A320628 -id:A320628 - cp's OEIS frontend

A331916 Numbers with exactly one distinct prime prime index.

3, 5, 6, 9, 10, 11, 12, 17, 18, 20, 21, 22, 24, 25, 27, 31, 34, 35, 36, 39, 40, 41, 42, 44, 48, 50, 54, 57, 59, 62, 63, 65, 67, 68, 69, 70, 72, 77, 78, 80, 81, 82, 83, 84, 87, 88, 95, 96, 100, 108, 109, 111, 114, 115, 117, 118, 119, 121, 124, 125, 126, 127

Offset: 1

Gus Wiseman, Feb 08 2020

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.

			The sequence of terms together with their prime indices begins:
    3: {2}           40: {1,1,1,3}       81: {2,2,2,2}
    5: {3}           41: {13}            82: {1,13}
    6: {1,2}         42: {1,2,4}         83: {23}
    9: {2,2}         44: {1,1,5}         84: {1,1,2,4}
   10: {1,3}         48: {1,1,1,1,2}     87: {2,10}
   11: {5}           50: {1,3,3}         88: {1,1,1,5}
   12: {1,1,2}       54: {1,2,2,2}       95: {3,8}
   17: {7}           57: {2,8}           96: {1,1,1,1,1,2}
   18: {1,2,2}       59: {17}           100: {1,1,3,3}
   20: {1,1,3}       62: {1,11}         108: {1,1,2,2,2}
   21: {2,4}         63: {2,2,4}        109: {29}
   22: {1,5}         65: {3,6}          111: {2,12}
   24: {1,1,1,2}     67: {19}           114: {1,2,8}
   25: {3,3}         68: {1,1,7}        115: {3,9}
   27: {2,2,2}       69: {2,9}          117: {2,2,6}
   31: {11}          70: {1,3,4}        118: {1,17}
   34: {1,7}         72: {1,1,1,2,2}    119: {4,7}
   35: {3,4}         77: {4,5}          121: {5,5}
   36: {1,1,2,2}     78: {1,2,6}        124: {1,1,11}
   39: {2,6}         80: {1,1,1,1,3}    125: {3,3,3}

These are numbers n such that A279952(n) = 1.
Prime-indexed primes are A006450, with products A076610.
The number of prime prime indices is A257994.
Numbers with at least one prime prime index are A331386.
The set S of numbers with exactly one prime index in S are A331785.
The set S of numbers with exactly one distinct prime index in S are A331913.
Numbers with at most one prime prime index are A331914.
Numbers with at most one distinct prime prime index are A331995.
Cf. A000040, A000720, A007097, A007821, A018252, A112798, A289509, A320628, A330944, A330945, A331784.

Select[Range[100],Count[PrimePi/@First/@FactorInteger[#],_?PrimeQ]==1&]

A320631 Products of odd primes of nonprime squarefree index.

Original entry on oeis.org

13, 29, 43, 47, 73, 79, 101, 113, 137, 139, 149, 163, 167, 169, 181, 199, 233, 257, 269, 271, 293, 313, 317, 347, 349, 373, 377, 389, 397, 421, 439, 443, 449, 467, 487, 491, 499, 557, 559, 571, 577, 601, 607, 611, 619, 631, 647, 653, 673, 677, 727, 733, 751

Offset: 1

Gus Wiseman, Oct 18 2018

nonn

			The sequence of terms begins:
   13 = prime(6)
   29 = prime(10)
   43 = prime(14)
   47 = prime(15)
   73 = prime(21)
   79 = prime(22)
  101 = prime(26)
  113 = prime(30)
  137 = prime(33)
  139 = prime(34)
  149 = prime(35)
  163 = prime(38)
  167 = prime(39)
  169 = prime(6)^2
  181 = prime(42)
  199 = prime(46)
  233 = prime(51)
  257 = prime(55)
  269 = prime(57)
  271 = prime(58)
  293 = prime(62)
  313 = prime(65)
  317 = prime(66)
  347 = prime(69)
  349 = prime(70)
  373 = prime(74)
  377 = prime(6)*prime(10)

Cf. A000040, A006450, A007821, A018252, A056239, A076610, A112798, A302242, A302478, A320628, A320629, A320630, A320633.

Select[Range[1,100,2],With[{f=PrimePi/@First/@FactorInteger[#]},And[And@@SquareFreeQ/@f,And@@Not/@PrimeQ/@f]]&]

A330949 Odd nonprime numbers whose prime indices are not all prime numbers.

Original entry on oeis.org

21, 35, 39, 49, 57, 63, 65, 69, 77, 87, 91, 95, 105, 111, 115, 117, 119, 129, 133, 141, 143, 145, 147, 159, 161, 169, 171, 175, 183, 185, 189, 195, 203, 207, 209, 213, 215, 217, 219, 221, 231, 235, 237, 245, 247, 253, 259, 261, 265, 267, 273, 285, 287, 291

Offset: 1

Gus Wiseman, Jan 14 2020

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.

Also MM-numbers of multiset partitions with at least two parts, not all of which are singletons (see example).

			The sequence of terms together with their prime indices of prime indices begins:
   21: {{1},{1,1}}
   35: {{2},{1,1}}
   39: {{1},{1,2}}
   49: {{1,1},{1,1}}
   57: {{1},{1,1,1}}
   63: {{1},{1},{1,1}}
   65: {{2},{1,2}}
   69: {{1},{2,2}}
   77: {{1,1},{3}}
   87: {{1},{1,3}}
   91: {{1,1},{1,2}}
   95: {{2},{1,1,1}}
  105: {{1},{2},{1,1}}
  111: {{1},{1,1,2}}
  115: {{2},{2,2}}
  117: {{1},{1},{1,2}}
  119: {{1,1},{4}}
  129: {{1},{1,4}}
  133: {{1,1},{1,1,1}}
  141: {{1},{2,3}}

Complement of A106092 in A330945.
Including even numbers gives A330948.
Including primes gives A330946.
The number of prime prime indices is given by A257994.
The number of nonprime prime indices is given by A330944.
Primes of prime index are A006450.
Primes of nonprime index are A007821.
Products of primes of prime index are A076610.
Products of primes of nonprime index are A320628.
The set S of numbers whose prime indices do not all belong to S is A324694.
Cf. A000040, A000720, A001222, A018252, A056239, A112798, A302242, A320629, A320633, A330943, A330947.

Select[Range[1,100,2],!PrimeQ[#]&&!And@@PrimeQ/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]&]

A360327 a(n) is the sum of divisors of n that have only prime-indexed prime factors.

Original entry on oeis.org

1, 1, 4, 1, 6, 4, 1, 1, 13, 6, 12, 4, 1, 1, 24, 1, 18, 13, 1, 6, 4, 12, 1, 4, 31, 1, 40, 1, 1, 24, 32, 1, 48, 18, 6, 13, 1, 1, 4, 6, 42, 4, 1, 12, 78, 1, 1, 4, 1, 31, 72, 1, 1, 40, 72, 1, 4, 1, 60, 24, 1, 32, 13, 1, 6, 48, 68, 18, 4, 6, 1, 13, 1, 1, 124, 1, 12

Offset: 1

Amiram Eldar, Feb 03 2023

Equivalently, a(n) is the sum of divisors of the largest divisor of n that has only prime-indexed prime factors.

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

Cf. A000203, A006450, A076610, A320628, A360325, A360326.

f[p_, e_] := If[PrimeQ[PrimePi[p]], (p^(e+1)-1)/(p-1), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, if(isprime(primepi(p[i])), (p[i]^(e[i]+1)-1)/(p[i]-1), 1));}

a(n) = 1 if and only if n is in A320628.
a(n) = A000203(n) if and only if n is in A076610.
a(n) = A000203(A360325(n)).
Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if p is a prime-indexed prime (A006450), and 1 otherwise.

A360331 a(n) is the sum of divisors of n that have only prime factors that are not prime-indexed primes.

Original entry on oeis.org

1, 3, 1, 7, 1, 3, 8, 15, 1, 3, 1, 7, 14, 24, 1, 31, 1, 3, 20, 7, 8, 3, 24, 15, 1, 42, 1, 56, 30, 3, 1, 63, 1, 3, 8, 7, 38, 60, 14, 15, 1, 24, 44, 7, 1, 72, 48, 31, 57, 3, 1, 98, 54, 3, 1, 120, 20, 90, 1, 7, 62, 3, 8, 127, 14, 3, 1, 7, 24, 24, 72, 15, 74, 114, 1

Offset: 1

Amiram Eldar, Feb 03 2023

Equivalently, a(n) is the sum of divisors of the largest divisor of n that has only prime factors that are not prime-indexed primes.

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

Cf. A000203, A006450, A007821, A076610, A320628, A360329, A360330.

a:= n-> mul(`if`(isprime(numtheory[pi](i[1])), 1,
   (i[1]^(i[2]+1)-1)/(i[1]-1)), i=ifactors(n)[2]):
seq(a(n), n=1..75);  # Alois P. Heinz, Feb 03 2023

f[p_, e_] := If[PrimeQ[PrimePi[p]], 1, (p^(e+1)-1)/(p-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, if(isprime(primepi(p[i])), 1, (p[i]^(e[i]+1)-1)/(p[i]-1)));}

a(n) = 1 if and only if n is in A076610.
a(n) = A000203(n) if and only if n is in A320628.
a(n) = A000203(A360329(n)).
Multiplicative with a(p^e) = 1 if p is a prime-indexed prime (A006450), and (p^(e+1)-1)/(p-1) otherwise (A007821).

A295665 Fully multiplicative with a(prime(m)) = prime(k) when m = prime(k), and a(prime(m)) = 1 when m is not a prime.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 26 2017

The number of applications to reach 1 is A322027(n). Positions of first appearances are A076610. - Gus Wiseman, Jan 17 2020

			For n = 360 = 2^3 * 3^2 * 5 = prime(1)^3 * prime(2)^2 * prime(3), 1 is not a prime, but 2 and 3 are, thus a(360) = 2^2 * 3 = 12.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Cf. A000040, A000720, A010051, A295666.
Cf. also A003963, A257538.
Positions of 1's are A320628.
Positions of terms > 1 are A331386.
Primes of prime index are A006450.
Primes of nonprime index are A007821.
Products of primes of prime index are A076610.
Products of primes of nonprime index are A320628.
The number of prime prime indices is A257994.
The number of nonprime prime indices is A330944.
Numbers whose prime indices are not all prime are A330945.
Cf. A001222, A007097, A018252, A056239, A112798, A320633, A322027, A330947, A330948.

Table[Times@@Cases[FactorInteger[n],{p_?(PrimeQ[PrimePi[#]]&),k_}:>PrimePi[p]^k],{n,40}] (* Gus Wiseman, Jan 17 2020 *)

(definec (A295665 n) (if (= 1 n) 1 (let ((k (A055396 n))) (* (if (zero? (A010051 k)) 1 k) (A295665 (A032742 n))))))

Multiplicative with a(p^e) = A000720(p)^(e*A010051(A000720(p))).
a(1) = 1; for n > 1, if A055396(n) is a prime, then a(n) = A055396(n) * a(A032742(n)), otherwise a(n) = a(A032742(n)).
Other identities. For all n >= 1:
a(A006450(n)) = A000040(n).
a(A007097(n)) = A007097(n-1).
a(A294876(n)) = A295666(n).

A360325 a(n) is the largest divisor of n that has only prime-indexed prime factors.

Original entry on oeis.org

1, 1, 3, 1, 5, 3, 1, 1, 9, 5, 11, 3, 1, 1, 15, 1, 17, 9, 1, 5, 3, 11, 1, 3, 25, 1, 27, 1, 1, 15, 31, 1, 33, 17, 5, 9, 1, 1, 3, 5, 41, 3, 1, 11, 45, 1, 1, 3, 1, 25, 51, 1, 1, 27, 55, 1, 3, 1, 59, 15, 1, 31, 9, 1, 5, 33, 67, 17, 3, 5, 1, 9, 1, 1, 75, 1, 11, 3, 1

Offset: 1

Amiram Eldar, Feb 03 2023

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

Cf. A006450, A076610, A320628, A360326, A360327, A360329.

f[p_, e_] := If[PrimeQ[PrimePi[p]], p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(isprime(primepi(f[i, 1])), f[i, 1]^f[i, 2], 1)); }

a(n) = 1 if and only if n is in A320628.
a(n) = n if and only if n is in A076610.
a(n) = n/A360329(n).
Multiplicative with a(p^e) = p^e if p is a prime-indexed prime (A006450), and 1 otherwise.

A360329 a(n) is the largest divisor of n that has only prime factors that are not prime-indexed primes.

Original entry on oeis.org

1, 2, 1, 4, 1, 2, 7, 8, 1, 2, 1, 4, 13, 14, 1, 16, 1, 2, 19, 4, 7, 2, 23, 8, 1, 26, 1, 28, 29, 2, 1, 32, 1, 2, 7, 4, 37, 38, 13, 8, 1, 14, 43, 4, 1, 46, 47, 16, 49, 2, 1, 52, 53, 2, 1, 56, 19, 58, 1, 4, 61, 2, 7, 64, 13, 2, 1, 4, 23, 14, 71, 8, 73, 74, 1, 76, 7

Offset: 1

Amiram Eldar, Feb 03 2023

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

Cf. A006450, A007821, A076610, A302590, A320628, A360325, A360330, A360331.

f[p_, e_] := If[PrimeQ[PrimePi[p]], 1, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); for(i = 1, #f~, if(isprime(primepi(f[i,1])), f[i,1]=1)); factorback(f);}

a(n) = 1 if and only if n is in A076610.
a(n) = n if and only if n is in A320628.
a(n) = n/A360325(n).
Multiplicative with a(p^e) = 1 if p is a prime-indexed prime (A006450), and p^e otherwise (A007821).
Sum_{k=1..n} a(k) ~ (1/2) * c * n^2, where c = Product_{p in A006450} p/(p+1) < 0.4 (see A302590 for an estimate of 1/c).

A371443 Numbers whose binary indices are nonprime numbers.

Original entry on oeis.org

1, 8, 9, 32, 33, 40, 41, 128, 129, 136, 137, 160, 161, 168, 169, 256, 257, 264, 265, 288, 289, 296, 297, 384, 385, 392, 393, 416, 417, 424, 425, 512, 513, 520, 521, 544, 545, 552, 553, 640, 641, 648, 649, 672, 673, 680, 681, 768, 769, 776, 777, 800, 801, 808

Offset: 1

Gus Wiseman, Mar 30 2024

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The terms together with their binary expansions and binary indices begin:
    1:          1 ~ {1}
    8:       1000 ~ {4}
    9:       1001 ~ {1,4}
   32:     100000 ~ {6}
   33:     100001 ~ {1,6}
   40:     101000 ~ {4,6}
   41:     101001 ~ {1,4,6}
  128:   10000000 ~ {8}
  129:   10000001 ~ {1,8}
  136:   10001000 ~ {4,8}
  137:   10001001 ~ {1,4,8}
  160:   10100000 ~ {6,8}
  161:   10100001 ~ {1,6,8}
  168:   10101000 ~ {4,6,8}
  169:   10101001 ~ {1,4,6,8}
  256:  100000000 ~ {9}
  257:  100000001 ~ {1,9}
  264:  100001000 ~ {4,9}
  265:  100001001 ~ {1,4,9}
  288:  100100000 ~ {6,9}
  289:  100100001 ~ {1,6,9}
  296:  100101000 ~ {4,6,9}

For powers of 2 instead of nonprime numbers we have A253317.
For prime indices instead of binary indices we have A320628.
For prime instead of nonprime we have A326782.
For composite numbers we have A371444.
An opposite version is A371449.
A000040 lists prime numbers, complement A018252.
A000961 lists prime-powers.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A001222, A005117, A326781, A368109, A368533, A371289, A371452.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],And@@Not/@PrimeQ/@bpe[#]&]

A371444 Numbers whose binary indices are composite numbers.

Original entry on oeis.org

8, 32, 40, 128, 136, 160, 168, 256, 264, 288, 296, 384, 392, 416, 424, 512, 520, 544, 552, 640, 648, 672, 680, 768, 776, 800, 808, 896, 904, 928, 936, 2048, 2056, 2080, 2088, 2176, 2184, 2208, 2216, 2304, 2312, 2336, 2344, 2432, 2440, 2464, 2472, 2560, 2568

Offset: 1

Gus Wiseman, Mar 30 2024

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The terms together with their binary expansions and binary indices begin:
     8:           1000 ~ {4}
    32:         100000 ~ {6}
    40:         101000 ~ {4,6}
   128:       10000000 ~ {8}
   136:       10001000 ~ {4,8}
   160:       10100000 ~ {6,8}
   168:       10101000 ~ {4,6,8}
   256:      100000000 ~ {9}
   264:      100001000 ~ {4,9}
   288:      100100000 ~ {6,9}
   296:      100101000 ~ {4,6,9}
   384:      110000000 ~ {8,9}
   392:      110001000 ~ {4,8,9}
   416:      110100000 ~ {6,8,9}
   424:      110101000 ~ {4,6,8,9}
   512:     1000000000 ~ {10}
   520:     1000001000 ~ {4,10}
   544:     1000100000 ~ {6,10}
   552:     1000101000 ~ {4,6,10}
   640:     1010000000 ~ {8,10}
   648:     1010001000 ~ {4,8,10}
   672:     1010100000 ~ {6,8,10}

For powers of 2 instead of composite numbers we have A253317.
For prime indices we have the even case of A320628.
For prime instead of composite we have A326782.
This is the even case of A371444.
An opposite version is A371449.
A000040 lists prime numbers, complement A018252.
A000961 lists prime-powers.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A001222, A005117, A055887, A320629, A326781, A368109, A368533, A371289, A371452.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],EvenQ[#]&&And@@Not/@PrimeQ/@bpe[#]&]

cp's OEIS Frontend

A331916 Numbers with exactly one distinct prime prime index.

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A320631 Products of odd primes of nonprime squarefree index.

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A330949 Odd nonprime numbers whose prime indices are not all prime numbers.

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A360327 a(n) is the sum of divisors of n that have only prime-indexed prime factors.

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A360331 a(n) is the sum of divisors of n that have only prime factors that are not prime-indexed primes.

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A295665 Fully multiplicative with a(prime(m)) = prime(k) when m = prime(k), and a(prime(m)) = 1 when m is not a prime.

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A360325 a(n) is the largest divisor of n that has only prime-indexed prime factors.

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A360329 a(n) is the largest divisor of n that has only prime factors that are not prime-indexed primes.

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