Jason G. Wurtzel - cp's OEIS frontend

A175730 Numbers with deficiency of 42.

43, 69, 99, 165, 1168, 1365, 2139136, 32062485, 33722368, 132701205, 8592621568

Offset: 1

Jason G. Wurtzel, Aug 24 2010

Numbers with an abundance of -42.
7454198513685 is also in the sequence. - Alexander Violette, Jan 10 2021
Numbers of the form 2^k * (2^(k + 1) + 41) are in the sequence if and only if 2^(k + 1) + 41 is prime. - David A. Corneth, Jan 15 2021

Cf. A033879 (deficiency).

for(n=1, 100000000, if(((sigma(n)-2*n)==-42), print1(n, ", ")))

is(n)=2*n-sigma(n)==42 \\ Charles R Greathouse IV, Jan 11 2021

list(lim)=my(v=List()); forfactored(n=1,lim\1, if(2*n[1]-sigma(n)==42, listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Jan 11 2021

a(10) from Alexander Violette, Jan 10 2021

a(11) from Charles R Greathouse IV, Jan 15 2021

A175728 Decimal expansion of e written as a sequence of distinct positive integers.

Original entry on oeis.org

2, 7, 1, 8, 28, 18, 284, 5, 90, 4, 52, 3, 53, 60, 287, 47, 13, 526, 6, 24, 9, 77, 57, 2470, 93, 69, 99, 59, 574, 96, 696, 76, 27, 72, 40, 766, 30, 35, 354, 75, 94, 571, 38, 21, 78, 525, 16, 64, 274, 2746, 63, 91, 93200, 305, 992, 181, 74, 135, 966, 290, 43, 572, 900, 33, 42

Offset: 1

Jason G. Wurtzel, Aug 19 2010

Inverse: 3, 1, 12, 10, 8, 19, 2, 4, 21, 119, 86, 166, 17, 154, 160, 47, 192, 6, 83, 214, ..., . - Robert G. Wilson v, Aug 22 2010

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A001113, A064809.

rd = RealDigits[E, 10, 200][[1]]; lst = {}; f[n_] := Block[{k = 1}, While[a = FromDigits[ Take[ rd, k]]; MemberQ[lst, a] || rd[[k + 1]] == 0, k++ ]; AppendTo[ lst, a]; rd = Drop[rd, k]]; Array[f, 70]; lst (* Robert G. Wilson v, Aug 22 2010 *)

from itertools import islice
from sympy import exp
def diggen():
    yield from map(int, str(exp(1).n(10**5))[:-1].replace(".", ""))
def agen(): # generator of terms
    g = diggen()
    aset, nextd = set(), next(g)
    while True:
        an, nextd = nextd, next(g)
        while an in aset or nextd == 0:
            an, nextd = int(str(an) + str(nextd)), next(g)
        yield an
        aset.add(an)
print(list(islice(agen(), 65))) # Michael S. Branicky, Mar 31 2022

More terms from Robert G. Wilson v, Aug 22 2010

A180231 Prime partial sums of digits of decimal expansion of e.

Original entry on oeis.org

2, 29, 37, 47, 79, 103, 173, 191, 257, 269, 331, 491, 523, 547, 547, 547, 641, 673, 677, 701, 701, 739, 751, 797, 823, 853, 881, 907, 907, 919, 977, 977, 1013, 1039, 1051, 1063, 1091, 1093, 1097, 1153, 1163, 1201, 1213, 1237, 1259, 1279, 1373, 1427, 1427, 1433, 1487

Offset: 1

Jason G. Wurtzel, Aug 18 2010

A046975 INTERSECT A000040.

Select[Accumulate[First[RealDigits[N[E,500]]]],PrimeQ] (* Harvey P. Dale, Aug 19 2010 *)

More terms from Harvey P. Dale, Aug 19 2010

Further terms from R. J. Mathar, Aug 20 2010

A180152 Numbers k such that the sum of the first k semiprimes is a prime.

Original entry on oeis.org

3, 4, 5, 7, 8, 15, 21, 22, 37, 56, 59, 62, 82, 85, 89, 91, 114, 119, 121, 129, 139, 146, 168, 169, 189, 195, 197, 214, 227, 258, 286, 295, 312, 333, 341, 352, 360, 361, 387, 400, 419, 426, 434, 437, 440, 466, 470, 483, 495, 497, 504, 556, 595, 610, 619, 629, 636

Offset: 1

Jason G. Wurtzel, Aug 13 2010

nonn

			21 is a term because the sum of the first 21 semiprimes is 647, which is prime.

K. Brockhaus, Table of n, a(n) for n = 1..10000

Cf. A062198, A000040.

SP:=[ n: n in [2..3000] | &+[ k[2]: k in Factorization(n) ] eq 2 ]; V:=[]; s:=0; for j in [1..640] do s+:=SP[j]; if IsPrime(s) then Append(~V, j); end if; end for; V; // Klaus Brockhaus, Aug 14 2010

A062198 := proc(n) add( A001358(i),i=1..n) ; end proc:
isA180152 := proc(n) isprime( A062198(n)) ; end proc:
for n from 1 to 1000 do if isA180152(n) then printf("%d,",n) ; end if; end do: # R. J. Mathar, Aug 14 2010

Position[Accumulate[Select[Range[10000],PrimeOmega[#]==2&]],?PrimeQ] // Flatten (* _Harvey P. Dale, Feb 17 2021 *)

More terms from Klaus Brockhaus and R. J. Mathar, Aug 14 2010

A179659 Digital root of the n-th weird number.

Original entry on oeis.org

7, 8, 7, 7, 1, 1, 2, 8, 4, 1, 1, 7, 8, 5, 2, 7, 5, 1, 2, 7, 1, 4, 5, 1, 2, 8, 4, 1, 2, 8, 5, 2, 7, 4, 5, 1, 8, 7, 8, 4, 5, 4, 1, 8, 4, 5, 2, 4, 1, 7, 8, 5, 7, 8, 1, 8, 4, 2, 7, 4, 5, 2, 4, 5, 1, 2, 5, 7, 8, 1, 2, 8, 2, 7, 7, 4, 2, 8, 5, 1, 7, 5, 2, 8, 4, 1, 7, 8, 4, 7, 5, 1, 2, 5, 8, 5, 1, 4, 5, 2, 4, 2, 4, 2, 4, 1

Offset: 1

Jason G. Wurtzel, Jul 23 2010

			The 9th weird number is 10570. 1+0+5+7+0=13, and 1+3=4.

Cf. A006037

a(n) = A010888(A006037(n)).

Extended, keyword:base and formula added - R. J. Mathar, Jul 27 2010

A179679 Smaller of each pair of consecutive primes which sum to a practical number.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 29, 37, 41, 43, 47, 53, 59, 61, 71, 79, 97, 101, 103, 107, 113, 137, 139, 149, 151, 157, 163, 167, 173, 179, 191, 193, 197, 223, 227, 229, 239, 257, 263, 269, 277, 283, 293, 311, 313, 317, 337, 347, 349, 359, 367, 397, 401, 409, 419, 431

Offset: 1

Jason G. Wurtzel, Jul 24 2010

nonn

Complement is: 2, 23, 31, 67, 73, 83, 89, 109, 127, 131, 181, 199, 211, ..., . - Robert G. Wilson v, Aug 03 2010

			19 is a term because the next consecutive prime is 23, and 19 + 23 = 42, which is a practical number.

Cf. A000040, A005153.

PracticalQ[n_] := Module[{f, p, e, prod = 1, ok = True}, If[n < 1 || (n > 1 && OddQ[n]), False, If[n == 1, True, f = FactorInteger[n]; {p, e} = Transpose[f]; Do[If[p[[i]] > 1 + DivisorSigma[1, prod], ok = False; Break[]]; prod = prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]]; First@# & /@ Select[ Partition[ Prime@ Range@ 85, 2, 1], PracticalQ[Plus @@ # ] &] (* Robert G. Wilson v, Aug 03 2010 *)

More terms from Robert G. Wilson v, Aug 03 2010

A179656 prime(n) mod (digital root(prime(n))).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 0, 3, 1, 3, 0, 1, 1, 1, 5, 4, 5, 3, 7, 0, 2, 1, 1, 6, 1, 3, 3, 0, 3, 0, 1, 1, 3, 4, 4, 1, 0, 2, 1, 3, 0, 1, 1, 5, 0, 3, 6, 1, 1, 1, 4, 3, 3, 2, 1, 5, 0, 4, 1, 3, 3, 0, 1, 5, 1, 2, 1, 2, 6, 1, 7, 3, 1, 0, 3, 1, 0, 1, 1, 4, 1, 7, 0, 5, 1, 1, 2, 1, 3, 3, 1, 0, 1, 3, 7, 4, 1, 0, 0, 1, 5, 3, 1, 3

Offset: 1

Jason G. Wurtzel, Jul 23 2010

For the first one million primes, the distribution of the values (0..8) is {166572, 361136, 69399, 194537, 69405, 69460, 27798, 41693, 0} . - Robert G. Wilson v, Aug 02 2010

			For n=9 : prime(9)=23, and digital root of 23 = 2+3 = 5, so 23 mod 5 = 3.
For n=16 : prime(16)=53, and digital root of 53 = 5+3 = 8, so 53 mod 8 = 5.

Cf. A000040, A038194.

f[n_] := Block[{p = Prime@n}, Mod[p, Mod[p, 9]]]; Array[f, 111] (* Robert G. Wilson v, Aug 02 2010 *)

More terms from Robert G. Wilson v, Aug 02 2010

A179655 Digital root of n-th abundant number.

Original entry on oeis.org

3, 9, 2, 6, 3, 9, 4, 6, 3, 9, 2, 6, 3, 7, 9, 6, 8, 3, 7, 9, 6, 1, 3, 5, 9, 4, 6, 3, 9, 6, 3, 5, 9, 6, 3, 7, 9, 6, 3, 5, 9, 6, 3, 7, 9, 2, 6, 1, 3, 9, 4, 6, 8, 3, 9, 6, 3, 9, 6, 8, 3, 9, 2, 6, 1, 3, 9, 6, 3, 7, 9, 2, 6, 3, 5, 9, 6, 3, 7, 9, 6, 8, 1, 3, 9, 4, 6, 8, 3, 9, 2, 6, 3, 5, 9, 4, 6, 3, 9, 2, 6, 3, 9, 6, 8

Offset: 1

Jason G. Wurtzel, Jul 22 2010

			The 11th abundant number is 56. 5+6=11. 1+1=2. The digital root is 2.

Cf. A005101.

abQ[n_] := DivisorSigma[1, n] > 2 n; Mod[ Select[ Range@ 500, abQ], 9] /. 0 -> 9 (* Robert G. Wilson v, Aug 23 2010 *)

More terms from Robert G. Wilson v, Aug 23 2010

A179650 a(n) = (n-th prime) mod (n-th nonprime).

Original entry on oeis.org

0, 3, 5, 7, 2, 3, 5, 5, 8, 13, 13, 17, 20, 21, 23, 3, 7, 7, 11, 11, 9, 13, 15, 19, 25, 25, 25, 27, 25, 25, 37, 39, 41, 41, 49, 49, 1, 1, 2, 5, 8, 7, 11, 7, 8, 7, 16, 25, 23, 22, 23, 23, 19, 26, 29, 32, 35, 31, 34, 35, 31, 38, 49, 50, 49, 47, 58, 61, 68, 67, 68, 71, 73, 76, 79, 77, 77

Offset: 1

Jason G. Wurtzel, Jul 22 2010

nonn

In the first 100000 terms, the integers {4, 36, 40, 54} are missing. - Robert G. Wilson v, Jul 23 2010

			For n=4, 7 mod 8 = 7.

Cf. A000040, A018252.

A179650 := proc(n) ithprime(n) mod A018252(n) ; end proc: seq(A179650(n),n=1..100) ; # R. J. Mathar, Jul 23 2010

NonPrime[n_Integer] := FixedPoint[n + PrimePi@# &, n + PrimePi@n]; f[n_] := Mod[Prime@n, NonPrime@n]; Array[f, 70] (* Robert G. Wilson v, Jul 23 2010 *)

More terms from R. J. Mathar, Robert G. Wilson v and Carl R. White, Jul 23 2010

A179657 Digital root of n-th practical number.

Original entry on oeis.org

1, 2, 4, 6, 8, 3, 7, 9, 2, 6, 1, 3, 5, 9, 4, 6, 3, 9, 2, 6, 1, 3, 9, 6, 8, 3, 7, 9, 6, 1, 5, 9, 4, 3, 9, 2, 6, 5, 9, 6, 3, 7, 9, 6, 5, 9, 3, 7, 9, 2, 6, 1, 3, 9, 4, 8, 3, 9, 6, 9, 4, 8, 3, 9, 2, 6, 1, 9, 6, 3, 7, 9, 2, 6, 5, 9, 6, 3, 7, 9, 6, 1, 9, 4, 8, 9, 2, 6, 3, 5, 9, 4, 3, 9, 2, 6, 9, 8, 7, 9, 6, 1, 3, 5, 9

Offset: 1

Jason G. Wurtzel, Jul 23 2010

			For n=11, the 11th practical number is 28. As 2+8 = 10 and 1+0 = 1, the digital root is 1.

Cf. A005153, A010888.

PracticalQ[n_] := Module[{f, p, e, prod = 1, ok = True}, If[n < 1 || (n > 1 && OddQ[n]), False, If[n == 1, True, f = FactorInteger[n]; {p, e} = Transpose[f]; Do[ If[ p[[i]] > 1 + DivisorSigma[1, prod], ok = False; Break[]]; prod = prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]]; Mod[ Select[ Range@ 500, PracticalQ], 9] /. {0 -> 9} (* Robert G. Wilson v, Aug 02 2010 *)

isok(n) = bittest(n, 0) && return(n==1); my(P=1); n && !for(i=2, #n=factor(n)~, n[1, i]>1+(P*=sigma(n[1, i-1]^n[2, i-1])) && return);
for(n=1, 1e3, if(isok(n), print1((n-1)%9+1", "))) \\ Altug Alkan, Nov 12 2015

a(n) = A010888(A005153(n)). - Michel Marcus, Nov 12 2015

More terms from Robert G. Wilson v, Aug 02 2010

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User: Jason G. Wurtzel

A175730 Numbers with deficiency of 42.

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A175728 Decimal expansion of e written as a sequence of distinct positive integers.

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A180231 Prime partial sums of digits of decimal expansion of e.

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A180152 Numbers k such that the sum of the first k semiprimes is a prime.

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A179659 Digital root of the n-th weird number.

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A179679 Smaller of each pair of consecutive primes which sum to a practical number.

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A179656 prime(n) mod (digital root(prime(n))).

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A179655 Digital root of n-th abundant number.

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