221645, A228812, A228813, A229940, A229942, A228944, A229950, A228951.
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Oct 03 2013
STATUS
approved
A228812 Triangle read by rows: T(n,k), n>=1, k>=1, in which row n lists m terms, where m = A055086(n). If k divides n and k < n^(1/2) then T(n,k) = k and T(n,m-k+1) = n/T(n,k). Also, if k^2 = n then T(n,k) = k. Other terms are zeros. +10
4
1, 1, 2, 1, 3, 1, 2, 4, 1, 0, 5, 1, 2, 3, 6, 1, 0, 0, 7, 1, 2, 4, 8, 1, 0, 3, 0, 9, 1, 2, 0, 5, 10, 1, 0, 0, 0, 11, 1, 2, 3, 4, 6, 12, 1, 0, 0, 0, 0, 13, 1, 2, 0, 0, 7, 14, 1, 0, 3, 5, 0, 15, 1, 2, 0, 4, 0, 8, 16, 1, 0, 0, 0, 0, 0, 17, 1, 2, 3, 0, 6, 9, 18 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
The number of positive terms of row n is A000005(n).
The positive terms of row n are the divisors of n in increasing order.
Row n has length A055086(n).
Column k starts in row A002620(k+1).
The number of zeros in row n equals A078152(n).
The sum of row n is A000203(n).
Positive terms give A027750.
It appears that there are only eight rows that do not contain zeros. The indices of these rows are 1, 2, 3, 4, 6, 8, 12, 24, the divisors of 24, see A018253.
For another version see A228814.
LINKS
Table of n, a(n) for n=1..83.
EXAMPLE
For n = 60 the 60th row of triangle is [1, 2, 3, 4, 5, 6, 0, 0, 10, 12, 15, 20, 30, 60]. The row length is A055086(60) = 14. The number of zeros is A078152(60) = 2. The number of positive terms is A000005(60) = 12. The positive terms are the divisors of 60. The row sum is A000203(60) = 168.
Triangle begins:
1;
1, 2;
1, 3;
1, 2, 4;
1, 0, 5;
1, 2, 3, 6;
1, 0, 0, 7;
1, 2, 4, 8;
1, 0, 3, 0, 9;
1, 2, 0, 5, 10;
1, 0, 0, 0, 11;
1, 2, 3, 4, 6, 12;
1, 0, 0, 0, 0, 13;
1, 2, 0, 0, 7, 14;
1, 0, 3, 5, 0, 15;
1, 2, 0, 4, 0, 8, 16;
1, 0, 0, 0, 0, 0, 17;
1, 2, 3, 0, 6, 9, 18;
1, 0, 0, 0, 0, 0, 19;
1, 2, 0, 4, 5, 0, 10, 20;
1, 0, 3, 0, 0, 7, 0, 21;
1, 2, 0, 0, 0, 0, 11, 22;
1, 0, 0, 0, 0, 0, 0, 23;
1, 2, 3, 4, 6, 8, 12, 24;
...
CROSSREFS
Column 1 is A000012.
Right border gives A000027.
Cf. A000005, A000203, A002620, A004526, A018253, A027750, A055086, A078152, A127093, A147861, A161904, A196020, A210959, A212119, A212120, A221645, A228813, A228814.
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Oct 03 2013
STATUS
approved
A328948 Number of primes that are a concatenation of two positive integers whose product is n. +10
1
1, 0, 2, 1, 0, 2, 2, 0, 1, 1, 0, 1, 2, 0, 2, 0, 0, 2, 1, 0, 3, 1, 0, 2, 1, 0, 2, 2, 0, 1, 2, 0, 3, 0, 0, 0, 1, 0, 2, 1, 0, 2, 1, 0, 1, 2, 0, 1, 2, 0, 3, 1, 0, 2, 0, 0, 3, 1, 0, 1, 0, 0, 4, 1, 0, 3, 1, 0, 2, 2, 0, 1, 1, 0, 1, 2, 0, 3, 1, 0, 2, 2, 0, 3, 0, 0, 1, 2, 0, 1, 3, 0, 3, 1, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Records: 1, 3, 21, 63, 231, 924, 4389, 5187, 51051, 69069, 127281, 245973, 302841, 969969, 1312311, 1716099. - Corrected by Robert Israel, Dec 14 2023
This is not always the same as the number of divisors d of n such that the concatenation of d and n/d is prime, because the same prime could occur for more than one divisor. For example, 1140678 = 14*81477 = 14814*77 with 1481477 prime, and this prime is counted only once in a(1140678) = 7. - Robert Israel, Dec 14 2023
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
FORMULA
a(3n + 2) = 0.
EXAMPLE
1(11), 2(-), 3(13, 31), 4(41), 5(-), 6(23, 61), 7(17, 71), 8(-), 9(19), 10(101), 11(-), 12(43), 13(113, 131), 14(-), 15(53, 151), 16(-).
MAPLE
f:= proc(n)
if n mod 3 = 2 then return 0 fi;
nops(select(isprime, {seq(dcat(t, n/t), t = numtheory:-divisors(n))})
end proc:
map(f, [$1..200]); # Robert Israel, Dec 14 2023
PROG
(PARI) a(n) = sumdiv(n, d, isprime(eval(concat(Str(d), Str(n/d))))); \\ Michel Marcus, Nov 05 2019
(Magma) [#[a: d in Divisors(n)| IsPrime(a) where a is Seqint(Intseq(d) cat Intseq(n div d))]:n in [1..100]]; // Marius A. Burtea, Nov 05 2019
CROSSREFS
Cf. A000040, A016789, A161904, A328903.
KEYWORD
nonn,base,easy
AUTHOR
Juri-Stepan Gerasimov, Nov 01 2019
STATUS
approved
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