Displaying 1-10 of 13 results found.
Sum of the odd-indexed parts (odd bisection) of the multiset of prime indices of n.
+10
23
0, 1, 2, 1, 3, 1, 4, 2, 2, 1, 5, 3, 6, 1, 2, 2, 7, 3, 8, 4, 2, 1, 9, 2, 3, 1, 4, 5, 10, 4, 11, 3, 2, 1, 3, 3, 12, 1, 2, 2, 13, 5, 14, 6, 5, 1, 15, 4, 4, 4, 2, 7, 16, 3, 3, 2, 2, 1, 17, 3, 18, 1, 6, 3, 3, 6, 19, 8, 2, 5, 20, 4, 21, 1, 5, 9, 4, 7, 22, 5, 4, 1
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
EXAMPLE
The prime indices of 1100 are {1,1,3,3,5}, so a(1100) = 1 + 3 + 5 = 9.
The prime indices of 2100 are {1,1,2,3,3,4}, so a(2100) = 1 + 2 + 3 = 6.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Total[First/@Partition[Append[primeMS[n], 0], 2]], {n, 100}]
CROSSREFS
The version for standard compositions is A209281(n+1) (even: A346633). The even reverse version is A346700. A000302 counts compositions with odd alternating sum, ranked by A053738.
A029837 adds up parts of standard compositions (alternating: A124754).
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A344606 counts alternating permutations of prime indices.
Cf. A000070, A025047, A120452, A341446, A344614, A344617, A344653, A344654, A345957, A345958, A345959.
Sum of the even-indexed parts (even bisection) of the multiset of prime indices of n.
+10
21
0, 0, 0, 1, 0, 2, 0, 1, 2, 3, 0, 1, 0, 4, 3, 2, 0, 2, 0, 1, 4, 5, 0, 3, 3, 6, 2, 1, 0, 2, 0, 2, 5, 7, 4, 3, 0, 8, 6, 4, 0, 2, 0, 1, 2, 9, 0, 2, 4, 3, 7, 1, 0, 4, 5, 5, 8, 10, 0, 4, 0, 11, 2, 3, 6, 2, 0, 1, 9, 3, 0, 3, 0, 12, 3, 1, 5, 2, 0, 2, 4, 13, 0, 5, 7, 14, 10, 6, 0, 5, 6, 1, 11, 15, 8, 4, 0, 4, 2, 4, 0, 2, 0, 7, 3
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
EXAMPLE
The prime indices of 1100 are {1,1,3,3,5}, so a(1100) = 1 + 3 = 4.
The prime indices of 2100 are {1,1,2,3,3,4}, so a(2100) = 1 + 3 + 4 = 8.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Total[Last/@Partition[Append[primeMS[n], 0], 2]], {n, 100}]
PROG
(PARI) A346698(n) = if(1==n, 0, my(f=factor(n), s=0, p=0); for(k=1, #f~, while(f[k, 2], s += (p%2)*primepi(f[k, 1]); f[k, 2]--; p++)); (s)); \\ Antti Karttunen, Nov 30 2021
CROSSREFS
The even reverse version is A346699. A029837 adds up parts of standard compositions (alternating: A124754).
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A344606 counts alternating permutations of prime indices.
Product of primes at even positions in the weakly increasing list (with multiplicity) of prime factors of n.
+10
17
1, 1, 1, 2, 1, 3, 1, 2, 3, 5, 1, 2, 1, 7, 5, 4, 1, 3, 1, 2, 7, 11, 1, 6, 5, 13, 3, 2, 1, 3, 1, 4, 11, 17, 7, 6, 1, 19, 13, 10, 1, 3, 1, 2, 3, 23, 1, 4, 7, 5, 17, 2, 1, 9, 11, 14, 19, 29, 1, 10, 1, 31, 3, 8, 13, 3, 1, 2, 23, 5, 1, 6, 1, 37, 5, 2, 11, 3, 1, 4, 9
EXAMPLE
The prime factors of 108 are (2,2,3,3,3), with even bisection (2,3), with product 6, so a(108) = 6.
The prime factors of 720 are (2,2,2,2,3,3,5), with even bisection (2,2,3), with product 12, so a(720) = 12.
MAPLE
f:= proc(n) local F, i;
F:= ifactors(n)[2];
F:= sort(map(t -> t[1]$t[2], F));
mul(F[i], i=2..nops(F), 2)
end proc:
MATHEMATICA
Table[Times@@Last/@Partition[Flatten[Apply[ConstantArray, FactorInteger[n], {1}]], 2], {n, 100}]
CROSSREFS
Positions of first appearances are A129597. The sum of prime indices of a(n) is A346698(n). A001221 counts distinct prime factors.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344606 counts alternating permutations of prime indices.
A344617 gives the sign of the alternating sum of prime indices.
A346633 adds up the even bisection of standard compositions.
Cf. A026424, A035363, A209281, A236913, A342768, A344653, A345957, A345958, A345960, A345961, A345962.
Product of primes at odd positions in the weakly increasing list (with multiplicity) of prime factors of n.
+10
15
1, 2, 3, 2, 5, 2, 7, 4, 3, 2, 11, 6, 13, 2, 3, 4, 17, 6, 19, 10, 3, 2, 23, 4, 5, 2, 9, 14, 29, 10, 31, 8, 3, 2, 5, 6, 37, 2, 3, 4, 41, 14, 43, 22, 15, 2, 47, 12, 7, 10, 3, 26, 53, 6, 5, 4, 3, 2, 59, 6, 61, 2, 21, 8, 5, 22, 67, 34, 3, 14, 71, 12, 73, 2, 15, 38
EXAMPLE
The prime factors of 108 are (2,2,3,3,3), with odd bisection (2,3,3), with product 18, so a(108) = 18.
The prime factors of 720 are (2,2,2,2,3,3,5), with odd bisection (2,2,3,5), with product 60, so a(720) = 60.
MATHEMATICA
Table[Times@@First/@Partition[Append[Flatten[Apply[ConstantArray, FactorInteger[n], {1}]], 0], 2], {n, 100}]
CROSSREFS
The even reverse version appears to be A329888. Positions of first appearances are A342768. A001221 counts distinct prime factors.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A209281 (shifted) adds up the odd bisection of standard compositions.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344606 counts alternating permutations of prime indices.
A344617 gives the sign of the alternating sum of prime indices.
A346633 adds up the even bisection of standard compositions.
A346698 gives the sum of the even bisection of prime indices.
A346700 gives the sum of the even bisection of reversed prime indices.
Cf. A025047, A027187, A027193, A053738, A097805, A106356, A341446, A344653, A345957, A345958, A345959.
Sum of even-indexed parts (even bisection) of the n-th composition in standard order.
+10
14
0, 0, 0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 3, 2, 1, 2, 0, 1, 2, 1, 3, 2, 1, 2, 4, 3, 2, 3, 1, 2, 3, 2, 0, 1, 2, 1, 3, 2, 1, 2, 4, 3, 2, 3, 1, 2, 3, 2, 5, 4, 3, 4, 2, 3, 4, 3, 1, 2, 3, 2, 4, 3, 2, 3, 0, 1, 2, 1, 3, 2, 1, 2, 4, 3, 2, 3, 1, 2, 3, 2, 5, 4, 3, 4, 2, 3, 4
COMMENTS
The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
EXAMPLE
Composition number 741 in standard order is (2,1,1,3,2,1), so a(741) = 1 + 3 + 1 = 5.
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[Total[Last/@Partition[Append[stc[n], 0], 2]], {n, 0, 100}]
CROSSREFS
Including odd-indexed parts gives A029837. Subtracting from the odd version gives A124754. The odd-indexed version is A209281(n+1). A097805 counts compositions by alternating (or reverse-alternating) sum.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A345197 counts compositions by sum, length, and alternating sum.
Sum of the odd bisection (odd-indexed parts) of the integer partition with Heinz number n.
+10
13
0, 1, 2, 1, 3, 2, 4, 2, 2, 3, 5, 3, 6, 4, 3, 2, 7, 3, 8, 4, 4, 5, 9, 3, 3, 6, 4, 5, 10, 4, 11, 3, 5, 7, 4, 3, 12, 8, 6, 4, 13, 5, 14, 6, 5, 9, 15, 4, 4, 4, 7, 7, 16, 4, 5, 5, 8, 10, 17, 4, 18, 11, 6, 3, 6, 6, 19, 8, 9, 5, 20, 4, 21, 12, 5, 9, 5, 7, 22, 5, 4
COMMENTS
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
EXAMPLE
The partition with Heinz number 1100 is (5,3,3,1,1), so a(1100) = 5 + 3 + 1 = 9.
The partition with Heinz number 2100 is (4,3,3,2,1,1), so a(2100) = 4 + 3 + 1 = 8.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Total[First/@Partition[Append[Reverse[primeMS[n]], 0], 2]], {n, 100}]
CROSSREFS
The version for standard compositions is A209281(n+1) (even: A346633). The non-reverse version (multisets instead of partitions) is A346697. The even non-reverse version is A346698. A103919 counts partitions by sum and alternating sum (reverse: A344612).
A344606 counts alternating permutations of prime indices.
Cf. A000070, A025047, A120452, A124754, A329888, A341446, A344617, A345957, A345958, A346701, A346703.
a(n) = A329900( A329602(n)); Heinz number of the even bisection (even-indexed parts) of the integer partition with Heinz number n.
+10
9
1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 4, 1, 3, 1, 2, 3, 2, 1, 4, 5, 2, 3, 2, 1, 3, 1, 4, 3, 2, 5, 6, 1, 2, 3, 4, 1, 3, 1, 2, 3, 2, 1, 4, 7, 5, 3, 2, 1, 6, 5, 4, 3, 2, 1, 6, 1, 2, 3, 8, 5, 3, 1, 2, 3, 5, 1, 6, 1, 2, 5, 2, 7, 3, 1, 4, 9, 2, 1, 6, 5, 2, 3, 4, 1, 6, 7, 2, 3, 2, 5, 8, 1, 7, 3, 10, 1, 3, 1, 4, 5
COMMENTS
Also the product of primes at even positions in the weakly decreasing list (with multiplicity) of prime factors of n. For example, the prime factors of 108 are (3,3,3,2,2), with even bisection (3,2), with product 6, so a(108) = 6.
Proof: A108951(n) gives a number with the same largest prime factor ( A006530) and its exponent ( A071178) as in n, and with each smaller prime p = 2, 3, 5, 7, ... < A006530(n) having as its exponent the partial sum of the exponents of all prime factors >= p present in n (with primes not present in n having the exponent 0). Then applying A000188 replaces each such "partial sum exponent" k with floor(k/2). Finally, A319626 replaces those halved exponents with their first differences (here the exponent of the largest prime present stays intact, because the next larger prime's exponent is 0 in n). It should be easy to see that if prime q is not present in n (i.e., does not divide it), then neither it is present in a(n). Moreover, if the partial sum exponent of q is odd and only one larger than the partial sum exponent of the next larger prime factor of n, then q will not be present in a(n), while in all other cases q is present in a(n). See also the last example. (End)
FORMULA
a(n^2) = n for all n >= 1.
EXAMPLE
The list of all numbers with image 12 and their corresponding prime factors begins:
144: (3,3,2,2,2,2)
216: (3,3,3,2,2,2)
240: (5,3,2,2,2,2)
288: (3,3,2,2,2,2,2)
336: (7,3,2,2,2,2)
360: (5,3,3,2,2,2)
(End)
The positions from the left are indexed as 1, 2, 3, ..., etc, so e.g., for 240 we pick the second, the fourth and the sixth prime factor, 3, 2 and 2, to obtain a(240) = 3*2*2 = 12. For 288, we similarly pick the second (3), the fourth (2) and the sixth (2) to obtain a(288) = 3*2*2 = 12. - Antti Karttunen, Oct 13 2021 Consider n = 11945934 = 2*3*3*3*7*11*13*13*17. Its primorial inflation is A108951(11945934) = 96478365991115908800000 = 2^9 * 3^8 * 5^5 * 7^5 * 11^4 * 13^3 * 17^1. Applying A000188 to this halves each exponent (floored down if the exponent is odd), leaving the factors 2^4 * 3^4 * 5^2 * 7^2 * 11^2 * 13^1 = 2497294800. Then applying A319626 to this number retains the largest prime factor (and its exponent), and subtracts from the exponent of each of the rest of primes the exponent of the next larger prime, so from 2^4 * 3^4 * 5^2 * 7^2 * 11^2 * 13^1 we get 2^(4-4) * 3^(4-2) * 5^(2-2) * 7^(2-2) * 11^(2-1) * 13^1 = 3^2 * 11^1 * 13^1 = 1287 = a(11945934), which is obtained also by selecting every second prime from the list [17, 13, 13, 11, 7, 3, 3, 3, 2] and taking their product. - Antti Karttunen, Oct 15 2021
MATHEMATICA
Table[Times@@Last/@Partition[Reverse[Flatten[Apply[ConstantArray, FactorInteger[n], {1}]]], 2], {n, 100}] (* Gus Wiseman, Oct 13 2021 *)
PROG
(PARI) A329888(n) = if(1==n, n, my(f=factor(n), m=1, p=0); forstep(k=#f~, 1, -1, while(f[k, 2], m *= f[k, 1]^(p%2); f[k, 2]--; p++)); (m)); \\ (After Wiseman's new interpretation) - Antti Karttunen, Sep 21 2021
CROSSREFS
The sum of prime indices of a(n) is A346700(n). The odd non-reverse version is A346703. The non-reverse version is A346704. A001221 counts distinct prime factors.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A346633 adds up the even bisection of standard compositions.
A346698 adds up the even bisection of prime indices.
Cf. A000097, A035363, A102750, A236913, A253553, A344606, A344617, A344653, A345957, A345958, A345959.
Heinz number of the odd bisection (odd-indexed parts) of the integer partition with Heinz number n.
+10
9
1, 2, 3, 2, 5, 3, 7, 4, 3, 5, 11, 6, 13, 7, 5, 4, 17, 6, 19, 10, 7, 11, 23, 6, 5, 13, 9, 14, 29, 10, 31, 8, 11, 17, 7, 6, 37, 19, 13, 10, 41, 14, 43, 22, 15, 23, 47, 12, 7, 10, 17, 26, 53, 9, 11, 14, 19, 29, 59, 10, 61, 31, 21, 8, 13, 22, 67, 34, 23, 14, 71
COMMENTS
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
EXAMPLE
The partition (2,2,2,1,1) has Heinz number 108 and odd bisection (2,2,1) with Heinz number 18, so a(108) = 18.
The partitions (3,2,2,1,1), (3,2,2,2,1), (3,3,2,1,1) have Heinz numbers 180, 270, 300 and all have odd bisection (3,2,1) with Heinz number 30, so a(180) = a(270) = a(300) = 30.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Times@@Prime/@First/@Partition[Append[Reverse[primeMS[n]], 0], 2], {n, 100}]
CROSSREFS
Positions of last appearances are A000290 without the first term 0. Positions of first appearances are A342768. The non-reverse version is A346703. The even non-reverse version is A346704. A001221 counts distinct prime factors.
A103919 counts partitions by sum and alternating sum, reverse A344612.
A209281 (shifted) adds up the odd bisection of standard compositions.
A344606 counts alternating permutations of prime indices.
A344617 gives the sign of the alternating sum of prime indices.
A346700 gives the sum of the even bisection of reversed prime indices.
Cf. A025047, A027187, A027193, A053738, A106356, A277103, A341446, A344653, A345957, A345958, A345959, A346698, A346702.
1, 2, 3, 8, 5, 12, 7, 32, 27, 20, 11, 48, 13, 28, 45, 128, 17, 108, 19, 80, 63, 44, 23, 192, 125, 52, 243, 112, 29, 180, 31, 512, 99, 68, 175, 432, 37, 76, 117, 320, 41, 252, 43, 176, 405, 92, 47, 768, 343, 500, 153, 208, 53, 972, 275, 448, 171, 116, 59, 720
COMMENTS
This sequence has similarities with A087019. These are the positions of first appearances of each positive integer in A346701, and also in A346703. - Gus Wiseman, Aug 09 2021
FORMULA
a(n) = n iff n = 1 or n is a prime number.
a(p^k) = p^(2*k-1) for any k > 0 and any prime number p.
If g = A006530(n) is the greatest prime factor of n, then a(n) = n^2/g. (End)
EXAMPLE
For n = 42:
- 42 = 2 * 3 * 7, so:
2 3 7
x 2 3 7
-------
2 3 7
2 3 3
+ 2 2 2
-----------
2 2 3 3 7
- hence a(42) = 2 * 2 * 3 * 3 * 7 = 252.
MATHEMATICA
Table[n^2/FactorInteger[n][[-1, 1]], {n, 100}] (* Gus Wiseman, Aug 09 2021 *)
PROG
(PARI) See Links section.
CROSSREFS
The version for even indices is A129597(n) = 2*a(n) for n > 1. These are the positions of first appearances in A346701 and in A346703. A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A209281 adds up the odd bisection of standard compositions (even: A346633).
A346697 adds up the odd bisection of prime indices (reverse: A346699).
Numbers whose division (or multiplication) by their greatest prime factor yields a perfect square. Numbers k such that k* A006530(k) is a perfect square.
+10
7
1, 2, 3, 5, 7, 8, 11, 12, 13, 17, 19, 20, 23, 27, 28, 29, 31, 32, 37, 41, 43, 44, 45, 47, 48, 52, 53, 59, 61, 63, 67, 68, 71, 73, 76, 79, 80, 83, 89, 92, 97, 99, 101, 103, 107, 108, 109, 112, 113, 116, 117, 124, 125, 127, 128, 131, 137, 139, 148, 149, 151, 153
COMMENTS
This is the sorted version of A342768(n) = position of first appearance of n in A346701 (but A346703 works also).
EXAMPLE
The terms together with their prime indices begin:
1: {} 31: {11} 71: {20}
2: {1} 32: {1,1,1,1,1} 73: {21}
3: {2} 37: {12} 76: {1,1,8}
5: {3} 41: {13} 79: {22}
7: {4} 43: {14} 80: {1,1,1,1,3}
8: {1,1,1} 44: {1,1,5} 83: {23}
11: {5} 45: {2,2,3} 89: {24}
12: {1,1,2} 47: {15} 92: {1,1,9}
13: {6} 48: {1,1,1,1,2} 97: {25}
17: {7} 52: {1,1,6} 99: {2,2,5}
19: {8} 53: {16} 101: {26}
20: {1,1,3} 59: {17} 103: {27}
23: {9} 61: {18} 107: {28}
27: {2,2,2} 63: {2,2,4} 108: {1,1,2,2,2}
28: {1,1,4} 67: {19} 109: {29}
29: {10} 68: {1,1,7} 112: {1,1,1,1,4}
MAPLE
filter:= proc(n) issqr(n/max(numtheory:-factorset(n))) end proc:
filter(1):= true:
MATHEMATICA
sqrQ[n_]:=IntegerQ[Sqrt[n]];
Select[Range[100], sqrQ[#*FactorInteger[#][[-1, 1]]]&]
PROG
(PARI) isok(m) = (m==1) || issquare(m/vecmax(factor(m)[, 1])); \\ Michel Marcus, Aug 12 2021
CROSSREFS
Removing 1 gives a subset of A026424. The unsorted even version is A129597. Except the first term, the even version is 2*a(n).
A001221 counts distinct prime factors.
A006530 gives the greatest prime factor.
A061395 gives the greatest prime index.
A027193 counts partitions of odd length.
A344606 counts alternating permutations of prime indices.
A346699 = odd bisection sum of reversed prime indices (weights of A346701).
Cf. A028260, A033942, A035363, A037143, A341446, A344653, A345957, A345958, A345959, A346698, A346700, A346704.
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