A261079 -id:A261079 - cp's OEIS frontend

A359362 a(n) = (A001222(n) + 1) * A056239(n), where A001222 counts prime indices and A056239 adds them up.

0, 2, 4, 6, 6, 9, 8, 12, 12, 12, 10, 16, 12, 15, 15, 20, 14, 20, 16, 20, 18, 18, 18, 25, 18, 21, 24, 24, 20, 24, 22, 30, 21, 24, 21, 30, 24, 27, 24, 30, 26, 28, 28, 28, 28, 30, 30, 36, 24, 28, 27, 32, 32, 35, 24, 35, 30, 33, 34, 35, 36, 36, 32, 42, 27, 32, 38

Offset: 1

Gus Wiseman, Dec 31 2022

nonn

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.

A055396 gives minimum prime index, maximum A061395.
A112798 list prime indices, length A001222, sum A056239.
Cf. A261079, A316413, A326836, A326837, A326844, A326846, A359358.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[(PrimeOmega[n]+1)*Total[primeMS[n]],{n,30}]

from sympy import primepi, factorint
def A359362(n): return (sum((f:=factorint(n)).values())+1)*sum(primepi(p)*e for p, e in f.items()) # Chai Wah Wu, Jan 01 2023

a(n) = (k + 1) * m, where m and k are the sum and length of the integer partition with Heinz number n.

a(n) = 2*A304818(n) - A261079(n).

A260737 Sum of Hamming distances between binary representations of prime factors of n, summed over all nonordered pairs of primes present (with multiplicity) in the prime factorization of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 2, 0, 2, 2, 0, 0, 2, 0, 6, 1, 2, 0, 3, 0, 4, 0, 4, 0, 6, 0, 0, 1, 3, 1, 4, 0, 2, 3, 9, 0, 4, 0, 4, 4, 3, 0, 4, 0, 6, 2, 8, 0, 3, 3, 6, 1, 5, 0, 10, 0, 4, 2, 0, 1, 4, 0, 6, 2, 6, 0, 6, 0, 4, 4, 4, 2, 8, 0, 12, 0, 4, 0, 7, 2, 3, 4, 6, 0, 9, 2, 6, 3, 4, 3, 5, 0, 4, 2, 12, 0, 6, 0, 12, 4, 5, 0, 6, 0, 8, 3, 8, 0, 4, 2, 10, 6, 4, 3, 14

Offset: 1

Antti Karttunen, Sep 22 2015

			For n = 1 the prime factorization is empty, thus there is nothing to sum, so a(1) = 0.
For n = 6 = 2*3, a(6) = 1 because the Hamming distance between 2 and 3 is 1 as 2 = "10" in binary and 3 = "11" in binary.
For n = 10 = 2*5, a(10) = 3 because the Hamming distance between 2 and 5 is 3 as 2 = "10" in binary (extended with a leading zero to make it "010") and 5 = "101" in binary.
For n = 12 = 2*2*3, a(12) = 2 because the Hamming distance between 2 and 3 is 1, and the pair (2,3) occurs twice as one can pick either one of the two 2's present in the prime factorization to be a pair of a single 3. Note that the Hamming distance between 2 and 2 is 0, thus the pair (2,2) of prime divisors does not contribute to the sum.
For n = 36 = 2*2*3*3, a(36) = 4 because the Hamming distance between 2 and 3 is 1, and the prime factor pair (2,3) occurs four times in total. Note that the Hamming distance is zero between 2 and 2 as well as between 3 and 3, thus the pairs (2,2) and (3,3) do not contribute to the sum.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A101080.
Cf. A000961 (positions of the zeros), A261077 (positions of the ones).
Cf. A072594, A072595, A235488.
Cf. also A261079.

A297173 Smallest difference between indices of prime divisors of n, or 0 if n is a prime power.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 3, 1, 0, 0, 1, 0, 2, 2, 4, 0, 1, 0, 5, 0, 3, 0, 1, 0, 0, 3, 6, 1, 1, 0, 7, 4, 2, 0, 1, 0, 4, 1, 8, 0, 1, 0, 2, 5, 5, 0, 1, 2, 3, 6, 9, 0, 1, 0, 10, 2, 0, 3, 1, 0, 6, 7, 1, 0, 1, 0, 11, 1, 7, 1, 1, 0, 2, 0, 12, 0, 1, 4, 13, 8, 4, 0, 1, 2, 8, 9, 14, 5, 1, 0, 3, 3, 2, 0, 1, 0, 5, 1

Offset: 1

Antti Karttunen, Mar 03 2018

nonn

			For n = 130 = 2*5*13 = prime(1)*prime(3)*prime(6), the smallest difference between indices is 3-1 = 2, thus a(130) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000720, A073490, A073491.

Cf. also A100573, A243055, A242411, A261079, A286470, A286471.

A297173(n) = if(omega(n)<=1,0,my(ps=factor(n)[,1]); vecmin(vector((#ps)-1,i,primepi(ps[i+1])-primepi(ps[i]))));

a(A073491(n)) <= 1.

A359397 Squarefree numbers with weakly decreasing first differences of 0-prepended prime indices.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 11, 13, 15, 17, 19, 21, 23, 29, 30, 31, 35, 37, 41, 43, 47, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 83, 89, 91, 97, 101, 103, 105, 107, 109, 113, 119, 127, 131, 133, 137, 139, 143, 149, 151, 157, 163, 167, 173, 179, 181, 187, 191, 193, 197

Offset: 1

Gus Wiseman, Dec 31 2022

nonn

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.

			715 has prime indices {3,5,6}, with first differences (2,1), which are weakly decreasing, so 715 is in the sequence.

This is the squarefree case of A325362.
These are the sorted Heinz numbers of rows of A359361.
A005117 lists squarefree numbers.
A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
A355536 lists first differences of prime indices, 0-prepended A287352.
A358136 lists partial sums of prime indices, row sums A318283.
Cf. A000009, A000720, A001221, A253566, A261079, A304818, A358137, A358169.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],SquareFreeQ[#]&&GreaterEqual@@Differences[Prepend[primeMS[#],0]]&]

Intersection of A325362 and A005117.

A261080 Semiprimes p*q for which p and q are successive primes and their binary representations differ from each other in one bit position only.

Original entry on oeis.org

6, 35, 323, 437, 899, 1763, 2021, 4757, 9797, 10403, 19043, 22499, 27221, 38021, 39203, 72899, 79523, 95477, 99221, 131753, 145157, 154433, 164009, 205193, 210677, 213443, 250997, 272483, 324899, 381923, 412163, 416021, 455621, 549077, 557993, 594437, 656099, 675683, 736163, 741317, 777923, 783221, 826277, 870473, 881717, 974153, 1022117, 1102499, 1127843, 1238753

Offset: 1

Antti Karttunen, Sep 23 2015

nonn

Numbers n for which A260737(n) = A261079(n) = 1.

			6 is included as 6 = 2*3, 2 and 3 are successive primes, and 2 (in binary "10") and 3 (in binary "11") differ by only one bit from each other.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A205302, A205511, A260737, A261079.

Intersection of A006094 and A261077.

brdQ[{a_,b_}]:=Module[{c=IntegerDigits[a,2],d=IntegerDigits[b,2]}, Length[ c] == Length[d]&&Count[Total/@Transpose[{c,d}],1]==1]; Times@@@ Select[ Partition[Prime[Range[200]],2,1],brdQ] (* Harvey P. Dale, Jan 29 2016 *)

a(n) = A205511(n) * A205302(n).

cp's OEIS Frontend

A359362 a(n) = (A001222(n) + 1) * A056239(n), where A001222 counts prime indices and A056239 adds them up.

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A260737 Sum of Hamming distances between binary representations of prime factors of n, summed over all nonordered pairs of primes present (with multiplicity) in the prime factorization of n.

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A297173 Smallest difference between indices of prime divisors of n, or 0 if n is a prime power.

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A359397 Squarefree numbers with weakly decreasing first differences of 0-prepended prime indices.

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A261080 Semiprimes p*q for which p and q are successive primes and their binary representations differ from each other in one bit position only.

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