A071403 -id:A071403 - cp's OEIS frontend

A379307 Positive integers whose prime indices include no squarefree numbers.

1, 7, 19, 23, 37, 49, 53, 61, 71, 89, 97, 103, 107, 131, 133, 151, 161, 173, 193, 197, 223, 227, 229, 239, 251, 259, 263, 281, 307, 311, 337, 343, 359, 361, 371, 379, 383, 409, 419, 427, 433, 437, 457, 463, 479, 497, 503, 521, 523, 529, 541, 569, 593, 613, 623

Offset: 1

Gus Wiseman, Dec 27 2024

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 terms together with their prime indices begin:
    1: {}
    7: {4}
   19: {8}
   23: {9}
   37: {12}
   49: {4,4}
   53: {16}
   61: {18}
   71: {20}
   89: {24}
   97: {25}
  103: {27}
  107: {28}
  131: {32}
  133: {4,8}
  151: {36}
  161: {4,9}
  173: {40}

Partitions of this type are counted by A114374, strict A256012.
Positions of zero in A379306.
For a unique squarefree part we have A379316, counted by A379308 (strict A379309).
A000040 lists the primes, differences A001223.
A005117 lists the squarefree numbers, differences A076259.
A008966 is the characteristic function for the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A061398 counts squarefree numbers between primes, zeros A068360.
A377038 gives k-th differences of squarefree numbers.
Other counts of prime indices:
- A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A330944 nonprime, see A000586, A000607, A076610, A330945.
- A379300 composite, see A023895, A034891, A036497, A302540, A379301.
- A379310 nonsquarefree, see A302478.
- A379311 old prime, see A204389, A320629, A379312-A379315.
Cf. A000720, A013928, A038550, A057627, A068361, A070321, A071403, A072284, A087436, A112929, A377430, A378086.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Length[Select[prix[#],SquareFreeQ]]==0&]

A379310 Number of nonsquarefree prime indices of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 1

Gus Wiseman, Dec 27 2024

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 prime indices of 39 are {2,6}, so a(39) = 0.
The prime indices of 70 are {1,3,4}, so a(70) = 1.
The prime indices of 98 are {1,4,4}, so a(98) = 2.
The prime indices of 294 are {1,2,4,4}, a(294) = 2.
The prime indices of 1911 are {2,4,4,6}, so a(1911) = 2.
The prime indices of 2548 are {1,1,4,4,6}, so a(2548) = 2.

Positions of first appearances are A000420.
Positions of zero are A302478, counted by A073576 (strict A087188).
No squarefree parts: A379307, counted by A114374 (strict A256012).
One squarefree part: A379316, counted by A379308 (strict A379309).
A000040 lists the primes, differences A001223.
A005117 lists the squarefree numbers, differences A076259.
A008966 is the characteristic function for the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A061398 counts squarefree numbers between primes, zeros A068360.
A377038 gives k-th differences of squarefree numbers.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.
- A330944 nonprime, see A000586, A000607, A076610, A330945.
- A379300 composite, see A023895, A034891, A036497, A302540, A379301.
- A379311 old prime, see A204389, A320629, A379312-A379315.
Cf. A000720, A013928, A038550, A057627, A068361, A070321, A071403, A087436, A112929, A120327, A378086.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[prix[n],Not@*SquareFreeQ]],{n,100}]

Totally additive with a(prime(k)) = A107078(k) = 1 - A008966(k).

A379306 Number of squarefree prime indices of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 0, 3, 2, 2, 1, 3, 1, 1, 2, 4, 1, 3, 0, 3, 1, 2, 0, 4, 2, 2, 3, 2, 1, 3, 1, 5, 2, 2, 1, 4, 0, 1, 2, 4, 1, 2, 1, 3, 3, 1, 1, 5, 0, 3, 2, 3, 0, 4, 2, 3, 1, 2, 1, 4, 0, 2, 2, 6, 2, 3, 1, 3, 1, 2, 0, 5, 1, 1, 3, 2, 1, 3, 1, 5, 4, 2, 1, 3, 2, 2, 2

Offset: 1

Gus Wiseman, Dec 25 2024

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 prime indices of 39 are {2,6}, so a(39) = 2.
The prime indices of 70 are {1,3,4}, so a(70) = 2.
The prime indices of 98 are {1,4,4}, so a(98) = 1.
The prime indices of 294 are {1,2,4,4}, a(294) = 2.
The prime indices of 1911 are {2,4,4,6}, so a(1911) = 2.
The prime indices of 2548 are {1,1,4,4,6}, so a(2548) = 3.

Positions of first appearances are A000079.
Positions of zero are A379307, counted by A114374 (strict A256012).
Positions of one are A379316, counted by A379308 (strict A379309).
A000040 lists the primes, differences A001223.
A005117 lists the squarefree numbers, differences A076259.
A008966 is the characteristic function for the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A061398 counts squarefree numbers between primes, zeros A068360.
A377038 gives k-th differences of squarefree numbers.
Other counts of prime indices:
- A087436 postpositive, see A038550.
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.
- A330944 nonprime, see A000586, A000607, A076610, A330945.
- A379300 composite, see A023895, A034891, A036497, A302540, A379301.
- A379310 nonsquarefree, see A302478.
- A379311 old prime, see A204389, A320629, A379312-A379315.
Cf. A000720, A013928, A057627, A068361, A070321, A071403, A072284, A112925, A112929, A120327, A377430, A378086.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[prix[n],SquareFreeQ]],{n,100}]

Totally additive with a(prime(k)) = A008966(k).

A277010 a(n) = A156552(A005117(n)); permutation of Fibbinary numbers.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 16, 32, 17, 10, 64, 128, 18, 33, 256, 65, 512, 21, 1024, 34, 129, 20, 2048, 257, 66, 4096, 37, 8192, 513, 16384, 130, 32768, 36, 258, 1025, 65536, 131072, 2049, 68, 69, 262144, 514, 41, 524288, 1048576, 4097, 40, 133, 2097152, 8193, 4194304, 132, 16385, 1026, 8388608, 72, 2050, 32769, 260, 16777216, 33554432, 261, 67108864, 42

Offset: 1

Antti Karttunen, Oct 07 2016

nonn

Permutation of A003714 (Fibbinary numbers).

Antti Karttunen, Table of n, a(n) for n = 1..1025

Cf. A000040, A003714, A005117, A071403, A156552.

Cf. also A277006, A277020, A277195.

from math import isqrt
from sympy import mobius, primepi, factorint
def A277010(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return sum(1<Chai Wah Wu, Aug 31 2024

(define (A277010 n) (A156552 (A005117 n)))

a(n) = A156552(A005117(n)).
Other identities. For all n >= 1:
a(A071403(n)) = A000079(n-1).

A277196 Permutation of natural numbers: a(n) = A107079(A277006(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 11, 8, 10, 14, 23, 19, 9, 15, 21, 34, 28, 48, 44, 65, 12, 17, 26, 40, 41, 57, 69, 101, 89, 94, 144, 233, 129, 13, 22, 32, 53, 49, 75, 80, 120, 115, 111, 168, 279, 203, 137, 176, 261, 438, 283, 609, 470, 703, 16, 25, 35, 60, 63, 82, 104, 157, 128, 148, 217, 363, 236, 152, 227, 342, 569, 334, 798, 554, 833, 199, 270

Offset: 0

Antti Karttunen, Oct 07 2016

nonn

Note the indexing: domain starts from 0, but the range starts from 1.

Antti Karttunen, Table of n, a(n) for n = 0..353
Index entries for sequences that are permutations of the natural numbers

Inverse: A277195.

Cf. A000045, A003714, A005940, A071403, A107079, A277006.

(define (A277196 n) (A107079 (A277006 n)))

a(n) = A107079(A277006(n)) = A107079(A005940(1+A003714(n))).
Other identities:
For all n >= 1, a(A000045(n+1)) = A071403(n).

A378615 Number of non prime powers <= prime(n).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 7, 10, 13, 14, 18, 21, 22, 25, 29, 34, 35, 39, 42, 43, 48, 50, 55, 62, 65, 66, 69, 70, 73, 84, 86, 91, 92, 101, 102, 107, 112, 115, 119, 124, 125, 134, 135, 138, 139, 150, 161, 164, 165, 168, 173, 174, 182, 186, 191, 196, 197, 202, 205

Offset: 1

Gus Wiseman, Dec 06 2024

nonn

			The non prime powers counted under each term:
  n=1  n=2  n=3  n=4  n=5  n=6  n=7  n=8  n=9  n=10
  -------------------------------------------------
   1    1    1    6   10   12   15   18   22   28
                  1    6   10   14   15   21   26
                       1    6   12   14   20   24
                            1   10   12   18   22
                                 6   10   15   21
                                 1    6   14   20
                                      1   12   18
                                          10   15
                                           6   14
                                           1   12
                                               10
                                                6
                                                1

Restriction of A356068 (first-differences A143731).
First-differences are A368748.
Maxima are A378616.
Other classes of numbers (instead of non prime powers):
- prime: A000027 (diffs A000012), restriction of A000720 (diffs A010051)
- squarefree: A071403 (diffs A373198), restriction of A013928 (diffs A008966)
- nonsquarefree: A378086 (diffs A061399), restriction of A057627 (diffs A107078)
- prime power: A027883 (diffs A366833), restriction of A025528 (diffs A010055)
- composite: A065890 (diffs A046933), restriction of A065855 (diffs  A005171)
A000040 lists the primes, differences A001223
A000961 and A246655 list the prime powers, differences A057820.
A024619 lists the non prime powers, differences A375735, seconds A376599.
A080101 counts prime powers between primes (exclusive), inclusive A366833.
A361102 lists the non powers of primes, differences A375708.
Cf. A027883, A053607, A304521, A343249, A345531, A377057, A377281, A377286, A377289, A377703, A377781, A378032.

Table[Length[Select[Range[Prime[n]],Not@*PrimePowerQ]],{n,100}]

from sympy import prime, primepi, integer_nthroot
def A378615(n): return int((p:=prime(n))-n-sum(primepi(integer_nthroot(p,k)[0]) for k in range(2,p.bit_length()))) # Chai Wah Wu, Dec 07 2024

a(n) = prime(n) - A027883(n). - Chai Wah Wu, Dec 08 2024

A378618 Sum of nonsquarefree numbers between prime(n) and prime(n+1).

Original entry on oeis.org

0, 4, 0, 17, 12, 16, 18, 20, 104, 0, 68, 40, 0, 89, 199, 110, 60, 127, 68, 72, 151, 161, 172, 278, 297, 0, 104, 108, 112, 849, 128, 403, 0, 579, 150, 461, 322, 164, 680, 351, 180, 561, 192, 196, 198, 819, 648, 449, 228, 232, 470, 240, 1472, 508, 521, 532, 270

Offset: 1

Gus Wiseman, Dec 09 2024

nonn

			The nonsquarefree numbers between prime(24) = 89 and prime(25) = 97 are {90, 92, 96}, so a(24) = 278.

For prime instead of nonsquarefree we have A001043.
For composite instead of nonsquarefree we have A054265.
Zeros are A068361.
A000040 lists the primes, differences A001223, seconds A036263.
A070321 gives the greatest squarefree number up to n.
A071403 counts squarefree numbers up to prime(n), restriction of A013928.
A120327 gives the least nonsquarefree number >= n.
A378086 counts nonsquarefree numbers up to prime(n), restriction of A057627.
For squarefree numbers (A005117, differences A076259) between primes:
- length is A061398, zeros A068360
- min is A112926, differences A378037
- max is A112925, differences A378038
- sum is A373197
For nonsquarefree numbers (A013929, differences A078147) between primes:
- length is A061399
- min is A377783 (differences A377784), union A378040
- max is A378032 (differences A378034), restriction of A378033 (differences A378036)
- sum is A378618 (this)
Cf. A007674, A045882, A053806, A067535, A072284, A073247, A179211, A224363, A373198, A377049, A377431.

Table[Total[Select[Range[Prime[n],Prime[n+1]],!SquareFreeQ[#]&]],{n,100}]

A071404 Which nonsquarefree number is a square number? a(n)-th nonsquarefree number equals n^2, the n-th square.

Original entry on oeis.org

1, 3, 5, 9, 13, 18, 25, 31, 39, 46, 55, 66, 76, 86, 99, 112, 125, 142, 157, 172, 187, 206, 225, 244, 264, 285, 307, 328, 353, 377, 400, 429, 453, 480, 507, 534, 562, 593, 623, 656, 691, 725, 762, 795, 831, 867, 904, 941, 977, 1019, 1059, 1101, 1145, 1185, 1226

Offset: 2

Labos Elemer, May 24 2002

nonn

			The first, 3rd, 5th, 9th, and 13th nonsquarefree numbers are 4, 9, 16, 25, and 36, respectively.

Amiram Eldar, Table of n, a(n) for n = 2..10001

Cf. A005117, A013929, A000040, A000290, A071403.

Position[Select[Range[3200], !SquareFreeQ[#] &], ?(IntegerQ[Sqrt[#]] &)][[;; , 1]] (* _Amiram Eldar, Mar 04 2024 *)

lista(nn) = {sqfs = select(n->(1-issquarefree(n)), vector(nn, i, i)); for (i = 1, #sqfs, if (issquare(sqfs[i]), print1(i, ", ")););} \\ Michel Marcus, Sep 12 2013

a(n)=n^2-sum(k=1, n, n^2\k^2*moebius(k)) \\ Charles R Greathouse IV, Sep 13 2013

from sympy import mobius
def A071404(n): return -sum(mobius(k)*(n**2//k**2) for k in range(2,n+1)) # Chai Wah Wu, Jul 20 2024

A013929(a(n)) = A000290(n) = n^2.
a(n) = kn^2 + O(n), where k = 1 - 6/Pi^2. - Charles R Greathouse IV, Sep 13 2013
a(n) = -Sum_{k=2..n} mu(k)*floor((n/k)^2). - Chai Wah Wu, Jul 20 2024

Offset changed by Charles R Greathouse IV, Sep 13 2013

A112045 Positions of primes (A000040) among nonsquares A000037.

Original entry on oeis.org

1, 2, 3, 5, 8, 10, 13, 15, 19, 24, 26, 31, 35, 37, 41, 46, 52, 54, 59, 63, 65, 71, 74, 80, 88, 91, 93, 97, 99, 103, 116, 120, 126, 128, 137, 139, 145, 151, 155, 160, 166, 168, 178, 180, 183, 185, 197, 209, 212, 214, 218, 224, 226, 236, 241, 247, 253, 255, 261

Offset: 1

Antti Karttunen, Aug 27 2005

nonn

Also: Distance of prime(n) from the integer part of its square root. - M. F. Hasler, Oct 19 2018

Cf. A000037, A000040, A028391, A071403 (the original name describes this sequence).

f[n_]:=n-IntegerPart[Sqrt[n]]; lst={};Do[p=Prime[n];AppendTo[lst,f[p]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 24 2009 *)

apply( A112045=n->(n=prime(n))-sqrtint(n), [1..200]) \\ M. F. Hasler, Oct 19 2018

from math import isqrt
from sympy import prime
def A112045(n): return (p:=prime(n))-isqrt(p) # Chai Wah Wu, Jun 05 2025

a(n) = A028391(A000040(n)), where A028391(x) = x - floor(sqrt(x)).

a(n) ~ 6/Pi^2 * n log n. - Charles R Greathouse IV, May 29 2013

Erroneous name corrected by Antti Karttunen, Jun 03 2014

A378620 Lesser prime index of twin primes with nonsquarefree mean.

Original entry on oeis.org

2, 5, 7, 17, 20, 28, 35, 41, 43, 45, 49, 52, 57, 64, 69, 81, 83, 98, 109, 120, 140, 144, 152, 171, 173, 176, 178, 182, 190, 206, 215, 225, 230, 236, 253, 256, 262, 277, 286, 294, 296, 302, 307, 315, 318, 323, 336, 346, 373, 377, 390, 395, 405, 428, 430, 444

Offset: 1

Gus Wiseman, Dec 10 2024

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.

This is a subset of A029707 (twin prime indices). The other twin primes are A068361, so A029707 is the disjoint union of A068361 and A378620.

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

The lesser of twin primes is A001359, index A029707 (complement A049579).
The greater of twin primes is A006512, index A107770 (complement appears to be A168543).
A subset of A029707 (twin prime lesser indices).
Prime indices of the primes listed by A061368.
Indices of twin primes with squarefree mean are A068361.
A000040 lists the primes, differences A001223, (run-lengths A333254, A373821).
A005117 lists the squarefree numbers, differences A076259.
A006562 finds balanced primes.
A013929 lists the nonsquarefree numbers, differences A078147.
A014574 is the intersection of A006093 and A008864.
A038664 finds the first position of a prime gap of 2n.
A046933 counts composite numbers between primes.
A120327 gives the least nonsquarefree number >= n.
Cf. A007674, A072284, A068360.
Cf. A057627, A061398, A061399, A067535, A070321, A071403, A112925.
Cf. A000720, A025584, A067774, A122535, A155752, A179067.

Select[Range[100],Prime[#]+2==Prime[#+1]&&!SquareFreeQ[Prime[#]+1]&]
PrimePi/@Select[Partition[Prime[Range[500]],2,1],#[[2]]-#[[1]]==2&&!SquareFreeQ[Mean[#]]&][[;;,1]] (* Harvey P. Dale, Jul 13 2025 *)

prime(a(n)) = A061368(n).

cp's OEIS Frontend

A379307 Positive integers whose prime indices include no squarefree numbers.

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A379310 Number of nonsquarefree prime indices of n.

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A379306 Number of squarefree prime indices of n.

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A277010 a(n) = A156552(A005117(n)); permutation of Fibbinary numbers.

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A277196 Permutation of natural numbers: a(n) = A107079(A277006(n)).

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A378615 Number of non prime powers <= prime(n).

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A378618 Sum of nonsquarefree numbers between prime(n) and prime(n+1).

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A071404 Which nonsquarefree number is a square number? a(n)-th nonsquarefree number equals n^2, the n-th square.

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